Diffusion Theory of Ion Permeation Through Protein

Diﬀusion Theory of Ion Permeation Through Protein Channels of Biological Membranes

Thesis submitted for the degree “Doctor of Philosophy”

Amit Singer

under the supervision of

Professor Zeev Schuss

Submitted to the senate of Tel Aviv University

June 2005 ii

This work was carried out under the supervision of Professor Zeev Schuss. Contents

Preface x

Abstract xii

I Diﬀusion of Independent Particles between Fixed Concentrations: Analysis and Computation of the Simulation Problem 1

1 Brownian Simulations and Unidirectional Flux in Diﬀusion 8 1.1 Introduction ...... 9 1.2 Derivation of the Fokker-Planck Equation from a Path Integral 14 1.3 The Unidirectional Flux of the Langevin Equation ...... 16 1.4 The Smoluchowski Approximation to the Unidirectional Current 17 1.5 The Unidirectional Current in the Smoluchowski Equation . . 18 1.6 Brownian Simulations ...... 20 1.7 Summary and Discussion ...... 24

2 Memoryless Control of Boundary Concentrations of Diﬀus- ing Particles 26 2.1 Introduction ...... 27 2.2 Formulation ...... 29 2.2.1 Equations of Motion ...... 31 2.3 Renewal Controls ...... 31 2.3.1 Probabilistic Control ...... 32 2.3.2 Rate Control ...... 36 2.3.3 The Renewal Control ...... 39 2.3.4 Calculation of PL and PR: the Albedo Problem . . . . 39 iv CONTENTS

2.3.5 Concentration Proﬁle and Net Flux ...... 40 2.4 Discussion ...... 41

3 Langevin Trajectories between Fixed Concentrations 44 3.1 Introduction ...... 44 3.2 Trajectories, Fluxes, and Boundary Concentrations ...... 46 3.3 Application to Simulation ...... 48

4 Recurrence Time of the Brownian Motion 52 4.1 Introduction ...... 53 4.2 Short Time Asymptotics of the Fokker-Planck Equation . . . . 56 4.2.1 Initial layer? ...... 59 4.2.2 The probability distribution of the RT ...... 60 4.2.3 Extrapolation length ...... 61 4.3 Integral Equation for the pdf of the RT: Short Time Asymptotics 64 4.3.1 Integral equation for the pdf of the RT ...... 65 4.3.2 The Laplace method for v > 0 ...... 66 4.3.3 Laplace method for v < 0 ...... 70 4.4 Mean Number of Returns ...... 74 4.5 Long Time Asymptotics ...... 78 4.6 Small v asymptotics ...... 81

5 Narrow Escape in Three Dimensions 84 5.1 Introduction ...... 84 5.2 General 3D Bounded Domain ...... 87 5.2.1 The Neumann function and integral equations . . . . . 88 5.2.2 Elliptic hole ...... 91 5.3 Explicit Computations for the Sphere ...... 93 5.3.1 Collins’ method ...... 95 5.3.2 The asymptotic expansion ...... 96 5.3.3 The MFPT ...... 104 5.4 Summary and Applications ...... 105

6 Narrow Escape: The Circular Disk 108 6.1 Introduction ...... 109 6.2 Solution of a Mixed Boundary Value Problem ...... 111 6.2.1 Small ε asymptotics ...... 115 6.2.2 Expected lifetime ...... 116 CONTENTS v

6.2.3 Boundary layers ...... 117 6.2.4 Flux proﬁle ...... 119

7 Narrow Escape: Riemann Surfaces and Non-Smooth Do- mains 122 7.1 Introduction ...... 123 7.2 Asymptotic Approximation to the MFPT on a Riemannian Manifold ...... 126 7.2.1 Expression of the MFPT using the Neumann function . 127 7.2.2 Leading order asymptotics ...... 129 7.3 The Annulus Problem ...... 132 7.4 Domains with Corners ...... 139 7.5 Domains with Cusps ...... 143 7.6 Diﬀusion on a 2-Sphere ...... 145 7.6.1 Small absorbing cap ...... 145 7.6.2 Mapping of the Riemann sphere ...... 147 7.6.3 Small cap with an absorbing arc ...... 148

8 Narrow Escape at Short Times 152 8.1 Absorbing Disk ...... 152 8.2 Narrow Escape from a Disk ...... 155 8.3 Narrow Escape in an Annulus ...... 156

II Non-Equilibrium Statistical Mechanics of Inter- acting Particles 163

9 From Langevin Equations to Partial Diﬀerential Equations 166 9.1 Introduction ...... 167 9.2 Standard Continuum Treatments ...... 170 9.3 Equilibrium Statistical Mechanics of Simple Fluids ...... 172 9.3.1 Equilibrium vs. Non-equilibrium Statistical Mechanics 174 9.4 Non-Equilibrium Statistical Mechanics ...... 175 9.4.1 A Trajectory Based Approach ...... 175 9.4.2 The Fokker-Planck Equation ...... 176 9.5 The C-PNP System ...... 177 9.5.1 The C-PNP Hierarchy ...... 180 9.5.2 PNP revisited ...... 182 vi CONTENTS

9.6 Summary and Discussion ...... 182

10 Maximum Entropy Formulation of the Kirkwood Superposi- tion Approximation 186 10.1 Introduction ...... 186 10.2 Maximum Entropy ...... 189 10.3 Two Examples ...... 192 10.3.1 Non-interacting particles in an external ﬁeld ...... 193 10.3.2 The Thermodynamic Limit ...... 193 10.4 Minimum Helmholtz Free Energy ...... 195 10.5 Probabilistic Interpretation of the Kirkwood Closure . . . . . 197 10.6 High Level Entropy Closure ...... 201 10.6.1 The thermodynamic limit ...... 202 10.6.2 Closure at the highest level n = N − 1 ...... 205 10.7 Conﬁned Systems ...... 206 10.8 Mixtures ...... 208 10.9 Discussion and Summary ...... 210

11 Attenuation of the Electric Potential and Field in Disordered Systems 211 11.1 Introduction ...... 212 11.2 A One-Dimensional Ionic Lattice ...... 215 11.3 One Dimensional Random Ionic Lattice ...... 218 11.3.1 Moments ...... 219 11.3.2 The electrical potential as a weighted i.i.d. sum . . . . 220 11.3.3 Large and small potentials. The saddle point approximation ...... 221 11.3.4 Tail asymptotics ...... 222 11.4 Random Distances ...... 226 11.5 Dimensions Higher than One ...... 227 11.5.1 The condition of global electroneutrality ...... 227 11.5.2 The condition of local electroneutrality ...... 229 11.5.3 The liquid state ...... 231 11.6 Summary and Discussion ...... 232

12 Boundary Conditions and a Closure Relation for the Pair Correlation Function in Non-Equilibrium Diﬀusion 233 12.1 Inﬁnite system in Steady-State ...... 233 CONTENTS vii

12.2 Quasi Steady State: The Neumann Problem in Finite Domains 237 12.3 The Electrical Current and the Ramo-Shockley Function . . . 238 12.4 The Connection between the Diﬀusion Current and the Elec- tric Current ...... 240 12.4.1 The Steady State ...... 241 12.4.2 Quasi Steady State ...... 242

13 Open Problems and Future Research 244

A Algebra Solution of the Albedo Problem 246 A.1 The Stationary Albedo Problem ...... 246 A.2 The Algebra Solution ...... 247

B Appendix of Chapter 5 250 B.1 Estimate of kKk2 ...... 250 B.1.1 Estimate of the kernel ...... 250 B.2 Elliptic Hole ...... 251 B.3 A Pathological Example ...... 253

C Appendix of Chapter 6 254 C.1 Maximal Exit Time for the Circular Disk ...... 254 C.2 Exit Times along the Ray ...... 257 C.3 Flux Proﬁle ...... 259

D Appendix of Chapter 7 267 D.1 Laplace Beltrami Operator on 2-Sphere ...... 267 List of Figures

1.1 Concentration profile of BD trajectories injected exactly at the boundary ...... 20 1.2 Concentration profile of BD trajectories injected with the residual distribution ...... 22 1.3 Concentration profile of BD trajectories, constant injection rate, different time steps ...... 23 1.4 Concentration profile BD trajectories, inverse square-root injection rate law, different time steps ...... 24

2.1 The concentration cell of experimental electrochemistry and molecular biophysics ...... 30

3.1 Concentration Proﬁle of LD trajectories: Residual vs. Maxwellian injection schemes ...... 51

4.1 Short time asymptotics of the RT PDF ...... 62 4.2 Short time asymptotics of the recurrence time pdf ...... 75

6.1 A circular disk with small absorbing boundary ...... 112 6.2 The boundary layer near the absorbing boundary ...... 121

7.1 Annulus with small absorbing boundary ...... 133 7.2 Rectangle with small absorbing boundary ...... 140 7.3 A small opening near a corner of angle α...... 142 7.4 Absorbing boundary near a cusp ...... 144 7.5 The decapitated sphere ...... 149 √ 8.1 Ray of length 2 2R2 in an annulus ...... 159 8.2 The geometry of a billiard ball in an annulus ...... 160 LIST OF FIGURES ix

8.3 The maximal hitting angle constraint in an annulus ...... 161 8.4 The shortest ray in annulus that connects the center of the hole and its antipodal point on the inner circle ...... 162

9.1 Time scales of ionic permeation processes ...... 183

10.1 Conﬁgurations of three particles (Kirkwood) ...... 198

11.1 1D semi inﬁnite lattice of alternating charges ...... 216 11.2 Integration Contour ...... 223 11.3 2D lattice of dipoles ...... 230

12.1 Inﬁnite domain separated by an inﬁnite membrane with a hole 234 Preface

It is a pleasure to thank the following people who made my graduate studies into an enjoyable and unique experience. First, I would like to thank my adviser Professor Zeev Schuss. I cannot imagine a better adviser than Zeev, who not only taught me the fundamentals of stochastic processes and the way of putting together good science and mathematics, but also shared with me his vast experience in a way that helped me in making career and life decisions. I cherish the time we spent together. I am mostly grateful to Professor Bob Eisenberg for his vision that the channel problem must be treated with mathematical rigor and for his belief in my capabilities as a mathematician. I had wonderful time working in Bob’s department for over six months during the period of my studies. I was lucky to be in an environment that encourages scientific independence, and at the same time I got motivated by Bob’s enthusiasm. I would like to thank Ardyth and Bob for making me feel at home in Chicago. Special thanks are due to Dr. Boaz Nadler, whose Ph.D. dissertation ends where mine starts. Our discussions were most helpful and constructive. I owe Boaz my connection to Yale University, where my academic path continues. This is the opportunity to thank Professor Amir Averbuch, Professor Philip Rosenau, and Professor Koby Rubinstein for their guidance and assistance along the way. I want to thank Dr. David Holcman for coming up with the Narrow Escape problem. I wish that someday I will also be able to make connections between biology and mathematics the way he does. I would like to express my appreciation to: Dr. Manor Mendel for calculating pair correlation functions of different closures; Dr. Duan P. Chen for running trajectory based simulations of the recurrence time problem; Dr. Shela Aboud for pointing out the depletion phenomenon in Brownian dy- xi namics simulations; Professor Douglas Henderson, Professor Roland Roth, and Professor Stuart A. Rice for discussing with me the statistical mechanics of liquids; and the anonymous referee of [175], for suggesting the running a Brownian dynamics simulation. Finally, I would like to thank my family for their never-ending support and encouragement. This achievement is due to their investment in my education throughout the years. Last but definitely not least, I would like to thank Or for accompanying me during the past few years here and abroad. She is the proof that life is best with the woman you love.

A.S.

Tel Aviv, Israel June 2005 Abstract

In this dissertation I answer some fundamental mathematical questions that arise in predicting the function of protein channels of biological membranes from their structure. The mathematical problems concern both the analytic description of the channel function as well as computer simulations. These problems arise from the molecular level description of the physics of ionic permeation through the channel. The prediction of the properties of ionic channels from their known structure is an interdisciplinary field, that involves biology, chemistry, physics, engineering, computer science, and mathematics. Also the inverse problem, of reverse engineering of the structure from measured channel properties, is a key problem in molecular biophysics. None of the existing continuum descriptions of ionic permeation captures the rich phenomenology of the patch clamp experiment of Neher and Sak- mann. It is therefore necessary to resort to particle simulations of the permeation process. Predicting the function of an ionic channel from its structure by a computer simulation raises the question of connecting a small discrete simulation volume to a continuum bath. Computer simulations are necessarily limited to a relatively small number of particles, due to computational difficulties. Therefore, I analyze both Brownian and Langevin dynamics simulations that describe the motion of mobile ions in solution. Both models reduce the interaction of ions with the solute (water) molecules into friction, a noise term, and a dielectric constant. A reasonable simulation can describe only a small portion of the experimental setup of the patch clamp experiment, the channel and its immediate surroundings. The inclusion in the simulation of a part of the bath and its connection to the surrounding bath are necessary, because the conditions at the boundaries of the channel are unknown, while they are measurable in the bath, away from the channel. The simulation of a test volume in an ionic solution, even without a membrane and a channel, is by itself an important field of chemical physics. xiii

There are many algorithms, procedures and protocols for particles at the interface between the simulation region and the continuum baths (see [139] for a complete list of references on simulations). However, none of them takes into account the actual physics at the interface. The failure of these attempts calls for a theory of the interface that is compatible with the physics and for the design of simulations based on such a theory. In Part I of this dissertation I develop such a theory of the interface between a small simulation volume and the surrounding continuum. In this theory I assume that the particles in the continuum region interact only through a mean field and are otherwise independent. This assumption is permissible, because away from the channel the effects of particle-particle interactions at biological concentrations are adequately captured by classical diffusion theory, as described by classical physical chemistry. The effects of particle-particle interactions are dominant in the confined geometry of the channel and its immediate neighborhood. A computer simulation in this neighborhood can include all interactions, because the number of particles in a reasonable simulation volume is manageable. The main results of Part I are (i) The discovery of the precise range of validity of diffusion theory (Brownian motion) as a description of ionic motion in solution. I found the correct way to use diffusion theory in simulations. Mathematically this is expressed in the determination of the range of validity of the Smoluchowski approximation to the Langevin equation; (ii) The design of Brownian and Langevin simulations that do not form spurious boundary layers, which are ubiquitous in molecular simulations in biology, chemistry, and physics. This design is based on two new mathematical insights into the theory of diffusion. One is the splitting of the probability flux of the Brownian motion into unidirectional components, which is the result of a new mathematical definition of probability flux in terms of path integrals. The other is the application of methods of renewal theory to the theory of diffusion. (iii) The solution of Wang and Uhlenbeck’s recurrence problem, open since 1945. The solution is based on the conversion of the problem from solving a boundary value problem for a partial differential equation to an integral equation and the development of a new ray method for the construction of its asymptotic solution. (iv) The discovery of how biology controls many key functions by narrow valves, which mathematically is expressed as the solution of the escape problem of Brownian motion through a small hole. I found an explicit asymptotic expression for the mean first passage time in different geometries. Also here all the results are new. xiv Abstract

In Part II I develop an analytical theory of diffusion of interacting particles, which describes the permeation process inside the channel and its immediate neighborhood. The particle-particle interactions in this neighborhood give rise to the rich nonlinear behavior of the channel current: blocking, saturation, selectivity, and more. The main mathematical problem here is to determine the pair correlation function of a non-equilibrium system of interacting particles diffusing between two concentrations. This function is the cornerstone of the statistical mechanics of simple liquids, including electrolytic solutions. This function determines the nonlinear phenomena mentioned above. Although this function is the subject of intense research in both equilibrium and non-equilibrium statistical mechanics, the only information about the non-equilibrium pair correlation function is the recently derived partial differential equation it satisfies, as a part of an infinite system of coupled partial differential equations. The main results of Part II contain, among others, (i) The discovery of the boundary conditions of the pair correlation function in a non-equilibrium system of diffusing interacting particles, (ii) A variational formulation of Kirkwood’s superposition approximation (closure relation) for short range interactions, that truncates the infinite coupled system, and (iii) The proof of attenuation (screening) of the electric field and potential in disordered systems under certain conditions, which renders the electric interactions short rather than long range. The proof is based on large deviations theory. In attenuated systems the results of (ii) are applicable. These results lay a part of the foundation of the diffusion theory of interacting particles that is presently in its infancy. I find it a fortunate coincidence that old and new tools of applied mathematics, including stochastic processes (Brownian motion and Langevin equation), path integrals, asymptotic methods in partial differential equations, short and long time expansions, the algebra of commutators, elliptic mixed boundary value problems, singular perturbations and boundary layer theory, operator theory, conformal mapping, probability theory, renewal processes, integral equations, calculus of variations and large deviations theory, combine to produce new insights that have not been gained by traditional methods of biology, physics, and chemistry. Even more fortunately, I find new and old unsolved mathematical problems in the classical theory of diffusion and make contributions toward their solution. Part I

Diﬀusion of Independent Particles between Fixed Concentrations: Analysis and Computation of the Simulation Problem 2

Ion channels are proteins with a hole down their middle, embedded in biological cell membranes. They allow ions to move through otherwise impermeable cell membranes, thereby controlling many biological processes of great importance in health and disease. Ion channels can conduct ions of one type much better than ions of another type and this selectivity between ions is one of the most important part of their function. The ionic current in a protein channel depends on the voltage and the concentrations of all ionic species on both sides of the membrane. These are measured in the patch clamp experiment of Neher and Sakmann, in which two baths of salt solutions of different concentrations are separated by a patch of membrane containing a single protein channel that connects the two baths. The current, voltage and concentration are measured in the surrounding bath, far away from the channel. Channology is a relatively young science, so the existing mathematical theories of ionic permeation in protein channels barely scratch the surface of the mathematical iceberg of the channel problem. Also the present theory is far from being exhaustive. In the absence of an adequate analytic description of channel function computer simulations are most valuable. Simulations, by virtue of their regime of applicability, often become the link between theory and experiment. Their applications to chemistry include the study of molten salts, dielectric interfaces, phase changes, and ionic solutions. Simulations are a valuable tool in solid state device engineering for the design of semiconductors, and in biology similar techniques are used for the study of large biological molecules such as proteins, enzymes, and nucleic acids, to name but a few. Part I of this dissertation is devoted to the development of a theory of connection of a small simulation volume to the surrounding continuum. Computer simulations of ions in solution have to be confined to to a relatively small number of mobile ions, due to computational limitations. Thus a reasonable simulation can describe only a small portion of the experimental setup of a patch clamp experiment, the channel and its immediate surroundings. The inclusion in the simulation of a part of the bath and its connection to the surrounding bath are necessary, because the conditions at the boundaries of the channel are unknown, while they are measurable in the bath, away from the channel. Thus the trajectories of the particles are described individually for each particle inside the simulation volume, while outside the simulation volume they can be described only by their statistical properties. It follows that on 3 one side of the interface between the simulation and the surrounding bath the description of the particles is discrete, while a continuum description has to be used on the other side. This poses the fundamental question of how to describe the particle trajectories at the interface. The interface between the simulation region and the bath is sufficiently far from the channel so that the effects of interactions between the mobile ions and the channel, as well as the short range ion-ion interactions are negligible. Therefore it suffices to consider the interfacing problem only for non-interacting particles or for particles interacting with a mean field. Thus, we assume that random ionic trajectories near and outside the interface are independent. Specifically, in Chapter 1 we address this problem for Brownian dynamics (BD) simulations, connected to a practically infinite surrounding bath by an interface that serves as both a source of particles that enter the simulation and an absorbing boundary for particles that leave the simulation. The same problem for Langevin dynamics (LD) simulations is addressed in Chapter 3. The mathematical model of the interface is expected to reproduce the physical conditions that actually exist on the boundary of the simulated volume. These physical conditions are not merely the average electrostatic potential and local concentrations at the boundary of the volume, but also their fluctuation in time. It is important to recover the correct fluctuation, because the stochastic dynamics of ions in solution are nonlinear, due to the coupling between the electrostatic field and the motion of the mobile charges, so that averaged boundary conditions do not reproduce correctly averaged nonlinear response. However, in a system of noninteracting particles incorrect fluctuation on the boundary may still produce the correct response outside a boundary layer in the simulation region. The boundary fluctuation consists of arrivals of new particles from the bath into, and of the recirculation of particles in and out of the simulation volume. The random motion of the mobile charges brings about fluctuations in both the concentrations and the electrostatic field. Since the simulation is confined to the volume inside the interface, the new and the recirculated particles have to be fed into the simulation by a source that imitates the random influx across the interface. The interface does not represent any physical device that feeds trajectories back into the simulation, but is rather an imaginary wall, which the physical trajectories of the diffusing particles cross and recross any number of times. The efflux of simulated trajectories through the interface is observed in the simulation, however, the influx of 4

new trajectories, which is the unidirectional flux (UF) of diffusion, has to be calculated so as to reproduce the random influx with the correct statistical properties of this stochastic process, as mentioned above. Thus the UF is the source strength of the influx, and also the stochastic process that counts the number of trajectories that cross the interface in one direction per unit time. We find the source strength needed to maintain average boundary concentrations in a manner consistent with the physical and mathematical assumptions of Brownian particles. Our new insight into the simulation algorithm is that the source strength has to be inversely proportional to the square root of the time step and also proportional to the average nominal concentration at the interface. An additional new insight is the observation that new simulated trajectories should not be started at the interface, but rather injected into the simulation volume with the residual distribution of the Brownian dynamics displacement, as though the simulation extends across the interface outside the simulation volume. Otherwise spurious boundary layers are formed, as has been painfully realized by simulators so far. The source strength of injection new Brownian trajectories into a discrete simulation is the UF of discrete diffusion. The concept of UF in diffusion is not well defined. Specifically, the diffusion equation is a conservation law for the net flux, but it does not define its unidirectional components, which have to be identified in order to run a discrete simulation. Our new insight into this problem is the splitting of the net diffusion flux into its unidirectional components by using the Wiener path integral description of diffusion, rather than Fick’s description in terms of a partial differential equation. The definition of the unidirectional diffusion flux is a natural outcome of the path integral formulation [121]. It is simply the probability density of Brownian trajectories that cross the given surface in one direction per unit time. The diffusion equation, whose solution is the Smoluchowski approximation to the solution of the Fokker-Planck equation for large values of the damping parameter γ, produces infinite UFs, because the Brownian motion is nowhere differentiable with probability 1. However, the unidirectional flux in the Fokker-Planck equation corresponding to the Langevin dynamics of the trajectories is finite for all γ, which leads to an apparent paradox. We resolve it by showing that the unidirectional fluxes of the discretized Langevin dynamics and Brownian dynamics coincide if the time step of a discretized Brownian dynamics is exactly twice the relaxation time, that is, only if ∆t = 2/γ. This observation has fundamental significance in the mathematical the- 5 ory of diffusion as a stochastic process. Namely, our analysis indicates that the mathematical Brownian motion (MBM) is an oversimplified mathematical description of the physical Brownian motion, that is, the MBM description is invalid on a time scale shorter than the relaxation time of the more realistic Langevin dynamics into Brownian dynamics. In 1905 Einstein [41] and, independently, in 1906 Smoluchowski [181] offered an explanation of the Brownian motion based on kinetic theory and demonstrated, theoretically, that the phenomenon of diffusion is the result of Brownian motion. Einstein’s theory was later verified experimentally by Perrin [143] and Svedberg [185]. That of Smoluchowski was verified by Smoluchowski [182], Svedberg [186], and Westgren [196]. To connect his mathematical theory with the “irregular movement which arises from thermal molecular movement”, Einstein made the following assumptions [41]: (1) the motion of each particle is independent of the others and (2) “the movement of one and the same particle after different intervals of time [are] mutually independent processes, so long as we think of these intervals of time as being chosen not too small.” Thus Einstein’s theory is based on the assumption that the diffusing particles are observed intermittently at time intervals that are short, but not too short. Smoluchowski’s theory was based on Langevin’s more refined description of the Brownian motion [111]. The study of the MBM on short time scales pushes diffusion theory beyond its range of applicability. Thus much of the mathematical phenomenology of the MBM, such as nowhere differentiabil- ity, local time, and so on, occurs on time scales that do not correspond to physical diffusion. In Chapter 2 we show that the popular memoryless (renewal) injecting schemes of Langevin dynamics [139, and references therein] lead to the cre- ation of spurious boundary layers, that spread out into the entire simulation volume due to the long range electric field. These boundary layers are inherited from the solution of the steady state Fokker-Planck equation with absorbing boundaries, also known as the steady-state albedo problem [20]. Thus, in designing a Langevin simulation, an algorithm has to be developed to eliminate these boundary layers, because they are spurious at an interface, as discussed above. This design problem is solved in Chapter 3, where the correct steady state influx and efflux of Langevin trajectories at the interface is replicated by injecting new Langevin trajectories into the simulation volume according to a new probability distribution, which is the phase space (two-dimensional) generalization of the apparently disconnected concept of residual time distribution of queueing (renewal) theory [97]. Our simulation 6

algorithm maintains average fixed concentrations without creating spurious boundary layers, consistently with the assumed Langevin dynamics. The analysis of Langevin trajectories near an absorbing boundary leads to the following theoretical question. What is the probability density function (pdf) of the recurrence time (RT) for the free particle motion? This problem was raised by Wang and Uhlenbeck (1945) [193], who considered a free particle that is initiated at the origin with a positive velocity, and asked when is the particle due to return to the origin for the first time. Marshall and Watson (1985) [123] and Hagan, Doering and Levermore (1989) [64] found the mean RT independently of each other. The original problem posed by Wang and Uhlenbeck, about the pdf of the recurrence time and the return velocity remained open and is solved in Chapter 4. Specifically, we construct a short time asymptotic expansion of the pdf of the RT with ”beyond all orders” accuracy by the ray method [25]. We generalize the method of images (reflection), that is well known for the one-dimensional diffusion equation, to the Fokker-Planck equation. This is done by deriving a new renewal-type integral equation for the pdf and its short time asymptotic solution. We also find the long time asymptotic expansion of the solution, and its small velocity asymptotic expansion. In Appendix A we present a simplified solution of the steady state albedo problem by using an algebra of commutators for the Fokker-Planck operator. The solution of the recurrence problem is the pdf of the residence time of free particles in the positive axis. This can represent the sojourn time of a free particle inside a long one-dimensional channel. Additional key problems in the theory of diffusion and its simulation are how frequently do Brownian particles, confined to a given volume, escape through a small hole in the boundary and how are the entrance points distributed on the surface of the hole. The hole may represent the mouth of an ionic channel embedded in an impermeable membrane. To answer these questions, we consider in Chapter 5 the mean exit time from a three-dimensional bounded domain whose boundary is reflecting, but for a small absorbing hole (window). The calculation of the mean exit time is formulated as a mixed Neumann-Dirichlet boundary value problem in potential theory. We find an asymptotic expansion for the mean exit time in terms of the geometry and the singular flux profile through the opening. This answers the above raised questions. In Chapter 6 we consider the escape problem from a two- dimensional simply connected bounded domain, and in Chapter 7 we discuss the escape problem from a bounded domain with corners and cusps at the boundary on a general two-dimensional Riemannian manifold as a model of 7 receptor diffusion on the surface of nerve cells. In Chapter 8 we show that the point from which it is the hardest to exit at short times may be different from the point where the MFPT is maximal. Chapter 1

Brownian Simulations and Unidirectional Flux in Diﬀusion

The contents of this chapter were published in [175]

The prediction of ionic currents in protein channels of biological membranes is one of the central problems of computational molecular biophysics. Existing continuum descriptions of ionic permeation fail to capture the rich phenomenology of the permeation process, so it is therefore necessary to resort to particle simulations. Brownian dynamics simulations require the connection of a small discrete simulation volume to large baths that are maintained at fixed concentrations and voltages. The continuum baths are connected to the simulation through interfaces, located in the baths sufficiently far from the channel. Average boundary concentrations have to be maintained at their values in the baths by injecting and removing particles at the interfaces. The particles injected into the simulation volume represent a unidirectional diffusion flux, while the outgoing particles represent the unidirectional flux in the opposite direction. The classical diffusion equation defines net diffusion flux, but not unidirectional fluxes. The stochastic formulation of classical diffusion in terms of the Wiener process leads to a Wiener path integral, which can split the net flux into unidirectional fluxes. These unidirectional fluxes are infinite, though the net flux is finite and agrees with classical theory. We find that the infinite unidirectional flux is an artifact caused by replacing the Langevin dynamics with its Smoluchowski approximation, which is classical diffusion. The Smoluchowski approximation fails on time scales shorter than the relaxation time 1/γ of the Langevin equation. 1.1 Introduction 9

We find that the probability of Brownian trajectories that cross an interface in one direction in unit time ∆t equals that of the probability of the corresponding Langevin trajectories if γ∆t = 2. That is, we find the unidirectional flux (source strength) needed to maintain average boundary concentrations in a manner consistent with the physics of Brownian particles. This unidirec- √tional flux is proportional to the concentration and inversely proportional to ∆t to leading order. We develop a BD simulation that maintains fixed average boundary concentrations in a manner consistent with the actual physics of the interface and without creating spurious boundary layers.

1.1 Introduction

The prediction of ionic currents in protein channels of biological membranes is one of the central problems of computational molecular biophysics. None of the existing continuum descriptions of ionic permeation captures the rich phenomenology of the patch clamp experiments [73]. It is therefore necessary to resort to particle simulations of the permeation process [83, 84, 165, 29, 200, 189]. Computer simulations are necessarily limited to a relatively small number of mobile ions, due to computational difficulties. Thus a reasonable simulation can describe only a small portion of the experimental setup of a patch clamp experiment, the channel and its immediate surroundings. The inclusion in the simulation of a part of the bath and its connection to the surrounding bath are necessary, because the conditions at the boundaries of the channel are unknown, while they are measurable in the bath, away from the channel. Thus the trajectories of the particles are described individually for each particle inside the simulation volume, while outside the simulation volume they can be described only by their statistical properties. It follows that on one side of the interface between the simulation and the surrounding bath the description of the particles is discrete, while a continuum description has to be used on the other side. This poses the fundamental question of how to describe the particle trajectories at the interface, which is the subject of this chapter. We address this problem for Brownian dynamics (BD) simulations, connected to a practically infinite surrounding bath by an interface that serves as both a source of particles that enter the simulation and an absorbing boundary for particles that leave the simulation. The interface is expected 10 Brownian Simulations and Unidirectional Flux in Diffusion

to reproduce the physical conditions that actually exist on the boundary of the simulated volume. These physical conditions are not merely the average electrostatic potential and local concentrations at the boundary of the volume, but also their fluctuation in time. It is important to recover the correct fluctuation, because the stochastic dynamics of ions in solution are nonlinear, due to the coupling between the electrostatic field and the motion of the mobile charges, so that averaged boundary conditions do not reproduce correctly averaged nonlinear response. In a system of noninteracting particles incorrect fluctuation on the boundary may still produce the correct response outside a boundary layer in the simulation region [180]. The boundary fluctuation consists of arrival of new particles from the bath into the simulation and of the recirculation of particles in and out of the simulation. The random motion of the mobile charges brings about the fluctuation in both the concentrations and the electrostatic field. Since the simulation is confined to the volume inside the interface, the new and the recirculated particles have to be fed into the simulation by a source that imitates the influx across the interface. The interface does not represent any physical device that feeds trajectories back into the simulation, but is rather an imaginary wall, which the physical trajectories of the diffusing particles cross and recross any number of times. The efflux of simulated trajectories through the interface is seen in the simulation, however, the influx of new trajectories, which is the unidirectional flux (UF) of diffusion, has to be calculated so as to reproduce the physical conditions, as mentioned above. Thus the UF is the source strength of the influx, and also the number of trajectories that cross the interface in one direction per unit time. The mathematical problem of the UF begins with the description of diffusion by the diffusion equation. The diffusion equation (DE) is often considered to be an approximation of the Fokker-Planck equation (FPE) in the Smoluchowski limit of large damping. Both equations can be written as the conservation law ∂p = −∇ · J. (1.1) ∂t The flux density J in the diffusion equation is given by 1 J(x, t) = − [ε∇p(x, t) − f(x)p(x, t)] , (1.2) γ k T where γ is the friction coefficient (or dynamical viscosity), ε = B , k is m B 1.1 Introduction 11

Boltzmann’s constant, T is absolute temperature, and m is the mass of the diffusing particle. The external acceleration field is f(x) and p(x, t) is the density (or probability density) of the particles [166]. The flux density in the FPE is given by where the net probability flux density vector has the components

Jx(x, v, t) = vp(x, v, t) (1.3) Jv(x, v, t) = − (γv − f(x)) p(x, v, t) − εγ∇vp(x, v, t).

The density p(x, t) in the diﬀusion equation (1.1) with (1.2) is the probability density of the trajectories of the Smoluchowski stochastic diﬀerential equation

1 r2ε x˙ = f(x) + w˙ , (1.4) γ γ where w(t) is a vector of independent standard Wiener processes (Brownian motions). The density p(x, v, t) is the probability density of the phase space trajectories of the Langevin equation

x¨ + γx˙ = f(x) + p2εγw˙ . (1.5)

In practically all conservation laws of the type (1.1) J is a net flux density vector. It is often necessary to split it into two unidirectional components across a given surface, such that the net flux J is their difference. Such splitting is pretty obvious in the FPE, because the velocity v at each point x tells the two UFs apart. Thus, in one dimension,

Z ∞ Z 0 JLR(x, t) = vp(x, v, t) dv, JRL(x, t) = − vp(x, v, t) dv 0 −∞ (1.6) Z ∞ Jnet(x, t) = JLR(x, t) − JRL(x, t) = vp(x, v, t) dv. −∞ In contrast, the net flux J(x, t) in the DE cannot be split this way, because velocity is not a state variable. Actually, the trajectories of a diffusion process do not have well defined velocities, because they are nowhere differentiable 12 Brownian Simulations and Unidirectional Flux in Diffusion

with probability 1 [39]. These trajectories cross and recross every point x infinitely many times in any time interval [t, t + ∆t], giving rise to infinite UFs. However, the net diffusion flux is finite, as indicated in eq.(1.2). This phenomenon was discussed in detail in [121], where a path integral description of diffusion was used to define the UF. The unidirectional diffusion flux, however, is finite at absorbing boundaries, where the UF equals the net flux. The UFs measured in diffusion across biological membranes by using radioactive tracer [73] are in effect UFs at absorbing boundaries, because the tracer is a separate ionic species [24]. An apparent paradox arises in the Smoluchowski approximation of the FPE by the DE, namely, the UF of the DE is infinite for all γ, while the UF of the FPE remains finite, even in the limit γ → ∞, in which the solution of the DE is an approximation of that of the FPE [47]. The “paradox” is resolved by a new derivation of the FPE for LD from the Wiener path integral. This derivation is different than the derivation of the DE or the Smoluchowski equation from the Wiener integral (see, e.g. [102, 164, 112, 54]) by the method of M. Kac [92]. The new derivation shows that the path integral definition of UF in diffusion, as first introduced in [121], is consistent with that of UF in the FPE. However, the definition of flux involves the limit ∆t → 0, that is, a time scale shorter than 1/γ, on which the solution of the DE is not a valid approximation to that of the FPE. This discrepancy between the Einstein and the Langevin descriptions of the random motion of diffusing particles was hinted at by both Einstein and Smoluchowski. Einstein [41] remarked that his diffusion theory is based on the assumption that the diffusing particles are observed intermittently at short time intervals, but not too short, while Smoluchowski [181] showed that the variance of the displacement of Langevin trajectories is quadratic in t for times much shorter than the relaxation time 1/γ, but is linear in t for times much longer that 1/γ, which is the same as in Einstein’s theory of diffusion [22]. The infinite unidirectional diffusion flux imposes serious limitations on BD simulations of diffusion in a finite volume embedded in a much larger bath. Such simulations are used, for example, in simulations of ion permeation in protein channels of biological membranes [73]. If parts of the bathing solutions on both sides of the membrane are to be included in the simulation, the UFs of particles into the simulation have to be calculated. Simulations with BD would lead to increasing influxes as the time step is refined. The method of resolution of the said “paradox” is based on the defini- 1.1 Introduction 13 tion of the UF of the Langevin dynamics (LD) in terms of the Wiener path integral, analogous to its definition for the BD in [121]. The UF JLR(x, t) is the probability per unit time ∆t of trajectories that are on the left of x at time t and are on the right of x at time t + ∆t. We show that the UF of BD coincides with that of LD if the time unit ∆t in the definition of the unidirectional diffusion flux is exactly

2 ∆t = . (1.7) γ

We find the strength of the source that ensures that a given concentration is maintained on the average at the interface in a BD simulation. The strength of the left source JLR is to leading order independent of the efflux and depends on the concentration CL, the damping coefficient γ, the temperature ε, and the time step ∆t, as given in eq.(1.27). To leading order it is

r ε 1 J = C + O . (1.8) LR πγ∆t L γ

We also show that the coordinate of a newly injected particle has the probability distribution of the residual of the normal distribution. Our simulation results show that no spurious boundary layers are formed with this scheme, while they are formed if new particles are injected at the boundary. The simulations also show that if the injection rate is fixed, there is depletion of the population as the time step is refined, but there is no depletion if the rate is changed according to eq.(1.8). In Section 1.2, we derive the FPE for the LD (1.5) from the Wiener path integral. In Section 1.3, we define the unidirectional probability flux for LD by the path integral and show that is indeed given by (1.6). In Section 1.4, we use the results of [47] to calculate explicitly the UF in the Smoluchowski approximation to the solution of the FPE and to recover the flux (1.2). In Section 1.5, we use the results of [121] to evaluate the UF of the BD trajectories (1.4) in a finite time unit ∆t. In the limit ∆t → 0 the UF diverges, but if it is chosen as in (1.7), the UFs of LD and BD coincide. In Section 1.6 we describe the a BD simulation of diffusion between fixed concentrations and give results of simulations. Finally, Section 1.7 is a summary and discussion of the results. 14 Brownian Simulations and Unidirectional Flux in Diffusion

1.2 Derivation of the Fokker-Planck Equation from a Path Integral

The LD (1.5) of a diﬀusing particle can be written as the phase space system

x˙ = v, v˙ = −γv + f(x) + p2εγ w.˙ (1.9)

This means that in time ∆t the dynamics progresses according to

x(t + ∆t) = x(t) + v(t)∆t + o(∆t) (1.10)

v(t + ∆t) = v(t) + [−γv(t) + f(x(t))]∆t + p2εγ ∆w + o(∆t),(1.11)

where ∆w ∼ N (0, ∆t), that is, ∆w is normally distributed with mean 0 and variance ∆t. This means that the probability density function evolves according to the propagator

Prob{x(t + ∆t) = x, v(t + ∆t) = v} = p(x, v, t + ∆t) = o(∆t) + 1 Z bZ ∞ √ p(ξ, η, t)δ(x − ξ − η∆t) (1.12) 4εγπ∆t a −∞ ( ) [v − η − [−γη + f(ξ)]∆t]2 × exp − dξ dη. 4εγ∆t

To understand (1.12), we note that given that the displacement and velocity of the trajectory at time t are x(t) = ξ and v(t) = η, respectively, then according to eq.(1.10), the displacement of the particle at time t+∆t is deter- ministic, independent of the value of ∆w, and is x = ξ + η∆t + o(∆t). Thus the probability density function (pdf) of the displacement is δ(x − ξ − η∆t + o(∆t)). It follows that the displacement contributes to the joint propagator (1.12) of x(t) and v(t) a multiplicative factor of the Dirac δ function. Simi- larly, eq.(1.11) means that the conditional pdf of the velocity at time t + ∆t, given x(t) = ξ and v(t) = η, is normal with mean η + [−γη + f(ξ)]∆t + o(∆t) and variance 2γ∆t + o(∆t), as reflected in the exponential factor of the propagator. If trajectories are terminated at the ends of an finite or infinite interval (a, b), the integration over the displacement variable extends only to that interval. 1.2 Derivation of the Fokker-Planck Equation from a Path Integral 15

The basis for our analysis of the UF is the following new derivation of the Fokker-Planck equation from eq.(1.12). Integration with respect to ξ gives 1 Z ∞ p(x, v, t + ∆t) = o(∆t) + √ p(x − η∆t, η, t) (1.13) 4εγπ∆t −∞ ( ) [v − η − [−γη + f(x − η∆t)]∆t]2 × exp − dη. 4εγ∆t Changing variables to v − η − [−γη + f(x − η∆t)]∆t −u = √ , 2εγ∆t and expanding in powers of ∆t, the integral takes the form ∞ 1 Z 2 p(x, v, t + ∆t) = √ e−u /2 du × (1.14) 2π(1 − γ∆t + o(∆t)) −∞

p(x − v(1 + γ∆t)∆t + o(∆t), v(1 + γ∆t) + up2εγ∆t − f(x)∆t(1 + γ∆t) + o(∆t), t) Reexpanding in powers of ∆t, we get p(x − v(1 + γ∆t)∆t + o(∆t), v(1 + γ∆t) + up2εγ∆t − f(x)∆t(1 + γ∆t) + o(∆t), t)

∂p(x, v, t) = p(x, v, t) − v∆t + ∂x ∂p(x, v, t) vγ∆t + up2εγ∆t − f(x)∆t + o(∆t) + ∂v ∂2p(x, v, t) εγu2∆t + o(∆t), ∂v2 so (1.14) gives p(x, v, t) 1 ∂p(x, v, t) p(x, v, t + ∆t) − = − v∆t + 1 − γ∆t 1 − γ∆t ∂x ∆t ∂p(x, v, t) (vγ − f(x)) + 1 − γ∆t ∂v εγ∆t ∂2p(x, v, t) + O ∆t3/2 . 1 − γ∆t ∂v2 16 Brownian Simulations and Unidirectional Flux in Diﬀusion

Dividing by ∆t and taking the limit ∆t → 0, we obtain the Fokker-Planck equation in the form

∂p(x, v, t) ∂p(x, v, t) ∂ ∂2p(x, v, t) = −v + [(γv − f(x)) p(x, v, t)] + εγ , ∂t ∂x ∂v ∂v2 (1.15) which is the conservation law (1.1) with the ﬂux components (1.3). The UF JLR(x1, t) is usually deﬁned as the integral of Jx(x1, v, t) over the positive velocities [47, and references therein], that is,

Z ∞ JLR(x1, t) = vp(x1, v, t) dv. (1.16) 0

To show that this integral actually represents the probability of the trajectories that move from left to right across x1 per unit time, we evaluate below the probability ﬂux from a path integral.

1.3 The Unidirectional Flux of the Langevin Equation

The instantaneous unidirectional probability ﬂux from left to right at a point x1 is deﬁned as the probability per unit time (∆t), of Langevin trajectories that are to the left of x1 at time t (with any velocity) and propagate to the right of x1 at time t + ∆t (with any velocity), in the limit ∆t → 0. This can be expressed in terms of a path integral on Langevin trajectories on the real line as

1 Z x1 Z ∞ Z ∞ Z ∞ 1 JLR(x1, t) = lim dξ dx dη dv √ p(ξ, η, t) × ∆t→0 ∆t −∞ x1 −∞ −∞ 4εγπ∆t ( ) [v − η − [−γη + f(ξ)]∆t]2 δ(x − ξ − η∆t) exp − . (1.17) 4εγ∆t 1.4 The Smoluchowski Approximation to the Unidirectional Current 17

Integrating with respect to v eliminates the exponential factor and integration with respect to ξ ﬁxes ξ at x − η∆t, so 1 ZZ JLR(x1, t) = lim p(x − η∆t, η, t) dη dx ∆t→0 ∆t x−η∆t

1.4 The Smoluchowski Approximation to the Unidirectional Current

The following calculations were done in [47] and are shown here for completeness. In the overdamped regime, as γ → ∞, the Smoluchowski approximation to p(x, v, t) is given by

e−v2/2 1 ∂p(x, t) 1 1 p(x, v, t) ∼ √ p(x, t) − − f(x)p(x, t) v + O , 2π γ ∂x γ2 (1.19) where the marginal density p(x, t) satisﬁes the Fokker-Planck-Smoluchowski equation

∂p(x, t) ∂2p(x, t) ∂ γ = ε − [f(x)p(x, t)] . (1.20) ∂t ∂x2 ∂x According to (1.16) and (1.19), the UF is Z ∞ JLR(x1, t) = vp(x1, v, t) dv 0 Z ∞ e−v2/2 1 ∂p(x, t) 1 1 √ = v p(x, t) − − f(x)p(x, t) v + O 2 dv 0 2π γ ∂x γ r ε 1 ∂p(x, t) 1 = p(x , t) − ε − f(x)p(x, t) + O . (1.21) 2π 1 2γ ∂x γ2 18 Brownian Simulations and Unidirectional Flux in Diﬀusion

Similarly, the UF from right to left is

Z 0 JRL(x1, t) = − vp(x1, v, t) dv −∞ (1.22) r ε 1 ∂p(x, t) 1 = p(x , t) + ε − f(x)p(x, t) + O . 2π 1 2γ ∂x γ2

Both UFs in (1.21) and (1.22) are finite and proportional to the marginal density at x1. The net flux is the difference

1 ∂p(x, t) J (x , t) = J (x , t) − J (x , t) = − ε − f(x)p(x, t) ,(1.23) net 1 LR 1 RL 1 γ ∂x

as in classical diﬀusion theory [47], [56].

1.5 The Unidirectional Current in the Smolu- chowski Equation

Classical diﬀusion theory, however, gives a diﬀerent result. In the overdamped regime the Langevin equation (1.9) is reduced to the Smoluchowski equation [166]

γx˙ = f(x) + p2εγ w.˙ (1.24)

As in Section 1.3, the unidirectional probability current (ﬂux) density at a point x1 can be expressed in terms of a path integral as

JLR(x1, t) = lim JLR(x1, t, ∆t), (1.25) ∆t→0

where

r γ Z ∞ Z ∞ γζ2 JLR(x1, t, ∆t) = dξ dζ exp − × (1.26) 4πε∆t 0 ξ 4ε √ ζf(x ) ∂p(x , t) ∆t p (x , t) − ∆t − 1 p (x , t) + (ζ − ξ) 1 + O . 1 2ε 1 ∂x γ 1.5 The Unidirectional Current in the Smoluchowski Equation 19

It was shown in [121] that r ε 1 ∂p(x , t) J (x , t, ∆t) = p(x , t) + f(x )p(x , t) − ε 1 LR 1 πγ∆t 1 2γ 1 1 ∂x √ ! ∆t +O . (1.27) γ3/2 Similarly, JRL(x1, t) = lim JRL(x1, t, ∆t), ∆t→0 where r γ Z ∞ Z ∞ γζ2 JRL(x1, t, ∆t) = dξ dζ exp − × 4πε∆t 0 ξ 4ε √ ζf(x ) ∂p(x , t) ∆t p (x , t) + ∆t − 1 p (x , t) + (ζ − ξ) 1 + O . 1 2ε 1 ∂x γ (1.28) √ ! r ε 1 ∂p(x , t) ∆t = p(x , t) − f(x )p(x , t) − ε 1 + O . πγ∆t 1 2γ 1 1 ∂x γ3/2

If p(x1, t) > 0, then both JLR(x1, t) and JRL(x1, t) are infinite, in contradiction to the results (1.21) and (1.22). However, the net flux density is finite and is given by

J (x1, t) = lim {JLR(x1, t, ∆t) − JRL(x1, t, ∆t)} net ∆t→0

1 ∂ = − ε p(x , t) − f(x )p(x , t) , (1.29) γ ∂x 1 1 1 which is identical to (1.23). The apparent paradox is due to the idealized properties of the Brown- ian motion. More specifically, the trajectories of the Brownian motion, and therefore also the trajectories of the solution of eq.(1.24), are nowhere differentiable, so that any trajectory of the Brownian motion crosses and recrosses the point x1 infinitely many times in any time interval [t, t + ∆t] [86]. Obvi- ously, such a vacillation creates infinite UFs. Not so for the trajectories of the Langevin equation (1.9). They have finite continuous velocities, so that the number of crossing and recrossing is finite. We note that setting γ∆t = 2 in equations (1.27) and (1.28) gives (1.21) and (1.22). 20 Brownian Simulations and Unidirectional Flux in Diffusion

Figure 1.1: The concentration proﬁle of Brownian trajectories that are initiated at x = 0 with a normal distribution, and terminated at either x = 0 or x = 1.

1.6 Brownian Simulations

Here we design and analyze a BD simulation of particles diﬀusing between ﬁxed concentrations. Thus, we consider the free Brownian motion (i.e., f = 0 in eq. (1.4)) in the interval [0, 1]. The trajectories were produced as follows: a) According to the dynamics (1.4), new trajectories that are started at r2ε x(−∆t) = 0 move to x(0) = |∆w|; b) The dynamics progresses accord- γ r2ε ing to the Euler scheme x(t + ∆t) = x(t) + ∆w; c) The trajectory is ter- γ minated if x(t) > 1 or x(t) < 0. The parameters are ε = 1, γ = 1000, ∆t = 1. We ran 10,000 random trajectories and collected their statistics with the results shown in Figure 1.1. 1.6 Brownian Simulations 21

The simulated concentration proﬁle is linear, but for a small depletion layer near the left boundary x = 0, where new particles are injected. This is inconsistent with the steady state DE, which predicts a linear concentration proﬁle in the entire interval [0, 1]. The discrepancy stems from part (a) of the numerical scheme, which assumes that particles enter the simulation interval exactly at x = 0. However, x = 0 is just an imaginary interface. Had the simulation been run on the entire line, particles would hop into the simulation across the imaginary boundary at x = 0 from the left, rather than exactly at the boundary. This situation is familiar from renewal theory [97]. The probability distribution of the distance an entering particle covers, not given its previous location, is not normal, but rather it is the residual of the normal distribution, given by

Z 0 (x − y)2 f(x) = C exp − 2 dy, (1.30) −∞ 2σ 2ε∆t where σ2 = and C is determined by the normalization condition γ Z ∞ f(x) dx = 1. This gives 0 r π x f(x) = erfc √ . (1.31) 2σ 2σ Rerunning the simulation with the entrance pdf f(x), we obtained the expected linear concentration profile, without any depletion layers (see Figure 1.2). Injecting particles exactly at the boundary makes their first leap into the simulation too large, thus effectively decreasing the concentration profile near the boundary. Next, we changed the time step ∆t of the simulation, keeping the injection rate of new particles constant. The population of trajectories inside the interval was depleted when the time step was refined (see Figure 1.3). A well behaved numerical simulation is expected to converge as the time step is refined, rather than to result in different profiles. This shortcoming of refining the time step is remedied by replacing the constant rate sources with time-step-dependent sources, as predicted by eqs.(1.27)-(1.28). Figure 1.4 describes the concentration profiles for√ three different values of ∆t and source strengths that are proportional to 1/ ∆t. The concentration profiles 22 Brownian Simulations and Unidirectional Flux in Diffusion

Figure 1.2: The concentration proﬁle of Brownian trajectories that are initiated at x = 0 with the residual of the normal distribution, and terminated at either x = 0 or x = 1. 1.6 Brownian Simulations 23

Figure 1.3: The concentration profile of Brownian trajectories that are initiated at x = 0 and terminated at either x = 0 or x = 1. Three different time steps (∆t = 4, 1, 0.25) were used, but the injection rate of new particles remained constant. Refining the time step decreases the concentration profile. 24 Brownian Simulations and Unidirectional Flux in Diffusion

Figure 1.4: The concentration profile of Brownian trajectories that are initiated at x = 0 and terminated at either x = 0 or x = 1. Three different time steps (∆t = 4, 1, 0.25)√ are shown, and the injection rate of new particles is proportional to 1/ ∆t. Refining the time step does not alter the concentration profile.

now converge when ∆t → 0. The key to this remedy is the calculation of the UF in diﬀusion.

1.7 Summary and Discussion

Both Einstein [41] and Smoluchowski [181] pointed out that BD is a valid description of diffusion only at times that are not too short. More specifically, the Brownian approximation to the Langevin equation breaks down at times shorter than 1/γ, the relaxation time of the medium in which the particles diffuse. 1.7 Summary and Discussion 25

In a BD simulation of a channel the dynamics in the channel region may be much more complicated than the dynamics near the interface, somewhere inside the continuum bath, suﬃciently far from the channel. Thus the net ﬂux is unknown, while the boundary concentration is known. It follow that the simulation should be run with source strengths (1.27), (1.28),

r ε 1 r ε 1 J ∼ C + J ,J ∼ C − J . LR πγ∆t L 2 net RL πγ∆t R 2 net

r ε However, J is unknown, so neglecting it relative to C will lead net πγ∆t L,R to steady state boundary concentrations that are close, but not necessarily equal to CL and CR. Thus a shooting procedure has to be adopted to adjust the boundary fluxes so that the output concentrations agree with CL and CR, and then the net flux can be readily found. According to (1.27) and (1.28), the efflux and influx remain finite at the boundaries, and agree with the fluxes of LD, if the time step in the BD 2 simulation is chosen to be ∆t = near the boundary. If the time step γ is chosen differently, the fluxes remain finite, but the simulation does not recover the UF of LD. At points away from the boundary, where correct UFs do not have to be recovered, the simulation can proceed in coarser time steps. The above analysis can be generalized to higher dimensions. In three dimensions the normal component of the UF vector at a point x on a given smooth surface represents the number of trajectories that cross the surface from one side to the other, per unit area at x in unit time. Particles cross the interface in one direction if their velocity satisfies v · n(x) > 0, where n(x) is the unit normal vector to the surface at x, thus defining the domain of integration for eq.(1.6). The time course of injection of particles into a BD simulation can be chosen with any inter injection probability density, as long as the mean time between injections is chosen so that the source strength is as indicated in (1.27) and (1.28). For example, these times can be chosen independently of each other, without creating spurious boundary layers. Chapter 2

Memoryless Control of Boundary Concentrations of Diﬀusing Particles

The contents of this chapter were published in [180]

Flux between regions of different concentration occurs in nearly every device involving diffusion, whether an electrochemical cell, a bipolar transistor, or a protein channel in a biological membrane. Diffusion theory has calculated that flux since the time of Fick (1855) [52], and the flux has been known to arise from the stochastic behavior of Brownian trajectories since the time of Einstein (1905) [41], yet the mathematical description of the behavior of trajectories corresponding to different types of boundaries is not complete. We consider the trajectories of non-interacting particles diffusing in a finite region connecting two baths of fixed concentrations. Inside the region, the trajectories of diffusing particles are governed by the Langevin equation. To maintain average concentrations at the boundaries of the region at their values in the baths, a control mechanism is needed to set the boundary dynamics of the trajectories. Different control mechanisms are used in Langevin and Brownian simulations of such systems. We analyze models of controllers and derive equations for the time evolution and spatial distribution of particles inside the domain. Our analysis shows a distinct difference between the time evolution and the steady state concentrations. While the time evolution of the density is governed by an integral operator, the spatial distribution is governed by the familiar Fokker-Planck operator. The bound- 2.1 Introduction 27 ary conditions for the time dependent density depend on the model of the controller; however, this dependence disappears in the steady state, if the controller is of a renewal type. Renewal-type controllers, however, produce spurious boundary layers that can be catastrophic in simulations of charged particles, because even a tiny net charge can have global effects. The design of a non-renewal controller that maintains concentrations of non-interacting particles without creating spurious boundary layers at the interface requires the solution of the time-dependent Fokker-Planck equation with absorption of outgoing trajectories and a source of ingoing trajectories on the boundary (the so called albedo problem).

2.1 Introduction

We consider particles that diffuse between two regions where average concentrations are maintained at constant unequal values (see fig. 2.1). Flux between regions of different concentration occurs in nearly every device involving diffusion, whether an electrochemical cell, a bipolar transistor, or a protein channel in a biological membrane. Continuum theories of such diffusive systems describe the concentration field by the (time independent) Nernst-Planck equation with fixed boundary concentrations [73, 42, 12, 140, 167, 89, 170]. The microscopic theory underlying diffusion describes motion of particles by Langevin’s equations [12, 167, 47, 109, 166] everywhere, except at the boundaries. The behavior of the Langevin trajectories at the boundaries depends on the interaction between the particles and the boundaries. Thus, for example, outgoing trajectories can be terminated (absorbed); reflected (or otherwise reinjected); delayed; and so on. None of this is described by the Langevin equations. Brownian dynamics cannot describe such boundary behavior, because Brownian particles have no definite velocity, being functions of infinite variation. Particles with positive (e.g., incoming) velocities can be distinguished from those with negative (e.g., outgoing) velocities, only if their velocity is well defined [47]. The Langevin equations are often directly integrated in simulations [3, 132, 28, 11, 16, 139, 134, 135, 82, 83, 84]. In devices, the interaction between the trajectories and the boundaries must be specified because the inputs, outputs, and power supplies of devices are at their boundaries; in physical systems, the boundaries are where charge, matter, and energy are injected into a device; in biological systems 28 Memoryless Control of Boundary Concentrations of Diffusing Particles

boundaries represent reservoirs maintained at a (nearly) fixed electrochemical potential by active processes of the cell. The formulation of boundary conditions for the particle concentration is obvious in macroscopic models, but formulation of boundary conditions for the underlying trajectories is not so clear cut, particularly because many different physical or computational control mechanisms can maintain a constant average density at prescribed locations, usually near the boundaries [3, 132, 28, 11, 16, 82, 165, 29, 200, 189, 190, 191, 157]. Many boundary conditions used in Brownian and Langevin simulations produce spurious boundary layers, that do not exist at those locations in the physical systems being simulated. Spurious boundary layers are particularly damaging to simulations of charged particles. A boundary layer leads to large fluctuations in the electrostatic field which spreads over the entire simulation region. This was clearly demonstrated in [139] for a problem with equal boundary concentrations in a simulation with a buffer zone. In this chapter we provide a general description of the concentration and flux of non-interacting particles diffusing between constant average concentrations near the boundaries. We study renewal-type controllers that maintain fixed concentrations near the boundaries, determining the time course both of concentration (in phase space) and current. This kind of controllers are oft en used in simulations. We show that the concentration is a weighted sum of “left” and “right” concentrations, each of which satisfies a different integro-partial-differential equation and different boundary conditions. In the steady state the phase space concentration is the weighted sum of the solutions of two stationary solutions of the so called albedo problem [96, 198, 199, 171, 20, 187]. The albedo problem was first posed by Wang and Uhlenbeck [193] in 1945 and its analytic solution was first found by Marshall and Watson [123]. Further progress was made by Hagan, Doering, and Levermore [64, 65], who used complex analysis to solve the half range expansion problem. The solution employed here was found by K losek [103]. The weights in the sum of “left” and “right” concentrations are the rates at which the controllers inject trajectories into the system. Different control mechanisms that maintain the same concentrations near the boundaries produce different time operators that govern the evolution of the “left” and “right” concentrations. Each evolution is non-Markovian. The removal and injection—or re-injection—of particles into the system by renewal-type boundary controllers are described by renewal-type integral operators that govern the time evolutions of these concentrations, in contrast 2.2 Formulation 29 to the Fokker-Planck or Nernst-Planck equations that are commonly used. The description of this simplified model of diffusion of non-interacting particles is apparently new: we include a detailed description of the physical mechanism that maintains the non-equilibrium state of the system. Similar descriptions are needed when particles interact.

2.2 Formulation

We consider a system composed of two finite macroscopic volumes containing electrolyte solutions of different ionic species, connected by a macroscopic or microscopic channel. A control mechanism keeps different average concentrations in the two volumes, so that a steady current flows through the system (see fig. 2.1), thus keeping it out of equilibrium. As seen in the figure, the control mechanism is located only on parts of the boundaries of the system, at macroscopic distances away from the connecting channel. The control mechanism re-injects exiting trajectories at one or the other boundaries in a way that maintains average fixed concentrations near the boundaries at all times. We have in mind, for example, a typical setup used to measure the diffusion of ions through a protein channel of a biological cell membrane that separates two solutions of different fixed concentrations [73]. Alter- natively, all trajectories are reflected at the boundary so that the system reaches equilibrium after a long time, but the long lasting transient regime is the non-equilibrium regime in which an almost steady current flows between the baths. This time behavior occurs when the number of particles that flow through the channel during the period of measurement is much smaller than the total number of ions in either bath. The problem at hand is to describe the steady diffusion current flowing between the two baths, in terms of the molecular properties of the diffusing ions, such as their radii and interaction forces, as a function of the experimentally controlled variables, such as the concentrations in the two baths and the external potential, and as a function of the system geometry, e.g., the geometry and charge distribution of the channel. The particles diffuse in a domain Ω that consists of the two macroscopic volumes and the connecting channel. We assume that there are N h ions of species h (h = Ca++, Na+, Cl−,... ) in Ω, which are numbered at time t = 0, P h h h h h h N = N, and we follow their trajectories, xj (t) = (xj (t), yj (t), zj (t)) at h all times t > 0 (xj (t) is the location of the j-th ion of species h at time t). 30 Memoryless Control of Boundary Concentrations of Diffusing Particles

Figure 2.1: The concentration cell of experimental electrochemistry and molecular biophysics. The region Ω typically consists of small parts of two large baths of eﬀectively constant concentrations, separated by a permeable membrane in experimental electrochemistry, or (in biophysics) an impermeable membrane containing one or more channels. 2.3 Renewal Controls 31

For future use, the coordinate and velocity vectors of all ions in the 3N- dimensional conﬁguration space, are denoted by x˜ = xh1 ,..., xh1 , xh2 ,..., xh2 ,... and x˜˙ or v˜. 1 N h1 1 N h2

2.2.1 Equations of Motion As in [167], we assume that the motion of an ion in the solution is overdamped diffusion in a field of force. The source of the noise and friction is the thermal motion of the solvent (e.g., water) and both are interrelated by Einstein’s fluctuation-dissipation principle [12]. More specifically, our starting point is a memoryless system of N coupled Langevin equations for the dynamics of all particles of the different species h = Ca++, Na+, Cl−,... , s h h h f (x˜) 2γ x kBT x¨h + γh xh x˙ h = j + j w˙ h, (j = 1, 2,...,N h), j j j M h M h j (2.1) where a dot on top of a variable means differentiation with respect to time, γh(xh) is the location dependent friction coefficient per unit mass and M h is h the effective mass of an ion of species h. The force f j (x˜) on the j-th ion of species h includes all ion-ion interactions and thus depends on the locations h of all ions. The functions w˙ j are, by assumption, independent standard Gaussian white noises. The parameter kB is Boltzmann’s constant and T is absolute temperature. As seen in fig. 2.1, some parts of the boundary ∂Ω are reflecting, while other parts contain the control mechanism. At the boundary ∂Ω, the random trajectories of the Langevin equations (2.1) are either reflected or are redirected by the external control mechanism.

2.3 Renewal Controls

The solution of (2.1) depends on the specific choice of control mechanism. We first analyze controls for one-dimensional non-interacting systems because the treatment of three-dimensional interacting particle systems is more complicated. In this section we show that renewal controls (to be defined in subsection 2.3.3) reproduce correct macroscopic properties such as total net flux and concentration profile, but also produce non-physical boundary layers for non-interacting diffusive particle systems. 32 Memoryless Control of Boundary Concentrations of Diffusing Particles

Consider particles diﬀusing in the interval Ω = [0, d]. The control mechanism maintains average concentrations CL and CR at 0 xL < xR d, respectively, away from the boundaries, where concentrations are actually measured. Each particle satisﬁes a Langevin equation

x¨ + γx˙ + U 0(x) = p2γ w.˙ (2.2)

In order to complete the description of the dynamics we have to describe the motion of particles at the boundaries, i.e., to describe the action of the control mechanism.

2.3.1 Probabilistic Control A possible control mechanism operates as follows: when a particle reaches either one of the boundaries, it tosses a Bernoulli coin with probabilities (L, R),L + R = 1, L, R ≥ 0. The control mechanism decides to re-enter the particle at the left boundary x = 0 with probability L, and to re-inject the particle to the bath at the right boundary x = d with probability R. The re-injections occur at random times; a particle that reached the boundary at time t, is delayed in the boundary a random time T and re-injected at time t + T . The random time T is a non-negative random variable with pdf

q(s) ds = Prob{s ≤ T < s + ds}. (2.3)

The velocity of injection is distributed according to pre-determined distributions sL(v) and sR(v) of the left and right sources, respectively. For example, if both sources are Maxwellian, then

2 −v2/2 sL(v) = √ e = sR(−v), v > 0 (2.4) 2π As shown below, the precise velocity distribution of the sources is unimportant for measurement of concentrations away from the boundaries. The dynamics (2.1) and the boundary behavior provide a complete description of the trajectories, and therefore determine the probability distribution of the random particle trajectories in the system at any time. Assuming, as we may, that the precise velocity distributions is unimportant, there are only two parameters to be determined, namely the ﬁxed number of particles in the system N and the re-injection probability R. These two parameters determine uniquely the two measured concentrations CL and CR. 2.3 Renewal Controls 33

Let pi(x, v, t) be the probability of finding the i − th particle at location x and velocity v at time t, given that it was injected to the bath at time t = 0, from either the left or right boundary with probabilities R and L, and the corresponding velocity distributions sL and sR. Since the particles are independent and interchangeable, we find that p1 = p2 = ... = pN , and set p(x, v, t) = p1(x, v, t). Let p(x, v) be the steady state density of a single particle, i.e., p(x, v) = limt→∞ p(x, v, t). The steady state concentration at location x is given by Z ∞ C(x) = N p(x, v) dv. (2.5) −∞ We use renewal theory [97] to calculate p(x, v). Suppose the device was turned on at time t = 0. Let t0 be the first time that the particle was injected into the system. Then the probability of finding the particle in location (x, v) of the phase space at time t is given by

Z t p(x, v, t) = p(x, v, t|t0 = s)q(s) ds. (2.6) 0

Let τ1 be the ﬁrst passage time of the particle to the boundary. Conditioning on τ1 yields Z t Z ∞ p(x, v, t) = q(s) ds p(x, v, t|t0 = s, τ1 = r)p(τ1 = r|t0 = s) dr, (2.7) 0 0 where p(τ1 = r|t0 = s) = p(τ = r−s) = 0 for r < s. We separate the integral into two parts

Z t Z t p(x, v, t) = q(s) ds p(x, v, t|t0 = s, τ1 = r)p(τ = r − s) dr 0 0 Z t Z ∞ + q(s) ds p(x, v, t|t0 = s, τ1 = r)p(τ = r − s) dr 0 t Z t Z t = q(s) ds p(x, v, t − r)p(τ = r − s) dr + f(x, v, t),(2.8) 0 0 where Z t Z ∞ f(x, v, t) = q(s) ds p(x, v, t|t0 = s, τ1 = r)p(τ = r − s) dr. (2.9) 0 t 34 Memoryless Control of Boundary Concentrations of Diﬀusing Particles

Changing the order of integration in (2.8) we obtain

Z t Z t p(x, v, t) = p(x, v, t − r) dr q(s)p(τ = r − s) ds + f(x, v, t) 0 0 Z t = p(x, v, t − r)(pτ ∗ q)(r) dr + f(x, v, t) 0 = (p ∗ pτ ∗ q)(t) + f(x, v, t), (2.10)

where ∗ denotes convolution. Taking the Laplace transform of the equation gives

fˆ(x, v, θ) pˆ(x, v, θ) = . (2.11) 1 − pˆτ (θ)ˆq(θ) The steady state distribution is given by

θfˆ(x, v, θ) p(x, v) = lim p(x, v, t) = lim θpˆ(x, v, θ) = lim . (2.12) t→∞ θ→0 θ→0 1 − pˆτ (θ)ˆq(θ) Both numerator and denominator of the right hand side vanish as θ tends to 0. Expanding the denominator in Taylor series, we ﬁnd that

fˆ(x, v, θ = 0) p(x, v) = , (2.13) hτi + hT i

where hτi is the mean ﬁrst passage time (MFPT), and hT i is the mean delay time before re-injection. To evaluate fˆ(x, v, θ = 0), we consider a Langevin particle in the interval [0, d] which is injected at time t = 0 at x = 0 with velocity distribution sL(v). When the particle reaches one of the boundaries, it is absorbed, and its trajectory is terminated at once. Letp ˜L(x, v, t) be the probability density function of the particle (p should not to be confused withp ˜L; the subscript L stands for left.) The densityp ˜L satisﬁes the Fokker-Planck equation

∂p˜ ∂p˜ ∂ ∂2p˜ L = L p˜ = −v L + [(γv + U 0(x))p ˜ ] + γ L , (2.14) ∂t x,v L ∂x ∂v L ∂v2 with the initial condition

+ p˜L(x, v, t = 0) = δ(x − 0 )sL(v), (2.15) 2.3 Renewal Controls 35 and the absorbing boundary conditions

− p˜L(x = 0 , v, t) = 0, v > 0, (2.16)

p˜L(x = d, v, t) = 0, v < 0. (2.17)

Equations (2.14)-(2.17) deﬁne the time dependent albedo problem. In the limit of high friction a new time scale is often used [166]

tˆ= t/γ, (2.18) so eq.(2.14) is rewritten as

1 ∂p˜L = Lx,vp˜L. (2.19) γ ∂tˆ We deﬁne the function Z ∞ 1 Z ∞ PL(x, v) = p˜L(x, v, tˆ) dtˆ= p˜L(x, v, t) dt. (2.20) 0 γ 0

The function γPL(x, v) is the average time that a particle spends at location (x, v) prior to its absorption, given that it was injected from the left electrode at time t = 0. It follows from equations (2.14)-(2.17) that PL, the solution of the steady state albedo problem, satisﬁes 1 L P = − δ(x − 0+)s (v), (2.21) x,v L γ L with the absorbing boundary conditions

− PL(x = 0 , v) = 0, v > 0, (2.22)

PL(x = d, v) = 0, v < 0.

The MFPT to the boundary hτLi of a particle that was injected from the left electrode is given by

Z d Z ∞ hτLi = γPL(x, v) dx dv. (2.23) 0 −∞ 36 Memoryless Control of Boundary Concentrations of Diﬀusing Particles

Similarly, we define γPR as the mean time spent by a trajectory at the point (x, v) prior to its absorbtion, given that it was injected to the bath from the right electrode at x = d at time t = 0. The function PR satisfies similar equations, and its integral is the MFPT hτRi. Using the definition of f(x, v, t), equation (2.9), and changing the order of integration, we find that

Z ∞ fˆ(x, v, θ = 0) = f(x, v, t) dt (2.24) 0 Z ∞ Z t Z ∞ = dt q(s) ds p(x, v, t|t0 = s, τ1 = r)p(τ = r − s) dr 0 0 t Z ∞ Z ∞ Z r = q(s) ds p(τ = r − s) dr p(x, v, t|t0 = s, τ1 = r) dt 0 s s We identify the inner two integrals as the mean total time that a particle had spent in the (x, v) location of phase space prior to its ﬁrst absorbtion. Z ∞ Since q(s) ds = 1, we ﬁnd that 0 ˆ f(x, v, θ = 0) = LγPL(x, v) + RγPR(x, v), (2.25)

and LγP (x, v) + RγP (x, v) p(x, v) = L R , (2.26) LhτLi + RhτRi + hT i from which the concentration in phase space is given by

LγP (x, v) + RγP (x, v) C(x, v) = Np(x, v) = N L R . (2.27) LhτLi + RhτRi + hT i

Equation (2.27) relates the probabilistic control mechanism to its resulting phase space steady state concentration, that satisﬁes the steady state Fokker- Planck equation with ﬂux boundary conditions (2.21).

2.3.2 Rate Control Another possible renewal control consists of two sources, placed at the left and right boundaries, which inject particles into the system. When a particle reaches the right or left boundary, its trajectory is terminated at once. 2.3 Renewal Controls 37

The sources inject particles at identical independent distributed (i.i.d) inter- arrival random times TL and TR, whose probability density functions are fL(t) and fR(t), respectively. The rates of injection are deﬁned as

1 1 λL = , λR = . (2.28) hTLi hTRi

Note that the number of particles in the system does not remain ﬁxed for this rate control mechanism. For any rectangle A ⊂ [0, d] × R, we denote by L NA (t) the number of particles in A at time t, that were originated at the left L source. Then NA (t) satisﬁes a set of renewal equations [97]

L Prob{NA (t) = 0} = Prob{(x(t), v(t)) ∈/ A} Z t Z ∞ L · fL(s)Prob{NA (t − s) = 0} ds + fL(s) ds . 0 t (2.29) L Prob{NA (t) = 1} = Prob{(x(t), v(t)) ∈ A} Z t Z ∞ L · fL(s)Prob{NA (t − s) = 0} ds + fL(s) ds 0 t

+Prob{(x(t), v(t)) ∈/ A}· Z t L fL(s)Prob{NA (t − s) = 1} ds. (2.30) 0

L Prob{NA (t) = n} = Prob{(x(t), v(t)) ∈ A}· Z t L fL(s)Prob{NA (t − s) = n − 1} ds 0

+Prob{(x(t), v(t)) ∈/ A}· Z t L fL(s)Prob{NA (t − s) = n} ds, n > 1. (2.31) 0 38 Memoryless Control of Boundary Concentrations of Diﬀusing Particles

L Thus, the expected value of NA (t) is given by ∞ L X L hNA (t)i = nProb{NA (t) = n} n=1 = Prob{(x(t), v(t)) ∈ A}· Z t Z ∞ L fL(s)hNA (t − s)i ds + fL(s) ds 0 0 Z t L +Prob{(x(t), v(t)) ∈/ A}· fL(s)hNA (t − s)i ds 0 Z t L = Prob{(x(t), v(t)) ∈ A} + fL(s)hNA (t − s)i ds. (2.32) 0 Dividing by the area |A| of A and taking the limit |A| → 0, we obtain the number of particles per unit length and per unit velocity, which we call the phase space density CL(x, v, t). It satisﬁes the renewal equation

L L C (x, v, t) =p ˜L(x, v, t) + (fL ∗ C )(x, v, t), (2.33) Taking the Laplace transform with respect to t, we ﬁnd that p˜ˆ (x, v, θ) CˆL(x, v, θ) = L , (2.34) ˆ 1 − fL(θ) and the steady state density is given by θp˜ˆ (x, v, θ) CL(x, v) = lim CL(x, v, t) = lim L t→∞ θ→0 ˆ 1 − fL(θ)

p˜ˆL(x, v, 0) = = λLp˜ˆL(x, v, 0). (2.35) hTLi We obtain from (2.20) that Z ∞ p˜ˆL(x, v, 0) = p˜L(x, v, t) dt = γPL(x, v). (2.36) 0 The linearity of the expectation implies that the steady state concentration is

L R C(x, v) = C (x, v) + C (x, v) = γλLPL(x, v) + γλRPR(x, v). (2.37) 2.3 Renewal Controls 39

2.3.3 The Renewal Control Even though the two control models described above are different, and have different time evolution (e.g., the number of particles inside the domain is bounded by N for the former, and unbounded for the latter), they have identical steady state phase space concentrations. Indeed, choosing NL NR λL = , λR = , (2.38) LhτLi + RhτRi + hT i LhτLi + RhτRi + hT i we find that eqs. (2.27) and (2.37) are identical. This is no mere coincidence: both controls are special cases of renewal controls. Definition 1. A source that injects particles into the domain at random times 0 = T0 ≤ T1 ≤ T2 ... ≤ Tn ≤ ..., such that Yn = Tn − Tn−1 are i.i.d with hY1i < ∞ is called a renewal source. Definition 2. A control made of renewal sources located at the absorbing boundary of the domain is called a renewal control. Theorem 1. The steady state phase space concentration of a renewal control 1 1 is given by equation (2.37), where λL = L , λR = R are the rates of hY1 i hY1 i the left and right renewal sources, respectively. Proof. The proof is given in the previous subsection.

2.3.4 Calculation of PL and PR: the Albedo Problem

As seen above, all renewal control mechanisms require the knowledge of PL and PR, which are the solutions of the steady state albedo problem. It was shown in [103] that PL is given by

1 −v2/2 −U(x)/ PL(x, v) = √ e e Q(x, v), (2.39) 2π

L R OUT L,R where Q = QBL + QBL + Q , with QBL the boundary layer solutions, which decay exponentially fast away from the boundaries, and QOUT the outer solution, given by Z x 1 QOUT (x, v) = C eU(z)/ dz − veU(x)/ + D + O(γ−2), (2.40) 0 γ 40 Memoryless Control of Boundary Concentrations of Diﬀusing Particles

with

eU(0)/ ζ 1 + BL C = √ 2 0 + O(γ−2), γ R d U(z)/ 0 e dz eU(0)/ 1 D = − √ ζ + BL + O(γ−2), (2.41) γ 2 0

L where B0 is a constant that depends on the velocity distribution of the left 1 source, and ζ denotes the Riemann zeta function (ζ 2 = −1.46035 ...). The outer solution QOUT approximates Q at distances O(γ−1) away from the boundaries. Similar expressions can be written for PR.

2.3.5 Concentration Proﬁle and Net Flux Equation (2.27) gives the concentration at x, which is established by the probabilistic control mechanism, as

Nγ (LP (x) + RP (x)) C(x) = L R . (2.42) LhτLi + RhτRi + hT i

Therefore, C LP (x ) + RP (x ) L = L 1 R 1 . (2.43) CR LPL(x2) + RPR(x2) We now solve equation (2.43) for the yet-undetermined parameter L that keeps constant concentrations CL and CR. Since L = 1 − R, the solution is given by

C P (x ) − C P (x ) L = R R 1 L R 2 . (2.44) CL [PL(x2) − PR(x2)] − CR [PL(x1) − PR(x1)]

Substituting in equation (2.42) we ﬁnd that

Nγ C = L , (2.45) hτi + hT i LPL(x1) + RPR(x1)

and the two parameters of the control mechanism, N and L, are uniquely determined. We assume that the left and right sources have the same velocity 2.4 Discussion 41

L R density distribution, sL(v) = sR(−v), v > 0, which guarantees B0 = B0 ≡ B0. The resulting concentration at x away from the boundary is given by

Z x2 Z x (U(x1)−U(x))/ U(z)/ (U(x2)−U(x))/ U(z)/ CLe e dz + CRe e dz x x1 C(x) = Z x2 , eU(z)/ dz x1 (2.46) which is the same as given in eq.(3.5) of [47]. Note that the constant factor 1 ζ 2 + B0 cancels out and therefore it cannot be seen in measuring concentrations. The total net ﬂux is given by

Z ∞ Nγ Z ∞ J(x) = N vp(x, v) dv = · vP (x, v) dv, (2.47) −∞ hτi + hT i −∞

−1 where P = LPL + RPR. The flux is constant and to leading order in γ is given by C eU(x1)/ − C eU(x2)/ J = L R . (2.48) γ Z x2 eU(z)/ dz x1 We see that the macroscopic net flux (2.48) is O(γ−1), and coincides with that given in eq.(3.7) of [47] and in [73]. Theorem 1 then implies that eqs.(2.46) and (2.48) describe the concentration and the flux for all renewal control mechanisms.

2.4 Discussion

The renewal controls studied here maintain systems of non interacting particles at constant average concentrations near the boundaries, and away from the boundaries they produce the stationary Nernst-Planck equation of classical diffusion theory. We have proven that all renewal controls produce the same steady state concentration and flux, even though their time evolutions can differ qual- itatively. However, renewal controls—that are widely used in computer simulations—are problematic because they produce spurious boundary layers. These boundary layers are expected to appear in interacting particle systems driven out of equilibrium by renewal controls. 42 Memoryless Control of Boundary Concentrations of Diffusing Particles

The existence of such boundary layers may be of little importance if the particles interact only through short range forces, such as Lennard-Jones forces, or the forces that prevent overlap of hard spheres. However, the boundary layers can have a catastrophic effect for particles that interact through long range forces, such as ions that interact electrostatically. The net charge carried by only a tiny fraction of the total number of ions is, after all, responsible for electrical signalling in the nervous system and the electrical potentials in electrochemical cells and these potentials extend over large distances, from micron to many meters, e.g., in the neurons of whales [194] as well as in inorganic applications from batteries to the trans-Atlantic cable [75, 161, 162, 163, 61]. The boundary behavior of diffusing particles has been studied for many types of boundaries, including absorbing, reflecting, sticky boundaries, and more [120, 97]. In [165] a sequence of Markovian jump processes is constructed such that their transition probability densities converge to the solution of the Nernst-Planck equation with given boundary conditions, including fixed concentrations and sticky boundaries. As mentioned above, replacing the baths with renewal sources is a mathematical idealization that can produce artificial boundary effects. The renewal control effectively terminates trajectories at boundaries and starts new trajectories there. Most experiments do not. In real physical systems, particles that reach the boundary usually move into a ‘guard’ region, from which they often return to the domain (with some probability), with a given time distribution. To capture this behavior by a mathematical model, the entire pdf of the first passage time for the albedo problem has to be found, not only its first moment. The spurious boundary layers will be avoided if the correct time course of recycling trajectories in and out of the domain is used. We postpone this calculation, which we could not find in the literature, to a future paper. The time evolution of systems whose average concentrations near the boundaries are maintained by renewal controls is complicated and cannot be described, in general, by a single partial differential equation. We have shown that the phase space concentration is a sum of two components, each of which satisfies a different integral-partial-differential equation with different boundary conditions. Only in the steady state does the concentration satisfy the Fokker-Planck equation with boundary conditions identical to those of the steady state albedo problem. Although the overdamped limit is a useful approximation inside the domain, it cannot be used near the boundaries, 2.4 Discussion 43 where the full Fokker-Planck equation has to be solved. For particle systems with only short range interactions, the outer solution—which is the solution to the Smoluchowski equation—determines the concentration and correlation functions away from the boundaries. One can hope that a simple boundary condition can be found for such systems, similar to the simple boundary condition that exists for non interacting systems. From the theoretical point of view, the absence of a rigorous mathematical theory of the boundary behavior of Brownian trajectories diffusing between fixed concentrations, based on the physical theory of the Brownian motion, is a serious gap in classical physics. This chapter is a step toward the bridging of this gap. Chapter 3

Langevin Trajectories between Fixed Concentrations

The contents of this chapter were published in [137] We consider the trajectories of particles diffusing between two infinite baths of fixed concentrations connected by a channel, e.g. a protein channel of a biological membrane. The steady state influx and efflux of Langevin trajectories at the boundaries of a finite volume containing the channel and parts of the two baths is replicated by termination of outgoing trajectories and injection according to a residual phase space density. We present a simulation scheme that maintains averaged fixed concentrations without creating spurious boundary layers, consistent with the assumed physics. This com- plements the results of [47] and comprises the solution of the problem of connecting a finite Langevin dynamics simulation to a continuum bath.

3.1 Introduction

We consider particles that diffuse in a domain Ω connecting two regions, where fixed, but possibly different concentrations are maintained by connection to practically infinite reservoirs. This is the situation in the diffusion of ions through a protein channel of a biological membrane that separates two salt solutions of different fixed concentrations [73]. Continuum theories of such diffusive systems describe the concentration field by the Nernst-Planck equation (NPE) with fixed boundary concentrations [73, 12, 140, 167]. On the other hand, the underlying microscopic theory 3.1 Introduction 45 of diffusion describes the motion of the diffusing particles by Langevin’s equations [12], [167, 47, 166]. This means that on a microscopic scale there are fluctuations in the concentrations at the boundaries. The question of the boundary behavior of the Langevin trajectories (LT), corresponding to fixed boundary concentrations, arises both in theory and in the practice of particle simulations of diffusive motion [3, 28, 11, 16, 139, 134, 135, 82]. When the concentrations are maintained by connection to infinite reservoirs, there are no physical sources and absorbers of trajectories at any definite location in the reservoir or in Ω. The boundaries in this setup can be chosen anywhere in the reservoirs, where the average concentrations are effectively fixed. Nothing unusual happens to the LT there. Upon reaching the boundary they simply cross into the reservoir and may cross the boundary back and forth any number of times. Limiting the system to a finite region necessarily puts sources and absorbers at the interfaces with the baths, as described in [175]. The boundary behavior of diffusing particles in a finite domain Ω has been studied in various cases, including absorbing, reflecting, sticky boundaries, and many other modes of boundary behavior [120], [97]. In [165] a sequence of Markovian jump processes is constructed such that their transition probability densities converge to the solution of the Nernst-Planck equation with given boundary conditions, including fixed concentrations and sticky boundaries. Brownian dynamics simulations with different boundary protocols seem to indicate that density fluctuations near the channels are independent of the boundary conditions, if the boundaries are moved sufficiently far away from the channel [29]. However, as shown in [180], many boundary protocols for maintaining fixed concentrations lead to the forma- tion of spurious boundary layers, which in the case of charged particles may produce large long range fluctuations in the electric field that spread throughout the entire simulation volume Ω. The analytic structure of these boundary layers was determined in [123, 64], following several numerical investigations (e.g, [20]). It seems that the boundary behavior of LT of particles diffusing between fixed concentrations has not been described mathematically in an adequate way. From the theoretical point of view, the absence of a rigorous mathematical theory of the boundary behavior of LT diffusing between fixed concentrations, based on the physical theory of the Brownian motion, is a serious lacuna in classical physics. It is the purpose of this letter to analyze the boundary behavior of LT 46 Langevin Trajectories between Fixed Concentrations

between fixed concentrations and to design a Langevin simulation that does not form spurious boundary layers. We find the joint probability density function of the velocities and locations, where new simulated LT are injected into a given simulation volume, while maintaining the fixed concentrations. As the time step decreases the simulated density converges to the solution of the Fokker-Planck equation (FPE) with the imposed boundary conditions without forming boundary layers.

3.2 Trajectories, Fluxes, and Boundary Con- centrations

We assume fixed concentrations CL and CR on the left and right interfaces between Ω and the baths B, respectively, with all other boundaries of Ω being impermeable walls, where the normal particle flux vanishes. We assume that the particles interact only with a mean field, whose potential is Φ(x), so the diffusive motion of a particle in the channel and in the reservoirs is described by the Langevin equation (LE) p x¨ + γ(x)x˙ + ∇xΦ(x) = 2γ(x)ε w˙ (3.1)

x(0) = x0, v(0) = v0,

where γ(x) is the (state-dependent) friction per unit mass, ε is a thermal factor, and w˙ is a vector of standard independent Gaussian δ-correlated white noises [166]. The probability density function (pdf) of finding the trajectory of the diffusing particle at location x with velocity v at time t, given its initial position, satisfies the Fokker-Planck equation (FPE) in the bath and in the reservoirs,

∂p = −v · ∇x p + γ(x)ε∆vp (3.2) ∂t h i +∇v · γ(x)v + ∇xΦ(x) p,

p(x, v, 0 | x0, v0) = δ(x − x0, v − v0).

In the Smoluchowski limit of large friction the stationary solution of (3.2) 3.2 Trajectories, Fluxes, and Boundary Concentrations 47 admits the form [47]

e−|v|2/2ε J (x) · v 1 p(x, v) = p(x) + + O (3.3) (2πε)3/2 ε γ2 where the ﬂux density vector J (x) is given by

1 1 J (x) = − ε∇p(x) + p(x) ∇Φ(x) + O , γ (x) γ2 and p(x) satisﬁes

1 −∇ · J (x) = ∇ · ε∇p(x) + p(x) ∇Φ(x) = 0. γ (x)

In one dimension, the stationary pdfs of velocities of the particles crossing the interface into the given volume are

e−v2/2ε J v √ 1 + 2πε εCL p (v) ∼ for v > 0, L 1 J + √ 2 CL 2πε (3.4) e−v2/2ε J v √ 1 − 2πε εCR p (v) ∼ for v < 0, R 1 J + √ 2 CR 2πε where J is the net probability ﬂux through the channel. The source strengths (unidirectional ﬂuxes at the interfaces) are given by [47]

r ε J 1 J = C − + O L 2π L 2 γ2 (3.5) r ε J 1 J = C + + O . R 2π R 2 γ2 48 Langevin Trajectories between Fixed Concentrations

3.3 Application to Simulation

Langevin simulations of ion permeation in a protein channel of a biological membrane have to include a part of the surrounding bath, because boundary conditions at the ends of the channel are unknown. The boundary of the simulation has to be interfaced with the bath in a manner that does not distort the physics. This means that new LT have to be injected into the simulation at the correct rate and with the correct distribution of displacement and velocity, for otherwise, spurious boundary layers will form [180]. Consider a single simulated trajectory that jumps according to the discretized LE (3.1)

x(t + ∆t) = x(t) + v(t)∆t, (3.6) p v(t + ∆t) = v(t)(1 − γ∆t) − ∇xΦ(x(t))∆t + 2εγ ∆w(t),

where ∆w is normally distributed with zero mean and variance ∆t. The trajectory is terminated when it exits Ω for the first time. The problem at hand is to determine an injection scheme of new trajectories into Ω such that the interface concentrations are maintained on the average at their nominal values CL and CR and the simulated density profile satisfies (3.3). To be consistent with (3.3), the injection rate has to be equal to the unidirectional flux at the boundary (3.5). New trajectories have to be injected with displacement and velocity as though the simulation extends outside Ω, consistently with the scheme (3.6), because the interface is a fictitious boundary. The scheme (3.6) can move the trajectory from the bath B into Ω from any point ξ ∈ B and with any velocity η. The probability that a trajectory, which is moved with time step ∆t from the bath into Ω, or from Ω into the bath will land exactly on the boundary is zero. It follows that the pdf of the point (x, v), where the trajectory lands in Ω in one time step, at time t0 = t + ∆t, say, given that it started at a bath point (ξ, η) (in phase space) is, according to (3.6),

Pr{x(t0) = x, v(t0) = v | x(t) = ξ, v(t) = η} (3.7) ( ) δ(x − ξ − η∆t) |v − η + (γη + ∇Φ(ξ))∆t|2 = exp − + o(∆t). (4πεγ∆t)3/2 4εγ∆t 3.3 Application to Simulation 49

The stationary pdf p(ξ, η) of such a bath point is given in (3.3). The conditional probability of such a landing is

Pr{x, v | x ∈ Ω, ξ ∈ B} = (3.8) Z Z dη dξ Pr{v(t0) = v, x(t0) = x | ξ, η}p(ξ, η) 3 R B , Pr{x ∈ Ω, ξ ∈ B}

where the denominator is a normalization constant such that Z Z dv dx Pr{x, v | x ∈ Ω, ξ ∈ B} = 1. 3 R Ω Thus the first point of a new trajectory is chosen according to the pdf (3.8) and the subsequent points are generated according (3.6), that is, with the transition pdf (3.7), until the trajectory leaves Ω. By construction, this scheme recovers the joint pdf (3.3) in Ω, so no spurious boundary layer is formed. As an example, we consider a one-dimensional Langevin dynamics simulation of diffusion of free particles between fixed concentrations on a given interval. Assuming that in a channel of length L √ (C − C ) ε L R C , γL L which means that γ is sufficiently large, the flux term in eq.(3.3) is negligible relative to the concentration term. The concentration term is linear with slope J and thus can be approximated by a constant, so that p(ξ) = p(0) + O (γ−1) in the left bath. Actually, the value of p(0) 6= 0 is unimportant, because it cancels out in the normalized pdf (3.8), which comes out to be

v2 exp − 2ε[1 + (γ∆t)2] Pr{x, v | x > 0, ξ < 0} = 2ε∆tp1 + (γ∆t)2 (3.9) s  1 + (γ∆t)2 x 1 − γ∆t ×erfc − v .  4εγ∆t ∆t 1 + (γ∆t)2  50 Langevin Trajectories between Fixed Concentrations

In the limit ∆t → 0 we obtain from eq.(3.9)

2δ(x)H(v) 2 Pr{x, v | x > 0, ξ < 0} → √ e−v /2ε, (3.10) 2πε where H(v) is the Heaviside unit step function. This means that with the said approximation, LT enter at x = 0 with a Maxwellian distribution of positive velocities. Without the approximation the limiting distribution of velocities is (3.4). Note, however, that injecting trajectories by any Markovian scheme, with the limiting distribution (3.10) and with any time step ∆t, creates a boundary layer [180]. A LD simulation with CL 6= 0,CR = 0, and the parameters γ = 100, ε = 1,L = 1, ∆t = 10−4 with 25000 trajectories, once with a Maxwellian distribution of velocities at the boundary x = 0 (red) and once with the pdf (3.9) (blue) shows that a boundary layer is formed in the former, but not in the latter (see Figure 3.1). An alternative way to interpret eq.(3.9) is to view the simulation (3.6) as a discrete time Markovian process (x(t), v(t)) that never enters or exits Ω exactly at the boundary. If, however, we run a simulation in which particles are inserted at the boundary, the time of insertion has to be random, rather than a lattice time n∆t. Thus the time of the ﬁrst jump from the boundary into the domain is the residual time ∆t0 between the moment of insertion and the next lattice time (n + 1)∆t. The probability density of jump size in both variables has to be randomized with ∆t0, with the result (3.9). 3.3 Application to Simulation 51

Figure 3.1: Left panel: Concentration against displacement of a LD simulation with injecting particles according to the residual distribution (3.9) (blue), and according to the Maxwellian velocity distribution (3.10) exactly at the boundary (red). The two graphs are almost identical, except for a small boundary layer near x = 0 in red. Right panel:√ Zoom in of the concentration proﬁle in the boundary layer x < 0.01 = ε/γ. Chapter 4

Recurrence Time of the Brownian Motion

Consider the motion of a diffusing particle governed by the Langevin equation. The particle is initiated at the origin x = 0 with a specific velocity v0 > 0 and its trajectory is terminated once it returns to the origin. In 1945, Wang and Uhlenbeck [193] posed the problem of finding the probability density function of return at time t, calling it “the recurrence time problem”. We partially answer this question by showing that the pdf of the recurrence time (RT) has a “beyond all orders” short time asymptotic expansion. We find that the asymptotic pdf is in good agreement with Langevin dynamics simulations. The Laplace transform of the pdf of the RT can be expanded in eigenfunctions using the half range expansion technique. However, this procedure does not reveal much about the pdf. Instead, we find an integral equation for the RT density and extract its short time asymptotics. We find a reflection rule for the Fokker-Planck equation for rays hitting an absorbing boundary, in a manner that extends the reflection rule of the diffusion equation (method of images in one dimension; eikonals in higher dimensions). A closely related problem is that of first passage times to x = 0 of particles initiated away from the boundary at x0 > 0, often called the Milne problem. Hagan, Doering and Levermore (1988) [65] solved this problem by constructing an initial time layer of size 2/γ, when particles almost do not see the absorbing boundary at the origin prior to that time. However, in the RT problem, the particle is initiated at the location of the absorbing boundary, so arbitrarily short return times are possible. Indeed, our analysis, as well as numerical simulations, show that a large percentage of the particles (depend- 4.1 Introduction 53 ing on the initial velocity) return at times shorter than 2/γ, even though the mean first passage time is infinite (as in the Gambler’s Ruin paradox [51]). We find the velocity distribution at the RT. We extend our results to the case of a Brownian motion with an external constant (gravitational) force field, and find that the beyond all orders short time asymptotic expansion is altered only by a change in the pre-exponential factor. In addition, we find the long time asymptotics of the RT pdf, and its asymptotic behavior for small velocities. We find that the mean number of crossings of the initiation point in the limit of small initial velocity is asymptotically logarithmic in v0.

4.1 Introduction

The one dimensional Brownian motion is described by the Langevin equation

x¨ + γx˙ = −g + p2γε w,˙ (4.1) where γ is the friction (damping) parameter, ε is the noise strength (temperature), g is the force field andw ˙ is a δ-correlated white (Gaussian) noise. Wang and Uhlenbeck (1945) [193] posed several problems in the theory of the Brownian motion, some of which are yet to be solved. The RT problem is to find the probability distribution of the first time that the Brownian motion returns to the location it was initiated at t = 0. A different problem, also posed in [193], is the first passage time problem, which is to find the probability time distribution of the Brownian motion for hitting a specific point located away from the location of initiation for the first time. Both problems remained open for many years, resisting all solution attempts, even for the free Brownian motion in the absence of external field, i.e., g = 0. Let p(x, v, t| x0, v0) be the probability density function (pdf) to find the diffusive particle at time t at position x with velocity v, given it was initiated at time t = 0 at location x0 ≥ 0 with initial velocity v0. In the absence of an external field, the pdf satisfies the Fokker-Planck equation

∂p ∂p ∂ ∂2p = −v + (γvp) + εγ , (4.2) ∂t ∂x ∂v ∂v2 with the initial condition

p(x, v, t = 0) = δ(x − x0)δ(v − v0), 54 Recurrence Time of the Brownian Motion

indicating the position and velocity at t = 0, and the absorbing boundary condition

p(x = 0, v, t) = 0, v > 0. (4.3)

The absorbing boundary condition reflects the fact that the first passage time of particles with and without absorption is the same. The motion after this time is irrelevant, so we may assume that the particle is absorbed, and there are no incoming trajectories from left to right (characterized by positive velocity) at the origin. In the RT problem x0 = 0, that is, the particle is initiated at the location of the absorbing boundary with a positive velocity v0 > 0, whereas the first passage time to the origin problems are identified with x0 > 0 and the initial velocity is not necessarily positive. In both problems the goal is to find the pdf p(x, v, t). What makes these problems hard is the form in which the absorbing boundary condition (4.3) is given for half the velocity space. Other boundary conditions, such as reflecting boundary conditions (for g = 0), or moving the absorbing boundary to infinity, are much simpler. For example, replacing the absorbing boundary condition (4.3) with boundary conditions at x = ±∞, oversimplifies the problem, whose explicit solution was given in [22]. The reflecting boundary problem is solved by the method of images, however, the absorbing boundary problem cannot be solved by this method. Integrating the Fokker-Planck equation (4.2) over time gives a time- Z ∞ independent problem for the steady state density P (x, v) = p(x, v, t) dt. 0 The steady state equation has a complete infinite set of eigenfunctions and eigenvalues. However, half the eigenfunctions are growing exponentials of the displacement variable and are ruled out of the expansion. Therefore, the problem is to find the expansion of a function, which is prescribed only for half the space v > 0 (4.3), in terms of half the eigenfunctions. Beals and Protopopescu [9] proved completeness of this half-range expansion problem, yet the determination of the expansion coefficients remained unresolved for two additional years. Titulaer et al. [96, 198, 199, 171, 20, 187] used several numerical approximation methods to solve the steady state problem. The RT and the first passage time problems were also renamed: the RT problem was renamed the albedo problem, whereas the first passage time problem was renamed the Milne problem, after the equivalent problem in transport theory. The 4.1 Introduction 55

Fokker-Planck equation is often approximated by the Smoluchowski (diffusion) equation. The absorbing boundary condition (4.3) is then replaced by the absorbing condition p(−xM ) = 0, where xM is the so called Milne extrapolation length. Attempts to calculate the Milne extrapolation length include numerical approximations [96, 198, 199, 171, 20, 187], moment closure assumptions [68, 149], and postulates about the form of the flux at the boundary [150]. These attempts yielded estimates of the Milne extrapolation length between 1.44 and 1.46. The half-range expansion problem was solved analytically by Marshall and Watson (1985) [123], who used the Wiener-Hopf method. Their analysis gave the exact value of the Milne extrapolation length xM = |ζ(1/2)| = 1.46035 ..., where ζ is the Riemann zeta function. Hagan, Doering and Lev- ermore (1989) [64] found an equivalent (somewhat) simplified solution by introducing a special complex variable function, and calculated the mean first passage time (MFPT) for a free Brownian motion initiated away from two absorbing boundaries. K losek [103] extended their analysis and calculated the mean RT in the overdamped limit γ 1 in the presence of an external force field, and in the presence of a second far absorbing boundary. Marshall and Watson [124] presented a simplified calculation of the MFPT and deduced the asymptotic distribution of passage times for large t in the presence of an attracting field. Marshall, Watson and Duck [37] developed a numerical scheme to calculate the MFPT in the presence of an arbitrary force field, and presented results for an harmonic oscillator, for which an analytic solution is unknown. The distribution of the first passage time (the time-dependent Milne problem) was first determined in [65], where a uniform expansion for both short and long times was found. The analysis in [65] is based on constructing an initial time layer of size t0 = 2/γ. The probability to exit before t0 is transcendentally small in γ, as the particle is initiated away from the absorbing boundary. This technique was extended in [104], where the first passage time distribution in the presence of an external field was found. These papers contain the solution to Wang and Uhlenbeck’s first passage time problem. Yet, the RT problem of Wang and Uhlenbeck remained unresolved. Mar- shall and Watson [123] computed the Laplace transform of the RT distribution in terms of infinite weighted sum of eigenfunctions. However, quoting [123, p. 3542] “The Laplace inversion ... appears to be out of question.” The initial time layer method [65] cannot be applied in the RT problem, because, as seen below, particles are absorbed in arbitrary small times with 56 Recurrence Time of the Brownian Motion

non-negligible probabilities. In this Chapter we solve Wang and Uhlenbeck’s problem by finding the short time asymptotics of the pdf of the recurrence time. Our approach differs from that of the above mentioned references in that instead of the traditional study of the Fokker-Planck equation, we derive an integral equation for the pdf of the RT. The derivation is based on the concept of unidirectional flux and on a renewal argument. We construct an asymptotic solution to the integral equation by developing an appropriate ray method for short times. We find that the short time asymptotic solution is a superposition of exactly two rays. The first ray is due to the short time asymptotics of the free Brownian motion [22] (that does not see the boundary), whereas the second ray, that has a a different (larger) exponent, is due to the absorbing boundary at the origin. The second ray is the main result of this Chapter. In principle, the second ray could have been recovered by re-summation of the Laplace transformed half range expansion (with respect to time). However, this re- summation seems to be unfeasible, as indicated by [123]. The long time asymptotics of the pdf of the RT, and the small velocity expansion are also calculated. A closely related problem is that of the recurrence of the integrated Brow- nian motion. This problem was studied by McKean [128] and its general- izations were studied by Sinai [173], Goldman [60], Groeneboom et al. [63], Isozaki and Watanabe [85], Lachal [110] and others. The Wang-Uhlenbeck RT problem reduces asymptotically to McKean’s winding problem at times t 1/γ. The short-time asymptotic expansion of McKean’s exact solution contains the two first terms in the expansion of the solution of the Wang- Uhlenbeck RT problem.

4.2 Short Time Asymptotics of the Fokker- Planck Equation

The phase space trajectories of the Langevin equation

x¨ + γx˙ + V 0(x) = p2εγ w˙ (4.4)

are the pair (x(t), v(t)), wherex ˙ = v. The joint transition probability density of the pair in the entire phase plane is the solution of the Fokker-Planck 4.2 Short Time Asymptotics of the Fokker-Planck Equation 57 equation [166] ∂p(x, v, t | x , v ) ∂p(x, v, t | x , v ) 0 0 = −v 0 0 ∂t ∂x ∂ + {[γv + V 0(x)] p(x, v, t | x , v )} ∂v 0 0 (4.5) ∂2 +εγ p(x, v, t | x , v ) for t > 0, (x, v, x , v ) ∈ 4, ∂v2 0 0 0 0 R with the initial condition 4 p(x, v, 0 | x0, v0) = δ(x − x0)δ(v − v0) for (x, v, x0, v0) ∈ R . For example, in the FPE for a free Brownian motion in a constant external ﬁeld V 0(x) = g, the solution is given by [22]

1 2 2 2 pc(x, v, t | x0, v0) = √ exp − GR − 2HRS + FS /2(FG − H ) , 2π FG − H2 (4.6) where −1 −γt −2 −γt R = x − x0 − γ v0(1 − e ) + gγ γt − 1 + e , −γt −1 −γt S = v − v0e + gγ 1 − e , (4.7) and F = εγ−2 2γt − 3 + 4e−γt − e−2γt , G = ε(1 − e−2γt), H = εγ−1(1 − e−γt)2. (4.8) The marginal pdf of the velocity is −γt −1 −γt 2 1 [v − v0e + gγ (1 − e )] p(v, t | v0) = exp − , p2πε(1 − e−2γt) 2ε(1 − e−2γt) (4.9) and the marginal pdf of the displacement is s γ2 p(x, t | x , v ) = (4.10) 0 0 2πε [2γt − 3 + 4e−γt − e−2γt] ( ) γ2 [x − x − v γ−1(1 − e−γt) − gγ−2(1 − e−γt − γt)]2 × exp − 0 0 . 2ε [2γt − 3 + 4e−γt − e−2γt] 58 Recurrence Time of the Brownian Motion

The solution (4.6) of the free particle problem has the short time asymptotic expansion is (the dependence on x0, v0 is suppressed) ψ (x) + tψ (x, v) + t2ψ (x, v) p (x, v, t | x , v ) ∼ exp − 0 1 2 c 0 0 εγt3 ∞ X n−3 × t Zn(x, v) for γt 1, (4.11) n=1

where the eikonals ψ0(x), ψ1(x, v), ψ2(x, v) are given by 2 ψ0(x) = 3(x − x0)

ψ1(x, v) = −3(x − x0)(v + v0) (4.12)

3 ψ (x, v) = v2 + vv + v2 + γ2(x − x )2, 2 0 0 10 0 and √ 3 γ(x − x )(v + v ) v2 − v2 (v − v) + γ(x − x) Z (x, v) = exp 0 0 + 0 + g 0 0 . 1 2πεγ 20ε 4ε 2εγ (4.13) In the general case (4.5), the asymptotic structure (4.11) gives the eikonal equations ∂ψ 0 = 0 ∂v

∂ψ 2 3ψ = 1 0 ∂v ∂ψ ∂ψ ∂ψ 2ψ = v 0 + 2 1 2 (4.14) 1 ∂x ∂v ∂v

∂ψ ∂ψ ∂2ψ ∂ψ 2 ∂ψ ∂ log Z ψ = v 1 − [γv + V 0(x)] 1 − εγ 1 + 2 − 2εγ 1 1 2 ∂x ∂v ∂v2 ∂v ∂v ∂v ∂ψ ∂ψ ∂2ψ ∂ψ ∂ log Z −2εγ = v 2 − [γv + V 0(x)] 2 − εγ 2 − 2εγ 2 1 ∂x ∂v ∂v2 ∂v ∂v 1 ∂ψ1 ∂Z2 ∂ log Z1 −2εγ − Z2 . Z1 ∂v ∂v ∂v 4.2 Short Time Asymptotics of the Fokker-Planck Equation 59

The coeﬃcients Zn(x, v) satisfy transport equations, as above. The ﬁrst three functions in (4.14) are given by (4.12). The function Z1(x, v) is given by √ 3 Z (x, v) = × (4.15) 1 2πεγ γ(x − x )(v + v ) v2 − v2 V 0(x)(v − v) V (x) − V (x ) exp 0 0 + 0 + 0 + 0 . 20ε 4ε 2εγ 2ε

In particular, setting x = x0 = 0 gives ψ0(0, v) = ψ1(0, v) = 0, and √ 3 v2 − v2 V 0(0)(v − v) ψ (0, v) = v2 + vv + v2,Z (x, v) = exp 0 + 0 . 2 0 0 1 2πεγ 4ε 2εγ (4.16)

4.2.1 Initial layer?

In the ﬁrst passage time (Milne) problem, the particle is initiated at x0, away from the origin x = 0. The short time behavior of the pdf (4.11) is governed by the ﬁrst eikonal ψ0,

3x2 p(x = 0, v, t) ∼ exp − 0 . (4.17) εγt3

Setting t = γ−1, we ﬁnd that

3γ2x2 p(x = 0, v, γ−1) ∼ exp − 0 , (4.18) ε

−1√ which is exponentially small for x0 γ ε. This observation enabled [65] −1 to construct an initial time layer of size t0 = 2γ . In [104] a similar initial layer was used for the first passage time problem of the Brownian motion in the presence of an external field. The particle undergoes thermalization at times t γ−1, its velocity distribution becomes Maxwellian, and its initial velocity is forgotten. However, in the RT (albedo) problem, the particle is initiated at the absorbing boundary x0 = 0. Setting x = x0 = 0 in eqs.(4.12), we find that both eikonals ψ0 and ψ1 vanish. Therefore, the short time asymptotics is 60 Recurrence Time of the Brownian Motion

2 2 governed by the ﬁrst non-vanishing eikonal ψ2(x = 0, v) = v + vv0 + v0, so that √ 3 v2 − v2 v − v p(x = 0, v, t | x = 0, v ) ∼ exp 0 + g 0 (4.19) 0 0 2πεγt2 4ε 2εγ v2 + vv + v2 × exp − 0 0 . εγt

In particular, setting t = γ−1 gives √ 3γ v − v 5v2 + 4vv + 3v2 p(x = 0, v, γ−1 | x = 0, v ) ∼ exp g 0 − 0 0 . 0 0 2πε 2εγ 4ε (4.20) Clearly, this expression is not exponentially small in γ. In fact, it is O(γ), indicating that short return times are probable. We conclude that the RT problem does not give rise to an initial time layer.

4.2.2 The probability distribution of the RT We have found that the particle is most likely to return to the origin before it is thermalized. We expect, as in the diffusion case [168], to find a ”beyond all orders” short time asymptotic expansion. In analogy to the diffusion case, we refer to Chandrasekhar’s solution (4.6) as a “ray solution”. The ray solution almost satisfies the absorbing boundary condition (4.3). Indeed, for γt 1 the pdf at the boundary (4.19) is approximately normally distributed v εγt N − 0 , . That is, at short times particles are likely to exit at half 2 2 rεγt their initial velocity, with a small standard deviation . Therefore, for 2 v2 γt 0 the absorbing boundary condition (4.3) is almost satisfied (it is ε satisfied up to a transcendentally small error). This analysis also shows that there are two non dimensional parameters in this problem, namely, γt (time v2 measured in relaxation time) and 0 (initial kinetic energy measured in k T ). ε B The RT of a trajectory x(t) is the first time it returns to the origin (or to any other point), τ = inf {t : x(t) = 0} . t>0 4.2 Short Time Asymptotics of the Fokker-Planck Equation 61

Replacing the pdf by the single ray (4.6) gives an approximation for the RT distribution at short times. Using the Taylor expansion of the marginal displacement pdf (4.10) and the large argument asymptotics of the error function [1], we ﬁnd the exit time probability distribution function (PDF) (Fig. 4.1) Z ∞ Pr {τ > t | v0} = p(x, t | x0, v0) dx x0 Z ∞ 1 = √ exp{−R2/2F } dx (4.21) x0 2πF 1 v γ−1(1 − e−γt) − gγ−2(γt − 1 + e−γt) = 1 − erfc 0 √ 2 2F √ εγt 3v2 3v g 3v2 ∼ 1 − √ exp 0 + 0 exp − 0 3π v0 16ε 4εγ 4εγt εγt × 1 + O 2 , v0 or √ 2 2 εγt 3v0 3v0g 3v0 Pr {τ < t | v0} ∼ √ exp + exp − (4.22) 3π v0 16ε 4εγ 4εγt εγt × 1 + O 2 . v0

We conclude that for short times the pdf of the RT has a beyond all orders approximation of the form e−α/t, with α given in equation (4.22). Note that the external force ﬁeld g appears only as a constant pre-exponential factor— the beyond all order behavior is driven by diﬀusion rather than drift.

4.2.3 Extrapolation length To compare the result (4.22) with the diffusion approximation, we note that in the diffusion approximation the velocity is not a state variable, therefore there is no distinction between incoming and outgoing particles. It follows that a particle initiated at an absorbing boundary is immediately absorbed. Thus, to obtain non trivial results, the diffusive particle must be initiated 62 Recurrence Time of the Brownian Motion

Figure 4.1: Short time asymptotics of the probability distribution function Pr{τ < t | v0} (4.21) for diﬀerent values of initial velocity v0. The ”beyond all order” behavior for short times γt 1 becomes noticeable even for short times. 4.2 Short Time Asymptotics of the Fokker-Planck Equation 63

away from the boundary, say at x0 > 0. In the Smoluchowski approximation of large damping, the pdf p(x, t) satisfies ∂p ∂2p g ∂p = D + , (4.23) ∂t ∂x2 γ ∂x ε where D = is the diffusion coefficient, with an absorbing boundary condi- γ tion p(x = 0, t) = 0 for all t > 0. (4.24) We recall that the solution is found by the method of images, which is a reflection principle, 1 g(x − x ) g2t p(x, t) = √ exp − 0 − (4.25) 4πDt 2γD 4γ2D (x − x )2 (x + x )2 × exp − 0 − exp − 0 . 4Dt 4Dt Integration yields the exit time distribution Z ∞ Pr{τ > t | x0} = p(x, t | x0) dx (4.26) 0 1 γx − gt gx gt + γx = 1 − erfc 0√ + exp 0 erfc √ 0 . 2 2γ Dt γD 2γ Dt The large argument asymptotic behavior of the error function [1] gives the short time asymptotics of the exit time pdf as √ 2 2 4Dt x0g x0 x0 Pr{τ < t | x0} ∼ √ exp exp − , for t . (4.27) πx0 2γD 4Dt 4D Comparing equations (4.22) and (4.27), we see that choosing √ −1 x0 = 3 γ v0 (4.28) results in a similar short time behavior of the Langevin dynamics and its diffusion approximation. Note that the pre-exponential factor is not recovered by this method. Recovering the pre-exponential factor is possible by using a different acceleration fieldg ˜ in the Smoluchowski approximation √ √ 3 g 3v γ 2εγ ln 2 g˜ = + 0 − √ . 2 8 3v0 64 Recurrence Time of the Brownian Motion

We conclude that extrapolation length that captures the short time asymptotics is diﬀerent than that of the steady state problem [123, 103]. The Smoluchowski approximation fails here twice: 1o near the absorbing boundary the velocity distribution is not Maxwellian (for all t and also for the steady-state problem), 2o it is doomed to failure for short times γt 1, as noted by Einstein and Smoluchowski.

4.3 Integral Equation for the pdf of the RT: Short Time Asymptotics

We have shown that the absorbing boundary condition (4.3) of the Fokker- Planck equation (4.5) is satisfied up to an exponentially small error for short times by the ray solution (4.6). That is, (4.6) is the first ray. Still, the exponentially small correction term has to be determined. In the diffusion (Smoluchowski) limit, the correction term is easily found by the method of images, and it consists of rays of different eikonals. However, there is no simple reflection rule for the Fokker-Planck equation with absorbing boundary conditions, so the problem should be tackled differently. The half range expansion method leads to an infinite sum in the Laplace time coordinate. In principle, the short time asymptotics could be found by the large argument asymptotics of the half range expansion series. However, this re-summation problem seems to be currently out of question [123]. Re-summation should come as no surprise: solutions of the diffusion equation also have two different representations [168]. Separation of variables gives an infinite series of time decaying exponentials and space eigenfunctions (e.g., trigonometric functions in one dimension)

∞ X −λnt p(x, t) = ane pn(x). n=0 This expansion is useful in determining the long time asymptotic behavior. However, it converges very slowly for short times, when all exponentials approach 1. Fortunately, the short time behavior is easily determined by a second representation of the solution, which is obtained by the method of images, that consists of rays. Here we generalize the concept of rays, developed for the diﬀusion equation in [25], to the Fokker-Planck equation. We introduce a new integral equation for the pdf of the RT, and construct 4.3 Integral Equation for the pdf of the RT: Short Time Asymptotics 65 its asymptotic solution by the ray method. The asymptotic solution consists of two rays, which we ﬁnd explicitly.

4.3.1 Integral equation for the pdf of the RT The probability density of returning to the origin for the ﬁrst time with a given velocity,

f(v, t | v0) = Pr {τ = t, v(τ) = v | x(0) = 0, v(0) = v0} , (4.29) is called the joint recurrence density. Obviously,

f(v, t | v0) = 0 for v > 0, v0 > 0, and for v < 0, v0 < 0. (4.30) Equation (4.30) means that a trajectory that starts at the origin with a positive velocity v0 returns to the origin for the first time with a negative velocity, so it cannot return to the origin for the first time with a positive velocity. It can, however, return to the origin for the second, fourth, and in general for the 2n-th time with positive velocities. It follows that for t > 0, v0 > 0, and for all v the recurrence density satisfies the integral equation Z t Z 0 vpc(0, v, t | 0, v0) = −f(v, t | v0) + v ds dη f(η, s | v0)pc(0, v, t − s | 0, η). 0 −∞ (4.31) Indeed, consider the unidirectional flux density of particles that cross the origin x = 0 with velocity v at time t. On the one hand, this unidirectional flux is vpc(0, v, t | 0, v0) (see (1.6)). On the other hand, it has two different contributions. The first contribution, given by f(v, t | v0), is due to particles that return to the origin for the first time exactly at time t with velocity v. The second contribution is due to particles that returned to the origin prior to time t. Due to the Markov property of the pair (x(t), v(t)), the unidirectional flux density of trajectories that cross the origin at time t with velocity v, given that it started with a positive velocity v0, is the probability density f(η, s | v0) that it returns to the origin for the first time at τ = s < t with some negative velocity η, and then it returns to the origin with velocity v (with probability density p(0, v, t − s | 0, η). The integral equation (4.31) can be written as

(I − L) f = −vpc, (4.32) 66 Recurrence Time of the Brownian Motion

where the linear integral operator L is deﬁned as

Z t Z 0 Lf = v ds dη f(η, s | v0)pc(0, v, t − s | 0, η). (4.33) 0 −∞

Obviously, kLk = O(t) < 1 for suﬃciently small t, so the Neumann series expansion gives 2 f = − I + L + L + ··· vpc, (4.34) that is,

f = −vpc (1 + O(L)) , as expected.

4.3.2 The Laplace method for v > 0 For v > 0 the integral equation (4.31) reduces to

Z t Z 0 pc(0, v, t | 0, v0) = ds dη f(η, s | v0)pc(0, v, t − s | 0, η), (4.35) 0 −∞

because particles cannot return to the origin for the ﬁrst time with a positive velocity (4.30). The integral equation (4.35) for γt 1 is asymptotically (neglecting all non exponentially small terms)

2 2 Z t Z 0 2 2 v + vv0 + v0 v + vη + η exp − ∼ ds f(η, s| v0) exp − dη. εγt 0 −∞ εγ(t − s) (4.36) We look for a solution in the form ψ(η, v ) f(η, s| v ) ∼ exp − 0 . (4.37) 0 εγt

Substituting s = tu, the integral takes the form

v2 + vv + v2 exp − 0 0 ∼ (4.38) εγt Z 1 Z 0 1 ψ(η, v ) v2 + vη + η2 du exp − 0 + dη. 0 −∞ εγt u 1 − u 4.3 Integral Equation for the pdf of the RT: Short Time Asymptotics 67

Asymptotic approximation of the integral is obtained by the Laplace method (γt is a small parameter). The integral is approximated by the value of ψ(η, v ) v2 + vη + η2 Ψ(η, u; v) = 0 + (4.39) u 1 − u at its minimum point (η∗, u∗) ∈ (−∞, 0] × [0, 1] (v is a parameter). Clearly, the minimum point depends on the values of v and v0, that is,

∗ ∗ ∗ ∗ η = η (v, v0), u = u (v, v0). Since (η∗, u∗) is a stationary point, it follows that ψ (η∗, v ) vη∗ + 2η∗ Ψ (η∗, u∗; v) = η 0 + = 0, (4.40) η u∗ 1 − u∗ and ψ(η∗, v ) v2 + vη∗ + η∗2 Ψ (η∗, u∗; v) = − 0 + = 0. (4.41) u u∗2 (1 − u∗)2 In addition, for the integral to have the asymptotic behavior (4.38), necessarily ψ(η∗, v ) v2 + vη∗ + η∗2 Ψ(η∗, u∗; v) = 0 + = v2 + vv + v2. (4.42) u∗ 1 − u∗ 0 0 Diﬀerentiating equation (4.42) with respect to v results in d Ψ(η∗, u∗; v) = Ψ (η∗, u∗; v)η∗ + Ψ (η∗, u∗; v)u∗ + Ψ (η∗, u∗; v) = 2v + v . dv η v u v v 0 (4.43) ∗ ∗ ∗ ∗ However, equations (4.40), (4.41) imply that Ψη(η , u ; v) = Ψu(η , u ; v) = 0. Therefore, 2v + η∗ Ψ (η∗, u∗; v) = = 2v + v . (4.44) v 1 − u∗ 0 ∗ Eliminating ψ(η , v0) between equations (4.41) and (4.42) gives another relation of η∗ and u∗,

2 ∗ ∗2 ∗ 2 2 2 v + vη + η = (1 − u ) (v + vv0 + v0). (4.45) The two solutions of eqs.(4.44), (4.45) are ∗ ∗ vv0 v0 (η , u ) = (v0, 0), − , . v0 + v v0 + v 68 Recurrence Time of the Brownian Motion

∗ The solution (v0, 0) is ruled out, because v0 > 0 and we require η < 0 (in fact, this solution represents the original trajectory that is initiated at ∗ ∗ vv0 v0 time t = 0 with velocity v0). Therefore, only (η , u ) = − , v0 + v v0 + v ∗ ∗ is possible. We note that, as required, −v0 < η < 0 and 0 < u < 1, because v > 0. This means that the main contribution to the probability of trajectories that cross the origin with a positive velocity v at time t, is due v t to trajectories that crossed the origin for the ﬁrst time at time u∗t = 0 v + v vv 0 with a velocity η∗ = − 0 . v0 + v ∗ ∗ η v0 ∗ The relations of η and v are invertible, so that v = − ∗ and u = η + v0 η∗ + v 0 . Therefore, eq.(4.42) gives v0

∗ ∗2 ∗ 2 ∗ ψ(η , v0) = η + η v0 + v0 for all − v0 < η < 0. (4.46)

We conclude that (4.37) is actually

η2 + ηv + v2 f(η, t | v ) ∼ exp − 0 0 for − v < η < 0. (4.47) 0 εγt 0

The evaluation of the integral by the Laplace method also gives the pre- exponential factor. Expanding

η2 + ηv + v2 η2 + ηv + v Ψ(η, u; v) = 0 0 + (4.48) u 1 − u

in Taylor series at the stationary point (η∗, u∗), gives

1 Ψ(η, u; v) = Ψ(η∗, u∗; v) + Ψ (η∗, u∗; v)(η − η∗)2 2 ηη

∗ ∗ ∗ ∗ +Ψηu(η , u ; v)(η − η )(u − u ) (4.49)

1 + Ψ (η∗, u∗; v)(u − u∗)2 + ··· 2 uu 4.3 Integral Equation for the pdf of the RT: Short Time Asymptotics 69 with the second order derivatives given by 2 ∗ ∗ 2(v + v0) Ψηη(η , u ; v) = , vv0 2 ∗ ∗ (v − v0)(v0 + v) Ψηu(η , u ; v) = , (4.50) vv0 2 2 2 ∗ ∗ 2(v + vv0 + v0)(v + v0) Ψuu(η , u ; v) = . vv0 The determinant of the bilinear form (4.49) is 6 2 (v + v0) ∆ = ΨηηΨuu − Ψηu = 3 2 . (4.51) (vv0) We look for a solution of the integral equation in the form A(η, v ) η2 + ηv + v2 f(η, s | v ) ∼ 0 exp − 0 0 , (4.52) 0 sn εγs where A(η, v0) and n are to be determined by the integral equation √ 3 v2 − v2 v − v v2 + vv + v2 exp 0 + g 0 exp − 0 0 2πεγt2 4ε 2εγ εγt

Z t Z 0 2 2 A(η, v0) η + ηv0 + v0 ∼ ds dη n exp − (4.53) 0 −∞ s εγs √ 3 η2 − v2 η − v v2 + vη + η2 × exp + g exp − . 2πεγ(t − s)2 4ε 2εγ εγ(t − s) Evaluating the integral by the Laplace method gives √ 3 v2 − v2 v − v exp 0 + g 0 = (4.54) 2πεγt2 4ε 2εγ √ 3 2πεγ A(η∗, v ) η∗2 − v2 η∗ − v √ 0 exp + g . 2πεγtn ∆ u∗n(1 − u∗)2 4ε 2εγ Therefore, n = 2 and √ 3 2 ∗2 ∗ ∗ 3 (v + v0) ∗2 ∗ 2 v0 − η v0 − η A(η , v0) = u (1 − u ) exp + g 2πεγvv0 4ε 2εγ √ 3 η∗ v2 − η∗2 v − η∗ = − exp 0 + g 0 . (4.55) 2πεγ 4ε 2εγ 70 Recurrence Time of the Brownian Motion

We conclude that for −v0 < v < 0, √ 3 v v2 − v2 v − v v2 + vv + v2 f(v, t | v ) = − exp 0 + g 0 exp − 0 0 0 2πεγt2 4ε 2εγ εγt

× [1 + O(γt)] . (4.56)

Note that for these velocities f(v, t | v0) ∼ −vpc(0, v, t | 0, v0). In view of (4.34), this is no mere coincidence, because the ray solution pc(x, v, t | 0, v0) satisﬁes the absorbing boundary condition (4.3) up to exponentially small error.

4.3.3 Laplace method for v < 0 The preceding analysis shows that for v < 0 the exponent Ψ(η, u; v) has no interior minima in the semi-infinite strip. Therefore, its single minimum is attained at the boundary, so a different exponent is recovered. Next, we n compute the terms L vpc (n ≥ 0) of the Neumann series (4.34). The first term is Z t Z 0 Lvpc = v ds ηpc(0, η, s | 0, v0)pc(0, v, t − s | 0, η) dη. 0 −∞

Evaluating Lvpc by the Laplace method, we see, again, that for v < 0 the function η2 + ηv + v2 η2 + ηv + v2 Ψ(η, u) = 0 0 + (4.57) u 1 − u does not have local minima inside the strip −∞ < η < 0, 0 < u < 1. Therefore, the minimum of Ψ is attained at the boundary of the strip. Since Ψ → +∞ as η → −∞ or u → 0+ or u → 1−, we conclude that the minimum is attained at η = 0. This means that trajectories that cross the origin with an arbitrarily small velocity η = 0− at time τ < t bring in the main contribution to pdf of the crossing event with a negative velocity at time t. The time at which this crossing occurs is determined by the minimum of the function v2 v2 Ψ(0, u) = 0 + . (4.58) u 1 − u v v The stationary condition Ψ = 0 is satisﬁed by u∗ = 0 and u∗ = 0 . u v + v v − v v 0 0 For v < 0, only u∗ = 0 satisﬁes 0 < u∗ < 1. Therefore, slow particles v0 − v 4.3 Integral Equation for the pdf of the RT: Short Time Asymptotics 71

v t that cross the origin at time 0 are most probable to cross (again and) v0 − v again at time t with a velocity v < 0. The minimum is

v2 v2 Ψ(0, u∗) = 0 + = (v − v)2. (4.59) u∗ 1 − u∗ 0

Therefore, neglecting non exponentially small terms, we have

(v − v)2 Lvp ∼ exp − 0 , (4.60) c εγt

2 2 2 which is the second ray. Note that (v0 − v) > v + vv0 + v0 for v < 0, ∗ therefore Lvpc vpc. Also Ψη(0, u ) = 0, so near the minimum the Taylor series of Ψ is

1 Ψ(η, u) = Ψ(0, u∗) + Ψ (0, u∗)η2 + Ψ (0, u∗)η(u − u∗) 2 ηη ηu 1 + Ψ (0, u∗)(u − u∗)2 + ··· , (4.61) 2 uu where the second order derivatives are given by

2 (v0 − v) Ψηη = −2 v0v 3 (v0 − v) Ψηu = (4.62) v0v 4 (v0 − v) Ψuu = −2 . v0v

The determinant of the bilinear form (4.61) is

6 2 (v0 − v) ∆ = ΨηηΨuu − Ψηu = 3 2 > 0 (4.63) (v0v) 72 Recurrence Time of the Brownian Motion

∗ and Ψηη, Ψuu > 0 (for v < 0), therefore (0, u ) is indeed a minimum. The Laplace method gives 3vt v2 − v2 v − v Lvp = exp 0 + g 0 c (2πεγ)2t4 4ε 2εγ Z 1 du Z 0 Ψ(η, u) × 2 2 η exp − dη 0 u (1 − u) −∞ εγt −3vt v2 − v2 v − v ∼ exp 0 + g 0 (2πεγ)2t4 4ε 2εγ Ψ(0, u∗) √ exp − 2πΨ εγt ×(εγt)3/2 uu (4.64) ∆ u∗2(1 − u∗)2

r 2 2 2 −v v0 − v v0 − v (v0 − v) = 3 3 exp + g exp − . 4π εγv0t 4ε 2εγ εγt

Note that the pre-exponential power of t in Lvpc is 3/2 rather than 2, as in η vp (eq.(4.56)), because the integrated pre-exponential function c u2(1 − u)2 vanishes at the minimum point η∗ = 0. 2 Next, we evaluate the term L vpc, Z t Z 0 2 L vpc = v ds (Lηpc)pc(0, v, t − s | 0, η) dη. (4.65) 0 −∞ Similar considerations show that (v − v)2 L2vp ∼ exp − 0 , (4.66) c εγt because the minimum of (η − v )2 η2 + vη + v2 Ψ(η, u) = 0 + (4.67) u 1 − u v is attained at the same boundary point (η∗, u∗) = 0, 0 . The point v0 − v ∗ ∗ ∗ ∗ ∗ ∗ (η , u ) is not stationary, because only Ψu(η , u ) = 0, while Ψη(η , u ) = −3(v0 − v) < 0. Therefore, the only second order derivative needed is 4 ∗ ∗ (v0 − v) Ψuu(η , u ) = −2 , (4.68) vv0 4.3 Integral Equation for the pdf of the RT: Short Time Asymptotics 73 and the Laplace method gives √ 2 2 2 3 1 v0 − v v0 − v L vpc ∼ vt exp + g 2 p 3 3 2πεγt 4π εγv0t 4ε 2εγ (v − v)2 π(εγt)2 1 × exp − 0 (4.69) p 3 ∗3/2 ∗ 2 εγt −2ΨuuΨη u (1 − u ) r 2 2 2 −1 εγ v0 − v v0 − v (v0 − v) = 3 3 exp + g exp − . 24 −π v0vt 4ε 2εγ εγt

2 Note that L vpc has an integrable singularity in v at the origin. Similarly, the Laplace method gives r 2 2 2 3 1 εγ v0 − v v0 − v (v0 − v) L vpc ∼ 3 3 exp + g exp − 48 −π v0vt 4ε 2εγ εγt 1 = − L2vp . (4.70) 2 c 2 The function L vpc is an asymptotic eigenfunction of the integral operator L (with eigenvalue −1/2). Therefore,

∞ n X 1 2 L2 + L3 + ... vp = − L2vp ∼ L2vp . (4.71) c 2 c 3 c n=0 Therefore, the Neumann series (4.34) collapses into 2 f ∼ − I + L + L2 vp . (4.72) 3 c We conclude that the short time expansion is given by " √ v2 − v2 v − v 3 v v2 + vv + v2 f(v, t | v ) ∼ exp 0 + g 0 − exp − 0 0 0 4ε 2εγ 2πεγt2 εγt (4.73) r r 2 # −v 1 εγ (v0 − v) + − 3 3 + 3 3 exp − . 4π εγv0t 36 −π v0vt εγt

Equation (4.73) is the beyond all order asymptotic solution of the recurrence density. The ﬁrst ray is the unidirectional ﬂux of the free Brownian motion. 74 Recurrence Time of the Brownian Motion

Therefore, the ﬁrst ray is an overestimate of the RT density f, because it takes into account the unidirectional ﬂux of particles that cross the origin and come back. Hence, the second ray has a negative contribution to the RT density as seen in Figure 4.2.

4.4 Mean Number of Returns

The unidirectional probability flux of a Langevin particle, vpc(0, v, t | 0, v0), is the probability that crossed the origin from left to right for v > 0 (from right to left for v < 0) at time t with velocity v in unit time per unit velocity, given that the particle was initiated from the origin x = 0 with velocity Z 0 v0 > 0. Therefore, − vpc(0, v, t | 0, v0) dv is the probability that crossed −∞ the origin from right to left at time t in unit time. In this Section we calculate the mean number of times a Langevin trajectory crosses the origin in the time interval [0,T ]. Note that in the overdamped limit the number of crossings is infinite for all T > 0, because the Brownian motion is nowhere differentiable with probability 1 [39]. However, the number of crossings is finite for the Langevin motion. We divide the interval [0,T ] into subintervals of size ∆t, where ∆t is so small such that the number of crossings at time interval of size ∆t is either 0 or 1 (the probability for more than 1 crossing is negligible). The number of crossings in the j-th time interval is a Bernoulli random variable that takes Z (j+1)∆t Z 0 the values 1, with probability − dt vpc(0, v, t | 0, v0) dv, and 0 j∆t −∞ otherwise. Therefore, in the limit ∆t → 0, the mean number of crossings from right to left in [0,T ] is Z T Z 0 E(NRL) = − dt vpc(0, v, t | 0, v0) dv. 0 −∞

For short times γT 1, the density pc is given by equation (4.19), and E(NRL) is asymptotically √ Z T Z 0 3 E(NRL) ∼ − dt dv v 2 × (4.74) 0 −∞ 2πεγt v2 − v2 v − v v2 + vv + v2 exp 0 + g 0 exp − 0 0 . 4ε 2εγ εγt 4.4 Mean Number of Returns 75

Figure 4.2: The short time asymptotics of the RT pdf (γ = 1, = 1, v0 = 1, v = −1/4): (a) First ray (red), (b) Second ray (green), (c) Sum of the two rays (eq.(4.73)) (blue). Note that the contribution of the second ray is negative, and that the particle is most probable to return before the relaxation time. 76 Recurrence Time of the Brownian Motion

Integrating ﬁrst with respect to t, gives √ Z 0 2 2 3v dv v0 − v v0 − v E(NRL) ∼ − 2 2 exp + g × −∞ 2π(v + vv0 + v0) 4ε 2εγ v2 + vv + v2 exp − 0 0 . (4.75) εγT

Setting v = −v0u, we obtain √ Z ∞ 2 2 3u du v0(1 − u ) v0(1 + u) E(NRL) ∼ 2 exp + g × 0 2π(u − u + 1) 4ε 2εγ v2(u2 − u + 1) exp − 0 . (4.76) εγT √ Z ∞ 3u du The integral in eq.(4.76) reduces for v0 = 0 to 2 , which 0 2π(u − u + 1) diverges, that is, the mean number of returns is infinite for any time interval [0,T ], if the trajectory begins at rest. Next, we find the rate at which E(NRL) tends to infinity as v0 → 0. First we consider the case g = 0 (no acceleration field), and to facilitate calculations we neglect the v2(1 − u2) exponential factor exp 0 , because it decays much slower than 4ε v2(u2 − u + 1) exp − 0 for γT 1. Hence, εγT √ Z ∞ 3u du 2 E(NRL) ∼ 2 exp −z(u − u + 1) , 0 2π(u − u + 1) v2 where z = 0 . This integral converges for z = 0 over any finite interval of εγT integration, so we replace the lower limit from 0 to 1 √ Z ∞ 3u du 2 E(NRL) ∼ 2 exp −z(u − u + 1) + O(1). 1 2π(u − u + 1) Furthermore, if the numerator u is replaced by a constant then the improper integral converges also for z = 0. Therefore, √ Z ∞ 3(u − 1/2) du 2 E(NRL) ∼ 2 exp −z(u − u + 1) + O(1). 1 2π(u − u + 1) 4.4 Mean Number of Returns 77

Setting w = z(u2 − u + 1) yields √ √ 3 Z ∞ e−w 3 E(NRL) ∼ dw + O(1) = Ei(z) + O(1), (4.77) 4π z w 4π where Ei(z) is the exponential integral. The small z asymptotics of the exponential integral [1] gives √ 3 E(N ) ∼ − log z + O(1). (4.78) RL 4π

The mean number of returns to the origin√ at short times has a logarithmic 3 v2 singularity at v = 0, and it diverges as − log 0 . It can be shown 0 4π εγT that the neglected exponential factor contributes only an O(γT ) term, but does not change the main logarithmic singularity. For each return event from left to right there is a preceding return event from right to left. Therefore, the mean number of crossings E(N) is √ 3 E(N) = E(N + N ) ∼ − log z + O(1). (4.79) RL LR 2π Expanding the mean number of returns in Taylor series in g gives

∂E(N , g = 0) E(N ) = E(N , g = 0) + RL g + ... (4.80) RL RL ∂g

We have already calculated the ﬁrst term E(NRL, g = 0) and showed that it has a logarithmic singularity. The second term √ Z ∞ ∂E(NRL, g = 0) v0 3u(1 + u) du 2 = 2 exp −z(u − u + 1) ∂g 2εγ 0 2π(u − u + 1) is calculated in a similar fashion, and we obtain s ! ∂E(N , g = 0) T RL = O , (4.81) ∂g εγ which is of order 1 in v0. Therefore, the small velocity asymptotics (4.78) remains even for small acceleration ﬁelds. 78 Recurrence Time of the Brownian Motion

4.5 Long Time Asymptotics

We ﬁnd the long time asymptotic behavior of the recurrence pdf by solving the integral equation (4.31) asymptotically for γt 1. We show that the recurrence pdf is to leading order g (v, v ) 1 f(v, t | v ) = 0 0 + O . 0 t3/2 t2

The t−3/2 decay law should come as no surprise, because it is also the tail asymptotics of the ﬁrst passage time of hitting a boundary away from the point of initiation for the Brownian motion. We rewrite the integral equation (4.31) as Z 0 vpc(0, v, t | 0, v0) = −f(v, t | v0) + v dηf(η, · | v0) ∗ pc(0, v, · | 0, η), (4.82) −∞ where Z t f(η, · | v0) ∗ pc(0, v, · | 0, η) = f(η, s | v0) ∗ pc(0, v, t − s | 0, η) ds 0 is the time convolution. The Laplace transform of the integral equation (4.82) with respect to t is Z 0 ˆ ˆ vpˆc(0, v, θ | 0, v0) = −f(v, θ | v0) + v dηf(η, θ | v0)p ˆc(0, v, θ | 0, η), (4.83) −∞ where θ is the Laplace time coordinate. We ﬁnd the long time behavior γt 1 by ignoring the exponentially decaying terms of the fundamental solution pc of eq. (4.6) and eqs.(4.7), (4.8). Therefore, introducing transcendentally small errors,

−1 R ∼ x − x0 − γ v0,S ∼ v, (4.84) where we have assumed g = 0 for the free Brownian motion, and

F ∼ εγ−2(2γt − 3),G ∼ ε, H ∼ εγ−1. (4.85)

Thus, for γt 1 γ v2 (v + v)2 p (0, v, t | 0, v ) ∼ √ exp − exp − 0 , (4.86) c 0 2πε 2γt − 4 2ε 2ε(2γt − 4) 4.5 Long Time Asymptotics 79 that is, the density is almost Maxwellian, as expected. Furthermore, eq. (4.86) gives

γ v2 1 p (0, v, t | 0, v ) = √ exp − 1 + O . (4.87) c 0 2πε 2γt 2ε γt

Hence, the small θ asymptotics of the Laplace transformp ˆc is

1 v2 rγ θ pˆc(0, v, θ | 0, v0) = √ exp − 1 + O . (4.88) 2 2πε 2ε θ γ

In view of eq.(4.88), we look for a solution of the integral equation (4.83) in s θ the form of a power series in γ

s θ θ fˆ(v, θ | v ) = f (v, v ) + f (v, v ) + f (v, v ) + ... (4.89) 0 0 0 1 0 γ 2 0 γ

The series (4.89) starts from a θ-independent term, because f(v, t | v0) is a, possibly defective, pdf and so

Z 0 Z ∞ Z 0 ˆ 0 < dη f(η, t | v0) ds = f(η, 0 | v0) dη ≤ 1. −∞ 0 −∞

rγ Equating the leading order singular terms in eq.(4.83), which are O , θ gives Z 0 f0(η, v0) dη = 1, (4.90) −∞ which means that the free particle is recurrent. Equating the O(1) terms gives v2 v exp − 2ε Z 0 f0(v, v0) = √ f1(η, v0) dη. (4.91) 2 2πε −∞ 80 Recurrence Time of the Brownian Motion

We integrate equation (4.91) and obtain by equation (4.90)

Z 0 1 = f0(v, v0) dv −∞ v2 v exp − Z 0 2ε Z 0 = √ dv f1(η, v0) dη −∞ 2 2πε −∞ 1 Z 0 = − √ f1(η, v0) dη. (4.92) 2 2π −∞

Therefore, v2 v exp − 2ε f (v, v ) = − . (4.93) 0 0 ε r θ Similarly, f (v, v ) is determined by the O , and we obtain 1 0 γ

v2 v exp − √ 2ε f (v, v ) = 2 2π . (4.94) 1 0 ε

We conclude that the half-integer power series (4.89) is self consistent. d 1 Note that fˆ ∼ √ is singular for θ = 0. Therefore, the mean time of dθ θ return is inﬁnite Z ∞ d ˆ tf(v, t | v0) dt = − f(v, 0 | v0) = ∞, 0 dθ

d 1 which is the Gambler’s ruin paradox [51]. Moreover, fˆ ∼ √ means dθ θ that the asymptotic behavior of tf(·, t | ·) for large t is the inverse Laplace 1 1 1 transform of √ , which is proportional to √ . That is, tf(·, t | ·) ∼ √ , or θ t t more precisely

f1(v, v0) f(v, t | v0) ∼ − , (4.95) 2pπγt3 4.6 Small v asymptotics 81 which, by eq.(4.94) is

v2 v exp − 2ε f(v, t | v0) ∼ − , for γt 1. (4.96) εp2γt3

Note that (4.96) means that at long times the return probability is to leading order independent of the initial velocity v0, because the initial velocity is thermalized for times longer than relaxation time.

4.6 Small v asymptotics

The Laplace method of Section 4.3 assumed that γt is a small parameter, smaller than all other parameters. This assumption does not hold for small velocities at ﬁxed time t, that is, for v2/ε γt. Therefore, the asymptotic expansion (4.73) holds for small γt, but it is not uniform in v2/ε. In this section we construct an asymptotic solution of the integral equation (4.31) for v2/ε γt 1. For γt 1 we have

1 v2 + vv + v2 p (0, v, t | 0, v ) ∼ exp − 0 0 . c 0 t2 εγt

Therefore, the left hand side (LHS) of the integral equation (4.31) is O(v). We show below that the integral term of the right hand side (RHS) of (4.31) is singular in v. Speciﬁcally,

Z 0 Z t v dη pc(0, v, t − s | 0, η) ds = O(1), (4.97) −∞ 0 Z 0 Z t v ηdη pc(0, v, t − s | 0, η) ds = O(v log v), (4.98) −∞ 0 Z 0 Z t n v η dη pc(0, v, t − s | 0, η) ds = O(v), n ≥ 2. (4.99) −∞ 0 Note that the right hand side terms of (4.31) have the same sign, hence, their leading order term in v cannot cancel out. It follows that f is not O(1), for otherwise equation (4.97) gives that the RHS is O(1), whereas the LHS is 82 Recurrence Time of the Brownian Motion

O(v). Similarly, f is not O(v), because eq.(4.98) gives an O(v log v) leading order term that cannot be balanced. We conclude that

2 f(v, t | v0) ∼ O(v ),

which, by eq.(4.99), gives balanced O(v) terms on both the RHS and the LHS. For example, we verify eq. (4.97)

Z 0 Z t v dη pc(0, v, t − s | 0, η) ds −∞ 0 Z 0 Z t ds v2 + vη + η2 ∼ v dη 2 exp − −∞ 0 (t − s) εγ(t − s) Z 0 dη v2 + vη + η2 = vεγ 2 2 exp − −∞ v + vη + η εγt Z ∞ du v2(1 + u + u2) = εγ 2 exp − = O(1), (4.100) 0 1 + u + u εγt

because the last integral converges for v = 0. Equations (4.98)-(4.99) are obtained in a similar fashion. The entire asymptotic series of (4.99) is (for n ≥ 2)

Z 0 Z t ∞ n X (n) j n v η dη pc(0, v, t − s | 0, η) ds ∼ aj v + bnv log v, (4.101) −∞ 0 j=1

(n) where aj and bn are coeﬃcients. For example, for n = 2

Z 0 Z t ∞ 2 X (2) j 2 v η dη pc(0, v, t − s | 0, η) ds ∼ aj v + b2v log v. (4.102) −∞ 0 j=1

P∞ j If we look for a solution in the form of merely a power series f ∼ n=2 fjv , then the logarithmic terms, such as v2 log v, cannot be balanced. This term can be balanced if we look for a solution that also includes powers of loga- rithms. In particular, the v2 log v term can be balanced by a v3 log v term. However, this term also produces a v log v term. Therefore, we include terms 4.6 Small v asymptotics 83 in the form vn log v in our asymptotic series. Similarly, the v3 log v term produces a v3 log2 v term, that can be balanced by a v4 log2 term. The procedure is now clear. We conclude that the asymptotic form of the solution is

∞ n−2 X X n m f ∼ anmv log v. (4.103) n=2 m=0 Chapter 5

Narrow Escape in Three Dimensions

The contents of this chapter were submitted for publication [177]

A Brownian particle with diffusion coefficient D is confined to a bounded 3 domain of volume V in R by a reflecting boundary, except for a small absorbing window. The mean time to absorption diverges as the window shrinks, thus rendering the calculation of the mean escape time a singular perturbation problem. We construct an asymptotic approximation for the case of an elliptical window of large semi axis a V 1/3 and show that the mean escape V time is Eτ ∼ K(e), where e is the eccentricity of the ellipse; and K(·) is 2πDa the complete elliptic integral of the first kind. In the special case of a circular V hole the result reduces to Lord Rayleigh’s formula Eτ ∼ , which was de- 4aD rived by heuristic considerations. For the special case of a spherical domain, V a R a we obtain the asymptotic expansion Eτ = 1 + log + O . 4aD R a R This problem is important in understanding the flow of ions in and out of narrow valves that control a wide range of biological and technological function.

5.1 Introduction

We consider the exit problem of a Brownian motion from a bounded domain, whose boundary is reﬂecting, except for a small absorbing window. 5.1 Introduction 85

The mean first passage time to the absorbing window (MFPT), Eτ, is the solution of a mixed Neumann-Dirichlet boundary value problem (BVP) for the Poisson equation, known as the corner problem, which has singularity at the boundary of the hole [33, 105, 106]. The MFPT grows to infinity as the window size shrinks to zero, thus rendering its calculation a singular perturbation problem for the mixed BVP, which we call the narrow escape problem. The narrow escape problem has been considered in the literature in only a few special cases, beginning with Lord Rayleigh (in the context of acoustics), who found the flux through a small hole by using a result of Helmholtz [69]. He stated [147] (p.176) “Among different kinds of channels an important place must be assigned to those consisting of simple apertures in unlimited plane walls of infinitesimal thickness. In practical applications it is sufficient that a wall be very thin in proportion to the dimensions of the aperture, and approximately plane within a distance from aperture large in proportion to the same quantity.” More recently, Rayleigh’s result was shown to fit the MFPT obtained from Brownian dynamics simulations [62]. Another result was presented in [78], where a two-dimensional narrow escape problem was considered and whose method is generalized here. A related problem is that of escape from a domain, whose boundary is reflecting, except for an absorbing sphere, disjoint from the reflecting part of the boundary [144, and references therein]. It differs from the narrow escape problem in that there is no singularity at the boundary and there is no boundary layer. The mixed boundary value problems of classical electrostatics (e.g., the electrified disk problem [88]), elasticity (punch problems), diffusion and conductance theory, hydrodynamics, and acoustics were solved, by and large, for special geometries by separation of variables. In axially symmetric geometries this method leads to a dual series or to integral equations that can be solved by special techniques [184, 49, 50, 116, 192]. The special case of asymptotic representation of the solution of the corner problem for small Dirichlet and large Neumann boundaries was not done for general domains. The first attempt in this direction seems to be [78]. The narrow escape problem does not seem to fall within the theory of large deviations [36]. It is different from Kol- mogorov’s exit problem [126] of a diffusion process with small noise from an attractor of the drift (e.g., a stable equilibrium or limit cycle) in that the narrow escape problem has no large and small coefficients in the equation. The singularity of Kolmogorov’s problem is the degeneration of a second order elliptic operator into a first order operator in the limit of small noise, whereas the singularity of the narrow escape problem is the degeneration of 86 Narrow Escape in Three Dimensions

the mixed BVP to a Neumann BVP on the entire boundary. There exist precise asymptotic expansions of Eτ for Kolmogorov’s exit problem, including error estimates (see, e.g., [66], [54]), which show that the MFPT grows exponentially with decreasing noise. In contrast, the narrow escape time grows algebraically rather than exponentially, as the window shrinks. Our first main result is a derivation of the leading order term in the expansion of the MFPT of a Brownian particle with diffusion coefficient D, from a general domain of volume V to an elliptical hole of large semi axis a that is much smaller than V 1/3, V Eτ ∼ K(e), (5.1) 2πDa where e is the eccentricity of the ellipse, and K(·) is the complete elliptic integral of the first kind. In the special case of a circular hole (5.1) reduces to V Eτ ∼ . (5.2) 4aD Equation (5.1) shows that the MFPT depends on the shape of the hole, and not just on its area. This result was known to Lord Rayleigh [147], who considered the problem of the electrified disk (which he knew was equivalent to finding the flow of an incompressible fluid through a channel and to the problem of finding the conductance of the channel), who reduced the problem to that of solving an integral equation for the flux density through the hole. The solution of the integral equation, which goes back to Helmholtz [69] and is discussed in [116], is proportional to (a2 −ρ2)−1/2 in the circular case, where ρ is the distance from the center of the hole [88, 184, 49]. Note that equations (5.1) and (5.2) are leading order approximations and do not contain an error estimate. We prove (5.1) by using the singularity properties of Neumann’s function for three-dimensional domains, in a manner similar to that used in [78] for two-dimensional problems. The leading order term is the solution of Helmholtz’s integral equation [69]. Our second main result is a derivation of the second term and error estimate for a ball of radius R with a small circular hole of radius a in the boundary, V a R a Eτ = 1 + log + O . (5.3) 4aD R a R Equation (5.3) contains both the second term in the asymptotic expansion of the MFPT and an error estimate. We use Collins’ method [26, 27] of solving 5.2 General 3D Bounded Domain 87 dual series of equations and expand the resulting solutions for small ε = a/R. The estimate of the error term, which turns out to be O(ε log ε), seems to be a new result. An error estimate for eq.(5.1) for a general domain is still an open problem. We conjecture that it is O(ε log ε), as is the case for the ball. If the absorbing window touches a singular point of the boundary, such as a corner or cusp, the singularity of the Neumann function changes and so do the asymptotic results. In three dimensions the class of isolated singularities of the boundary is much richer than in the plane, so the methods of [179] cannot be generalized in a straightforward manner to three dimensions. We postpone the investigation of the MFPT to windows at isolated singular points in three dimensions to a future paper. In Section 5.2 we derive a leading order approximation to the MFPT for a general domain with a general small window. The leading order term is expressed in terms of a solution to Helmholtz’s integral equation, which is solved explicitly for an elliptical window. In Section 5.3 we obtain two terms in the asymptotic expansion of the MFPT from a ball with a circular window and an error estimate. Finally, we present a summary and list some applications in Section 5.4. This is the first chapter in a series of three, the second of which considers the narrow escape problem from a bounded simply connected planar domain, and the third of which considers the narrow escape problem from a bounded domain with boundary with corners and cusps on a two-dimensional Riemannian manifold.

5.2 General 3D Bounded Domain

3 A Brownian particle diffuses freely in a bounded domain Ω ⊂ R , whose boundary ∂Ω is sufficiently smooth. The trajectory of the Brownian particle, denoted x(t), is reflected at the boundary, except for a small hole ∂Ωa, where it is absorbed. The reflecting part of the boundary is ∂Ωr = ∂Ω − ∂Ωa. The lifetime of the particle in Ω is the first passage time τ of the Brownian particle from any point x ∈ Ω to the absorbing boundary ∂Ωa. The MFPT,

v(x) = E[τ | x(0) = x], is finite under quite general conditions [166]. As the size (e.g., the diameter) of the absorbing hole decreases to zero, but that of the domain remains finite, we assume that the MFPT increases indefinitely. A measure of smallness can 88 Narrow Escape in Three Dimensions

be chosen as the ratio between the surface area of the absorbing boundary and that of the entire boundary,

|∂Ω | ε = a 1, |∂Ω|

(see, however, a pathological example in Appendix B.3). The MFPT v(x) satisﬁes the mixed boundary value problem [166]

1 ∆v(x) = − , for x ∈ Ω, (5.4) D

v(x) = 0, for x ∈ ∂Ωa, (5.5)

∂v(x) = 0, for x ∈ ∂Ω , ∂n(x) r

where D is the diffusion coefficient. According to our assumptions v(x) → ∞ as the size of the hole decreases to zero, e.g., as ε → 0, except in a boundary layer near ∂Ωa. Our purpose is to find an asymptotic approximation to v(x) in this limit.

5.2.1 The Neumann function and integral equations

To calculate the MFPT v(x), we use the Neumann function N(x, ξ) (see [78], [144]), which is a solution of the boundary value problem

∆xN(x, ξ) = −δ(x − ξ), for x, ξ ∈ Ω, (5.6)

∂N(x, ξ) 1 = − , for x ∈ ∂Ω, ξ ∈ Ω, ∂n(x) |∂Ω|

and is deﬁned up to an additive constant. The Neumann function has the form [55] 1 N(x, ξ) = + v (x, ξ), (5.7) 4π|x − ξ| S 5.2 General 3D Bounded Domain 89

where vS(x, ξ) is a regular harmonic function of x ∈ Ω and of ξ ∈ Ω. Green’s identity gives Z [N(x, ξ)∆v(x) − v(x)∆N(x, ξ)] dx = Ω Z ∂v(x(S)) ∂N(x(S), ξ) = N(x(S), ξ) − v(x(S)) dS ∂Ω ∂n ∂n Z ∂v(x(S)) 1 Z = N(x(S), ξ) dS + v(x(S)) dS. ∂Ω ∂n |∂Ω| ∂Ω On the other hand, equations (5.4) and (5.6) imply that Z 1 Z [N(x, ξ)∆v(x) − v(x)∆N(x, ξ)] dx = v(ξ) − N(x, ξ) dx, Ω D Ω hence 1 Z v(ξ) − N(x, ξ) dx = (5.8) D Ω Z ∂v(x(S)) 1 Z N(x(S), ξ) dS + v(x(S)) dS. ∂Ω ∂n |∂Ω| ∂Ω Note that the second integral on the right hand side of eq.(5.8) is an additive constant. Setting 1 Z C = v(x(S)) dS, (5.9) |∂Ω| ∂Ω we rewrite eq.(5.8) as 1 Z Z ∂v(x(S)) v(ξ) = N(x, ξ) dx + N(x(S), ξ) dS − C, (5.10) D Ω ∂Ωa ∂n which is an integral representation of v(ξ). We deﬁne the boundary ﬂux density ∂v(x(S)) g(S) = , (5.11) ∂n choose ξ ∈ ∂Ωa, and use the boundary condition (5.5) to obtain the equation 1 Z Z 0 = N(x, ξ) dx + N(x(S), ξ)g(S) dS − C, (5.12) D Ω ∂Ωa 90 Narrow Escape in Three Dimensions

for all ξ ∈ ∂Ωa. Equation (5.12) is an integral equation for g(S) and C. To construct an asymptotic approximation to the solution, we note that the ﬁrst integral in equation (5.12) is a regular function of ξ on the boundary. Indeed, due to symmetry of the Neumann function, we have from (5.6)

Z ∆ξ N(x, ξ) dx = −1 for ξ ∈ Ω (5.13) Ω

and ∂ Z |Ω| N(x, ξ) dx = − for ξ ∈ ∂Ω. (5.14) ∂n(ξ) Ω |∂Ω| Equation (5.13) and the boundary condition (5.14) are independent of the hole ∂Ωa, so they define the integral as a regular function, up to an additive constant, also independent of ∂Ωa. The assumption that for all x ∈ Ω, away from ∂Ωa, the MFPT v(x) increases to infinity as the size of the hole decreases and eq.(5.9) imply that C → ∞ as as the size of the hole decreases to zero. This means that for ξ ∈ ∂Ωa the second integral in eq.(5.12) must also become infinite in this limit, because the first integral is independent of ∂Ωa. Therefore, the integral equation (5.12) is to leading order

Z N(x(S), ξ))g0(S) dS = C0 for ξ ∈ ∂Ωa, (5.15) ∂Ωa

where g0(S) is the ﬁrst asymptotic approximation to g(S) and C0 is the ﬁrst approximation to the constant C. Furthermore, only the singular part of the Neumann function contributes to the leading order, so we obtain the integral equation Z 1 g0(x) dSx = C0, (5.16) 2π ∂Ωa |x − y|

where C0 is a constant, which represents the ﬁrst approximation to the mean ﬁrst passage time (MFPT). Note that the singularity of the Neumann function at the boundary is twice as large as it is inside the domain, due to the contribution of the regular part (the “image charge”) and therefore the factor 1 1 of equation (5.7) was replaced by . In general, the integral equation 4π 2π (5.16) has no explicit solution, and should be solved numerically. 5.2 General 3D Bounded Domain 91

5.2.2 Elliptic hole

When the hole ∂Ωa is an ellipse, the solution of the integral equation (5.16) is known [147], [116]. Speciﬁcally, assuming the ellipse is given by

x2 y2 + = 1, z = 0, (b ≤ a), a2 b2 the solution is g˜0 g0(x) = r , (5.17) x2 y2 1 − − a2 b2 whereg ˜0 is a constant (to be determined below). The proof, originally given in [69], is reproduced in Appendix B.2. To determine the value of the constant g˜0, we use the compatibility condition

Z |Ω| g0(x) dSx = , (5.18) ∂Ωa D obtained from the integration of eq.(5.4) over Ω. Using the value

q x2 Z Z a Z b 1− 2 a g˜0 dy g0(x) dSx = dx = 2πabg˜0 (5.19) q x2 r 2 2 ∂Ωa −a −b 1− x y a2 1 − − a2 b2 and the compatibility condition (5.18), we obtain

|Ω| g˜ = . (5.20) 0 2πDab Hence, by equation (B.6), the leading order approximation to C is Z 1 g0(x) |Ω| C0 = dSx = K(e), (5.21) 2π ∂Ωa |x − y| 2πDa where K(·) is the complete elliptic integral of the ﬁrst kind, and e is the eccentricity of the ellipse, r b2 e = 1 − . (5.22) a2 92 Narrow Escape in Three Dimensions

In other words, the MFPT from a large cavity of volume |Ω| through a small elliptic hole is to leading order |Ω| Eτ(a, b) ∼ K(e). (5.23) 2πDa π For example, in the case of a circular hole, we have e = 0 and K(0) = , so 2 that |Ω| 1 Eτ(a, a) ∼ = O , (5.24) 4Da ε provided |Ω|2/3 = O(1) for ε 1. |∂Ω| Equation (5.24) was used in [62], [32]. If the mouth of the channel is not circular, the MFPT is different. Equation (5.24) indicates that a Brownian particle that tries to leave the domain “sees” finer details in the geometry of the hole and the domain than just the quotient of the surface areas. The additional geometric features contained in the MFPT are illustrated by the two interesting limits e 1, where the ellipse is almost circular, and 1−e 1, where the ellipse is squeezed. In the case e 1, we use the expansion of the complete elliptic integral of the first kind [1] ( ) π 12 1 · 32 1 · 3 · 53 K(e) = 1 + e2 + e4 + e6 + ··· . (5.25) 2 2 2 · 4 2 · 4 · 6

In the second limit 1 − e 1, we ﬁnd from the asymptotic behavior [1] 1 16 lim K(e) − log = 0 (5.26) e→1 2 1 − e that |Ω| 16 Eτ ∼ log , for 1 − e 1. (5.27) 4πa 1 − e The area of the hole is given by √ S = πab = πa2 1 − e2, (5.28) or equivalently S1/2 a = , (5.29) π1/2 (1 − e2)1/4 5.3 Explicit Computations for the Sphere 93 and the MFPT has the asymptotic form √ 4 2 |Ω| (1 − e)1/4 16 Eτ ∼ √ log , for 1 − e 1. (5.30) 4 πS 1 − e

5.3 Explicit Computations for the Sphere

The analysis of Section 5.2 is not easily extended to the computation, or even merely the estimation of the next term in the asymptotic approximation of the MFPT. The explicit results for the particular case of escape from a ball through a small circular hole gives an idea of the order of magnitude of the second term and the error in the asymptotic expansion of the MFPT. If the domain Ω is a ball, the method of [184, 49, 50, 26, 27] can be used to obtain a full asymptotic expansion of the MFPT. We consider the motion of a Brownian particle inside a ball of radius R. The particle is reﬂected at the ε sphere, except for a small cap of radius a = εR and surface area 4πR2 sin2 , 2 where it exits the ball. We assume ε 1. The MFPT v(r, θ, φ) satisﬁes the mixed boundary value problem for Poisson’s equation in the ball [166],

∆v(r, θ, φ) = −1, for r < R, 0 ≤ θ ≤ π, 0 ≤ φ < 2π,

v(r, θ, φ) = 0, for 0 ≤ θ < ε, 0 ≤ φ < 2π, (5.31) r=R

∂v(r, θ, φ) = 0, for ε ≤ θ ≤ π, 0 ≤ φ < 2π, ∂r r=R

The diﬀusion coeﬃcient has been chosen to be D = 1. Due to the cylindrical symmetry of the problem, the solution is independent of the angle φ, that is, v(r, θ, φ) = v(r, θ), so the system (5.31) can be written as

∆v(r, θ) = −1, for r < R, 0 ≤ θ ≤ π,

v(r, θ) = 0, for 0 ≤ θ < ε, r=R

∂v(r, θ) = 0, for ε ≤ θ ≤ π, ∂r r=R 94 Narrow Escape in Three Dimensions

where the Laplacian is given by 1 ∂ ∂v 1 ∂ ∂v ∆v(r, θ) = r2 + sin θ . r2 ∂r ∂r r2 sin θ ∂θ ∂θ

R2 − r2 The function f(r, θ) = is the solution of the boundary value problem 6 ∆f = −1, for r < R,

f = 0. r=R In the decomposition v = u + f, the function u(r, θ) satisﬁes the mixed Dirichlet-Neumann boundary value problem for the Laplace equation

∆u(r, θ) = 0, for r < R, 0 ≤ θ ≤ π,

u(r, θ) = 0, for 0 ≤ θ < ε, (5.32) r=R

∂u(r, θ) R = , for ε ≤ θ ≤ π. ∂r r=R 3 Separation of variables suggests that

∞ X r n u(r, θ) = a P (cos θ), (5.33) n R n n=0

where Pn(cos θ) are the Legendre polynomials, and the coeﬃcients {an} are to be determined from the boundary conditions ∞ X u(r, θ) = anPn(cos θ) = 0, 0 ≤ θ < ε, (5.34) r=R n=0 ∞ 2 ∂u(r, θ) X R = nanPn(cos θ) = , ε ≤ θ ≤ π. (5.35) ∂r 3 r=R n=1 Equations (5.34), (5.35) are dual series equations of the mixed boundary value problem at hand, and their solution results in the solution of the bound- 5.3 Explicit Computations for the Sphere 95 ary value problem (5.32). Dual series equations of the form

∞ X anPn(cos θ) = 0, for 0 ≤ θ < ε, (5.36) n=0 ∞ X (2n + 1)anPn(cos θ) = G(θ), for ε ≤ θ ≤ π (5.37) n=0 are solved in [184, eqs.(5.5.12)-(5.5.14), (5.6.12)]. However, the dual series equations (5.36)-(5.37) are different from equations (5.34)-(5.35). The factor 2n + 1 that appears in equation (5.37) is replaced by n in equation (5.35). What seems as a slight difference turns out to make our task much harder. The factor 2n + 1 fits much more easily into the infinite sums (5.36)-(5.37), because it is the normalization constant of the Legendre polynomials.

5.3.1 Collins’ method

The solution of dual relations of the form (5.35) (see [184, (5.6.19)-(5.6.20)]) is discussed in [26, 27]. Speciﬁcally, assume that for given functions G(θ) and F (θ) we have the representation

∞ X −m (1 + Hn)bnTm+n(cos θ) = F (θ), for 0 ≤ θ < ε, n=0 ∞ X −m (2n + 2m + 1)bnTm+n(cos θ) = G(θ), for ε < θ ≤ π, n=0

−m where Tm+n are Ferrer’s associated Legendre polynomials [48, 5] and {Hn} is a given series that is O(n−1) as n → ∞. Then for m = 0, we have

∞ X (1 + Hn)bnPn(cos θ) = F (θ), for 0 ≤ θ < ε, (5.38) n=0 ∞ X (2n + 1)bnPn(cos θ) = G(θ), for ε < θ ≤ π. (5.39) n=0 96 Narrow Escape in Three Dimensions

2n + 1 Setting a = b , a = b , n ≥ 1 in equations (5.34)-(5.35) results in 0 0 n 2n n ∞ X (1 + Hn)bnPn(cos θ) = 0, for 0 ≤ θ < ε, (5.40) n=0 ∞ X 2R2 (2n + 1)b P (cos θ) = + b , for ε ≤ θ ≤ π. (5.41) n n 3 0 n=0

Equations (5.40)-(5.41) are equivalent to (5.38)-(5.39) with H0 = 0,Hn = 1 2R2 , n ≥ 1, F (θ) = 0, and G(θ) = + b . Collins’ method of solution 2n 3 0 consists in ﬁnding an integral equation for the function ∞ X h(θ) = (2n + 1)bnPn(cos θ), for 0 ≤ θ < ε, n=0 so that 1 Z ε 1 Z π bn = h(α)Pn(cos α) sin α dα + G(α)Pn(cos α) sin α dα. 2 0 2 ε Substituting into equation (5.38), with F (θ) ≡ 0, we ﬁnd for 0 ≤ θ < ε that ∞ 1 Z ε X 0 = h(α) (1 + H )P (cos α)P (cos θ) sin α dα 2 n n n 0 n=0 ∞ 1 Z π X + G(α) (1 + H )P (cos α)P (cos θ) sin α dα. (5.42) 2 n n n ε n=0

5.3.2 The asymptotic expansion

To facilitate the calculations, we consider ﬁrst the case Hn = 0 for all n. Then we will show that the leading order term obtained for this case is the same as that for the case Hn 6= 0. In the latter case, we obtain the ﬁrst correction to the leading order term and an estimate on the remaining error.

The leading order term when Hn ≡ 0

We will now sum the series (5.42) in the case Hn ≡ 0. First, we recall Mehler’s integral representation for the Legendre polynomials [1, 117], √ Z θ 1 2 cos(n + 2 )u du Pn(cos θ) = √ , (5.43) π 0 cos u − cos θ 5.3 Explicit Computations for the Sphere 97 and the identity [184] ∞ √ X 1 H(α − u) 2 P (cos α) cos n + u = √ , (5.44) n 2 cos u − cos α n=0 where H(x) is the Heaviside unit step function. Then we obtain for u < θ < ε < α, ∞ 1 Z π X G(α) P (cos α)P (cos θ) sin α dα = 2 n n ε n=0 ∞ √ 1 Z π X 2 Z θ cos(n + 1 )u du = G(α) P (cos α) √ 2 sin α dα 2 n π ε n=0 0 cos u − cos θ 1 Z θ du Z π G(α) sin α dα = √ √ . (5.45) 2π 0 cos u − cos θ ε cos u − cos α Similarly, ∞ 1 Z ε X h(α) P (cos α)P (cos θ) sin α dα = 2 n n 0 n=0 1 Z θ du Z ε h(α) sin α dα = √ √ . (5.46) 2π 0 cos u − cos θ u cos u − cos α Hence, Z θ du Z ε h(α) sin α dα √ √ = 0 cos u − cos θ u cos u − cos α Z θ du Z π G(α) sin α dα − √ √ . (5.47) 0 cos u − cos θ ε cos u − cos α Equation (5.47) means that the Abel transforms [197] of two functions are the same, so that Z ε h(α) sin α dα Z π G(α) sin α dα √ = − √ , (5.48) u cos u − cos α ε cos u − cos α because the Abel transform is uniquely invertible. Equation (5.48) is an Abel-type integral equation, whose solution is given by 1 d Z ε sin u du Z π G(α) sin α dα h(θ) sin θ = √ √ , (5.49) π dθ θ cos θ − cos u ε cos u − cos α 98 Narrow Escape in Three Dimensions

2 d Z ε H(u) sin u du h(θ) = − √ , (5.50) sin θ dθ θ cos θ − cos u

where

H(u) = −G(u, ε), (5.51)

and

1 Z π G(θ) sin θ dθ G(u, ε) = √ . (5.52) 2π ε cos u − cos θ

2R2 The dual integral equations (5.40)-(5.41) deﬁne G(θ) = + b , so that 3 0

1 Z π 2R2 sin θ dθ G(ψ, φ) = + b0 √ (5.53) 2π φ 3 cos ψ − cos θ 2 π 2R 1 p = + b0 cos ψ − cos θ 3 π θ=φ 2R2 1 √ ψ = + b 2 cos − pcos ψ − cos φ , for ψ < φ. 3 0 π 2

In particular, setting n = 0 in equation (5.42) and using equation (5.50), gives

1 Z ε 1 Z π 2R2 b0 = h(α) sin α dα + + b0 sin α dα (5.54) 2 0 2 ε 3 √ Z ε 2 ψ 2R 2 ε = 2 H(ψ) cos dψ + + b0 cos . 0 2 3 2 5.3 Explicit Computations for the Sphere 99

Integrating equation (5.53), we obtain

√ Z ε ψ 2 G(ψ, ε) cos dψ = (5.55) 0 2 √ 2 ε 2R 2 Z √ ψ p ψ = + b0 2 cos − cos ψ − cos ε cos dψ 3 π 0 2 2 2R2 ε + b0 2R2 4 Z sin s2 ds = 3 (ε + sin ε) − + b 2 π 3 0 π r ε 0 sin2 − s2 2 2R2 + b0 2R2 ε = 3 (ε + sin ε) − + b sin2 . π 3 0 2

Combining equations (5.54) and (5.55) gives

2R2 π 2R2 π |Ω| a b = − 1 = + O(1) = 1 + O , 0 3 ε + sin ε 3 2ε 4a R (5.56) 4πR3 where |Ω| = is the volume of the ball, and a = Rε is the radius of the 3 hole.

The case Hn 6= 0

The asymptotic expression (5.56) for b0, was derived under the simplifying assumption that Hn ≡ 0. However, we are interested in the value of b0 which is produced by the solution of the dual series equations (5.40)-(5.41), where 100 Narrow Escape in Three Dimensions

1 H = . We sum the series (5.42) by the identities n 2n

∞ 1 Z ε X h(α) H P (cos α)P (cos θ) sin α dα 2 n n n 0 n=0 ∞ √ √ 1 Z ε X 2 Z α cos(n + 1 )v dv 2 Z θ cos(n + 1 )u du = h(α) H √ 2 √ 2 sin α dα 2 n π cos v − cos α π 0 n=0 0 0 cos u − cos θ 1 Z ε Z α dv Z θ K(u, v) du = h(α) sin α dα √ √ 2π 0 0 cos v − cos α 0 cos u − cos θ 1 Z θ du Z ε Z ε h(α) sin α dα = √ K(u, v) dv √ , (5.57) 2π 0 cos u − cos θ 0 v cos v − cos α

where

∞ 2 X 1 1 K(u, v) = H cos n + u cos n + v (5.58) π n 2 2 n=0 1 cos 2 (v + u) 1 = − log 2 sin (v + u) 2π 2 1 cos 2 (v − u) 1 − log 2 sin (v − u) 2π 2 v + u − π 1 v − u − π 1 + sin (v + u) + sin (v − u). 4π 2 4π 2

Similarly,

∞ 1 Z π X G(α) H P (cos α)P (cos θ) sin α dα (5.59) 2 n n n ε n=0 1 Z π Z α dv Z θ K(u, v) du = G(α) sin α dα √ √ 2π ε 0 cos v − cos α 0 cos u − cos θ 1 Z θ du Z π Z α K(u, v) dv = √ G(α) sin α dα √ . 2π 0 cos u − cos θ ε 0 cos v − cos α 5.3 Explicit Computations for the Sphere 101

Substituting equations (5.45), (5.46), (5.57), and (5.59) into equation (5.42) yields 1 Z θ du Z ε h(α) sin α dα 0 = √ √ 2π 0 cos u − cos θ u cos u − cos α 1 Z θ du Z ε Z ε h(α) sin α dα + √ K(u, v) dv √ 2π 0 cos u − cos θ 0 v cos v − cos α 1 Z θ du Z π G(α) sin α dα + √ √ 2π 0 cos u − cos θ ε cos u − cos α 1 Z θ du Z π Z α K(u, v) dv + √ G(α) sin α dα √ , 2π 0 cos u − cos θ ε 0 cos v − cos α which is again an Abel-type integral equation. Inverting the Abel transform [197], we obtain 1 Z ε h(α) sin α dα 1 Z ε Z ε h(α) sin α dα 0 = √ + K(u, v) dv √ 2π u cos u − cos α 2π 0 v cos v − cos α (5.60) 1 Z π G(α) sin α dα 1 Z π Z α K(u, v) dv + √ + G(α) sin α dα √ . 2π ε cos u − cos α 2π ε 0 cos v − cos α Setting 1 Z ε h(α) sin α dα H(u) = √ , (5.61) 2π u cos u − cos α we invert the Abel transform (5.61) to obtain 2 d Z ε sin uH(u) du h(θ) = − √ . (5.62) sin θ dθ θ cos θ − cos u Writing J(u) = H(u) + G(u, ε), (5.63) equation (5.60) becomes Z ε J(u) + K(u, v)J(v) dv = M(u), (5.64) 0 where the free term M(u) is given by Z π M(u) = − K(u, v)G(v, v) dv. (5.65) ε Equation (5.64) is a Fredholm integral equation for J. 102 Narrow Escape in Three Dimensions

The second term and the remaining error: L2 estimates Equations (5.54), (5.55), and (5.63) give that √ 2R2 2R2 π 2π Z ε u b0 + = + J(u) cos du, (5.66) 3 3 ε + sin ε ε + sin ε 0 2 where J is the solution of the Fredholm equation (5.64). In this section we show that √ 2π Z ε u 2R2 1 J(u) cos du = b0 + ε log + O(ε) , ε + sin ε 0 2 3 ε therefore the last term in eq.(5.66) should be considered a small correction R2 π to the leading order term , obtained in Section 5.3.2. This confirms the 3 ε intuitive results of [62], [32] and gives an estimate on the error term. Due to the logarithmic singularity of the function K(u, v) (see (5.58)) the operator K, defined by Z ε Kf(u) = K(u, v)f(v) dv, (5.67) 0 maps L2[0, ε] into L2[0, ε]. In Appendix B.1 we derive the estimate √ 30 1 kKk ≤ ε log , (5.68) 2 2π ε for ε 1. Better estimates can be found; however we settle for this rough estimate that suffices for our present purpose.

Estimate of kJk2 In terms of the operator K, equation (5.64) can be written as J = M − KJ. (5.69) The triangle inequality yields

kJk2 ≤ kMk2 + kKJk2 ≤ kMk2 + kKk2kJk2, (5.70) which together with the estimate (5.68) gives kMk2 1 kJk2 ≤ ≤ 1 + ε log kMk2 for ε 1. (5.71) 1 − kKk2 ε 5.3 Explicit Computations for the Sphere 103

Estimate of kMk2

We proceed to ﬁnd an estimation for kMk2. First, we prove that the kernel satisﬁes the identity Z π v K(u, v) cos dv = 0, for all u. (5.72) 0 2 Indeed, by changing the order of summation and integration, we obtain

π ∞ n+1 π Z v 1 X cos u Z 1 v K(u, v) cos dv = 2 cos n + v cos dv 2 π n 2 2 0 n=1 0 ∞ n+1 π 1 X cos u Z = 2 (cos(n + 1)v + cos nv) dv 2π n n=1 0 = 0. (5.73) Equations (5.53), (5.65), and (5.72) imply that √ 2 2R2 Z ε v M(u) = + b0 K(u, v) cos dv. (5.74) π 3 0 2 The estimate (5.68) gives √ √ 2 2R2 √ 15 2R2 1 kMk ≤ + b kKk ε ≤ + b ε3/2 log . (5.75) 2 π 3 0 2 π2 3 0 ε Combining the estimates (5.71) and (5.75), we obtain for ε 1 4 2R2 1 2R2 kJk ≤ + b ε3/2 log = + b O(ε3/2 log ε). (5.76) 2 π2 3 0 ε 3 0

The second term and error estimate The Cauchy-Schwartz inequality implies that √ Z ε 2 2π u 2R 1 J(u) cos du ≤ + b0 ε log , (5.77) ε + sin ε 0 2 3 ε for ε 1, which together with (5.66) gives πR2 |Ω| b = (1 + O(ε log ε)) = (1 + O(ε log ε)) . (5.78) 0 3ε 4a 104 Narrow Escape in Three Dimensions

To obtain the explicit expression for the term O(ε log ε), we write the Fred- holm integral equation (5.64) as (I + K)J = M. (5.79)

The estimate (5.68) implies that kKk2 < 1 for suﬃciently small ε, hence

J = M + O (kKk2kMk2) . (5.80) Thus, using equation (5.74) and the estimates (5.68) and (5.75), we write the last term in equation (5.66) as Z ε u Z ε u J(u) cos du = M(u) cos du + O (εkKk2kMk2) = (5.81) 0 2 0 2 √ 2 Z ε Z ε 2 2R u v 3 2 b0 + K(u, v) cos cos du dv + O ε log ε . π 3 0 0 2 2 Equation (5.58) gives the double integral as Z ε Z ε u v 1 1 K(u, v) cos cos du dv = ε2 log + O(ε2), 0 0 2 2 π ε hence √ 2π Z ε u 2R2 1 J(u) cos du = b0 + ε log + O(ε) . ε + sin ε 0 2 3 ε Now it follows from equation (5.66) that |Ω| 1 b = 1 + ε log + O(ε) . (5.82) 0 4a ε

5.3.3 The MFPT Using the explicit expression (5.82), we obtain the MFPT from the center of the ball as 2 2 R R |Ω| 1 v = u + = b0 + = 1 + ε log + O(ε) . (5.83) r=0 r=0 6 6 4a ε This is also the averaged MFPT for a uniform initial distribution, 1 Z 2π Z π Z R |Ω| 1 Eτ = dφ sin θ dθ v(r, θ)r2 dr = 1 + ε log + O(ε) . |Ω| 0 0 0 4a ε 5.4 Summary and Applications 105

5.4 Summary and Applications

The narrow escape problem for a Brownian particle leads to a singular perturbation problem for a mixed Dirichlet-Neumann (corner) problem with large Neumann part and small Dirichlet part of the boundary. The corner problem, that arises in classical electrostatics (e.g., the electrified disk), elasticity (punch problems), diffusion and conductance theory, hydrodynamics, acoustics, and more recently in molecular biophysics, was solved hitherto mainly for special geometries. In this chapter, we have constructed a leading order asymptotic approximation to the MFPT in the narrow escape problem for a general smooth domain and have derived a second term and an error estimate for the case of a sphere. Our derivation makes Lord Rayleigh’s qualitative observation into a quantitative one. Our leading order analysis of the general case uses the singularity property of the Neumann function for a 3 general domain in R . The special case of the sphere is analyzed by a method developed by Collins and yields a better result. A different approach to the calculation of the MFPT would be to use singular perturbation techniques. The vanishing escape time at the boundary would then be matched to the large outer escape time of order ε−1 by constructing a boundary layer near the boundary. The analysis of the MFPT to a small window at an isolated singular point of the boundary is postponed to a future paper. Brownian motion through narrow regions controls flow in many non-equilibrium systems, from fluidic valves to transistors and ion channels, the protein valves of biological membranes [73]. Indeed, one can view an ion channel as the ultimate nanovalve—nearly picovalve—in which macroscopic flows are controlled with atomic resolution. In this context, the narrow escape problem appeared in the calculation of the equilibration time of diffusion between two chambers connected by a capillary [32]. The equilibration time is the reciprocal of the first eigenvalue of the Neumann problem in this domain, which depends on the MFPT of a Brownian motion in each chamber to the narrow connecting channel. The first eigenfunction is constructed by piecing together the eigenfunctions of the narrow escape problem in each chamber and in the channel so that the function and the flux are continuous across the connecting interfaces. It was assumed in [32] that the flux profile in the connecting hole was uniform. The structure of the flux profile, which is proportional to (a2 − ρ2)−1/2, has been observed by Rayleigh in 1877 [147]. Rayleigh first assumed a radially uniform profile of flux and then refined the profile of flux going through the channel, allowing it to vary with the radial 106 Narrow Escape in Three Dimensions

distance from the center of the cross section of the channel, so as to minimize the kinetic energy. A calculation of the equilibration time was carried out in [98] by solving the same problem, and gave a result that differs from that of [147], which was obtained by heuristic means, by less than two percent. A different approximation, based on the Fourier-Bessel representation in the pore, was derived in [50]. Another application of the narrow escape problem concerns ionic channels [73], and particularly particle simulations of the permeation process [83, 84, 29, 200, 189] that capture much more detail than continuum models. Up to now, computer simulations are inefficient because an ion takes so long even to enter a channel and then so many of the ions return from where they came. From the present analysis, it becomes clear why ions take so long to enter the channel. According to (5.2) the mean time between arrival of ions at the channel is Eτ 1 τ¯ = = , (5.84) N 4DaC where N is the number of ions in the simulation and C is their concentration. A coarse estimate ofτ ¯ at the biological concentration of 0.1Molar, channel radius a = 20A˚, diffusion coefficient D = 1.5 × 10−9m2/sec isτ ¯ ≈ 1nsec. In a Brownian dynamics simulation of ions in solution with time step which is 10 times the relaxation time of the Langevin equation to the Smoluchowski (diffusion) equation at least 1000 simulation steps are needed on the average for the first ion to arrive at the channel. It should be taken into account that most of the ions that arrive at the channel do not cross it [47]. The narrow escape problem comes up in problems of the escape from a domain composed of a big subdomain with a small hole, connected to a thin cylinder (or cylinders) of length L. If ions that enter the cylinder do not return to the big subdomain, the MFPT to the far end of the cylinder is the sum of the MFPT to the small hole and the MFPT to the far end of the narrow cylinder. The latter can be approximated by a one-dimensional problem with one reflecting and one absorbing endpoint. If the domain has a volume V , the approximate expression for the MFPT is V L2 Eτ ≈ + . (5.85) 4εD 2D This method can be extended to a domain composed of many big subdomains with small holes connected by narrow cylinders. The case of one sphere of 4πR3 volume V = , with a small opening of size ε connected to a thin cylinder 3 5.4 Summary and Applications 107 of length L is relevant in biological micro-structures, such as dendritic spines in neurobiology. Indeed, the mean time for calcium ion to diffuse from the spine head to the parent dendrite through the neck controls the spine-dendrite coupling [80]. This coupling is involved in the induction of processes such as synaptic plasticity [118]. Formula (5.85) is useful for the interpretation of experiments and for the confirmation of the diffusive motion of ions from the spine head to the dendrite. Another significant application of the narrow escape formula is to provide a new definition of the forward binding rate constant in micro-domains [79]. Indeed, the forward chemical constant is really the flux of particles to a given portion of the boundary, depending on the substrate location. Up to now, the forward binding rate was computed using the Smoluchowski formula, which corresponds to the absorption flux of particles in a given sphere immersed in an infinite medium. The formula applies when many particles are involved. But to model chemical reactions in micro-structures, where a bounded domain contains only a few particles that bind to a given number of binding sites, the forward binding rate, 1 k = , forward τ¯ has to be computed withτ ¯ given in eq.(5.84). Chapter 6

Narrow Escape: The Circular Disk

The contents of this chapter were submitted for publication [178]

We consider Brownian motion in a circular disk Ω, whose boundary ∂Ω is reflecting, except for a small arc, ∂Ωa, which is absorbing. As ε = |∂Ωa|/|∂Ω| decreases to zero the mean time to absorption in ∂Ωa, denoted Eτ, becomes infinite. The narrow escape problem is to find an asymptotic expansion of Eτ for ε 1. We find the first two terms in the expansion and an estimate of the error. The results are extended in a straightforward manner to planar domains and two-dimensional Riemannian manifolds that can be mapped conformally onto the disk. Our results improve the previously de- |Ω| 1 rived expansion for a general smooth domain, Eτ = log + O(1) , Dπ ε (D is the diffusion coefficient) in the case of a circular disk. We find that the mean first passage time from the center of the disk is E[τ | x(0) = 0] = R2 1 1 log + log 2 + + O(ε) . The second term in the expansion is needed D ε 4 in real life applications, such as trafficking of receptors on neuronal spines, 1 because log is not necessarily large, even when ε is small. We also find the ε singular behavior of the probability flux profile into ∂Ωa at the endpoints of ∂Ωa, and find the value of the flux near the center of the window. 6.1 Introduction 109

6.1 Introduction

The expected lifetime of a Brownian motion in a bounded domain, whose boundary is reflecting, except for a small absorbing portion, increases indefinitely as the absorbing part shrinks to zero. The narrow escape problem is to find an asymptotic expansion of the expected lifetime of the Brownian motion in this limit. The narrow escape problem in three dimensions has been studied in the first chapter of this series [177], where is was converted to a mixed Dirichlet-Neumann boundary value problem for the Poisson equation in the domain. This is a well known problem of classical electrostatics (e.g., the electrified disk problem [88]), elasticity (punch problems), diffusion and conductance theory, hydrodynamics, and acoustics [184, 49, 50, 116, 192]. It dates back to Helmholtz [69] and Lord Rayleigh [147] and has been exten- sively studied in the literature for special geometries. The study of the two-dimensional narrow escape problem began in [78] in the context of receptor trafficking on biological membranes [14], where a leading order expansion of the expected lifetime was constructed for a general smooth planar domain. In this chapter we present a thorough analysis of the narrow escape problem for the circular disk and note that our calculations apply in a straightforward manner to any simply connected domain in the plane that can be mapped conformally onto the disk. According to Riemann’s mapping theorem [122], this covers all simply connected planar domains whose boundary contains at least one point. The same conclusion holds for the narrow escape problem on two-dimensional Riemannian manifolds that are conformally equivalent to a circular disk. The biological problem of receptor trafficking on membranes is locally planar, but globally it is a problem on a Riemannian manifold. The narrow escape problem of non-smooth domains that contain corners or cusp points at their boundary is treated in the third part of this series [179], where the conformal mapping method is demonstrated. The specific mathematical problem can be formulated as follows. A Brow- nian particle diffuses freely in a disk Ω, whose boundary ∂Ω is reflecting, except for a small absorbing arc ∂Ωa. The ratio between the arclength of the absorbing boundary and the arclength of the entire boundary is a small parameter |∂Ω | ε = a 1. |∂Ω|

The mean ﬁrst passage time to ∂Ωa, denoted Eτ, becomes inﬁnite as ε → 0. 110 Narrow Escape: The Circular Disk

The asymptotic expansion of Eτ for ε 1 was considered for the particular case when ∂Ωa is a disjoint component of ∂Ω in [144, and references therein]. This case differs from the case at hand in that the absorption probability flux density in the former is regular, while in the latter it is singular. It was shown in [78] that Eτ for the narrow escape problem in a general planar domain Ω has the asymptotic form |Ω| 1 Eτ = log + O(1) , (6.1) Dπ ε where |Ω| is the area of Ω, and D is the diffusion coefficient. This leading order asymptotics has the drawback that log ε can be O(1) when ε 1. Thus the second term in the expansion is needed. For the particular case of a circular disk an approximate value for the correction was given in [78]. In contrast, the asymptotics of Eτ for a three dimensional ball of radius R with an absorbing window of radius εR is [177] |Ω| Eτ = [1 + O(ε log ε)] , 4DεR so the leading order term is much larger than the correction term if ε is small. The difference in the asymptotic form of Eτ stems from the different singularities of the Neumann function in two and three dimensions: it is logarithmic in two dimensions and has a pole in three dimensions. Our computations are based on the mixed boundary value techniques of [184]. They reveal the singularity of the absorption flux in the absorbing 2 2 −1/2 arc ∂Ωa. Specifically, the singularity is (ε − s ) , where s is the (dimensionless) arclength measured from the center of ∂Ωa, and attains the values s = ±ε at the endpoints. The exit time vanishes at the absorbing boundary, and is small near the absorbing boundary, but it attains large and almost constant values of 1 order log inside the domain. We show that this “jump” occurs in a small ε 1 boundary layer of size O ε log . We calculate the average exit time, where ε the averaging is against a uniform initial distribution in the disk, the time to exit from the center, and the maximum mean exit time, attained at the antipodal point to the center of the absorbing window. The mean first passage time (MFPT) from the center of the disk is R2 1 1 E[τ | x(0) = 0] = log + log 2 + + O(ε) , (6.2) D ε 4 6.2 Solution of a Mixed Boundary Value Problem 111 the MFPT, averaged with respect to an initial uniform distribution in the disk is R2 1 1 Eτ = log + log 2 + + O(ε) , (6.3) D ε 8 and the maximal value of the MFPT is attained on the circumference, at the antipodal point to the center of the hole,

R2 1 max E[τ | x] = E[τ | r = 1, θ = 0] = log + 2 log 2 + O(ε) . (6.4) x∈Ω D ε

The boundary layer analysis of Eτ can be applied to the approximation of the first eigenfunction and eigenvalue of the mixed Neumann-Dirichlet boundary value problem with a small Dirichlet window on the boundary. This problem arises in the construction of the first eigenfunction and eigenvalue of the Neumann problem in a domain that consists of two domains (e.g., circular disks) connected by a narrow channel [62], [32]. Specifically, it is easy to see that

∞ X 1 1 Eτ = ∼ (6.5) λ λ n=0 n 0 where 0 < λ0 λ1 < ··· are the eigenvalues of the mixed problem and the MFPT is also averaged with respect to the initial point. The first eigenfunction u0 of the mixed problem is differs from the first eigenfunction of the Neumann problem, which is v0 = 1, only in a boundary layer about the small 2 window. Thus u0 is a small perturbation (in L norm) of v0 = 1. It follow that u0/λ0 differs from Eτ only in the boundary layer.

6.2 Solution of a Mixed Boundary Value Prob- lem

In non-dimensional variables the narrow escape problem concerns Brownian motion inside the unit disk, whose boundary is reﬂecting but for a small absorbing arc of length 2ε (see Fig.6.1). In polar coordinates x = (r, θ) the MFPT v(r, θ) = E[τ | x(0) = (r, θ)], 112 Narrow Escape: The Circular Disk

Figure 6.1: A circular disk of radius R. The arclength of the absorbing boundary (dashed line) is 2εR. The solid line indicates the reﬂecting boundary. 6.2 Solution of a Mixed Boundary Value Problem 113 is the solution to the mixed Neumann-Dirichlet inhomogeneous boundary value problem (see, e.g. [166])

∆v(r, θ) = −1, r < 1, for 0 ≤ θ < 2π,

v(r, θ) = 0, for |θ − π| < ε, r=1

∂v(r, θ) = 0, for |θ − π| > ε, (6.6) ∂r r=1 which is reduced by the substitution

1 − r2 u = v − (6.7) 4 to the mixed Neumann-Dirichlet problem for the Laplace equation

∆u(r, θ) = 0, for r < 1, 0 ≤ θ < 2π,

u(r, θ) = 0, for |θ − π| < ε, (6.8) r=1

∂u(r, θ) 1 = , for |θ − π| > ε. ∂r r=1 2

We adapt the method of [184] to the solution of (6.8). Separation of variables suggests that

∞ a0 X u(r, θ) = + a rn cos nθ, (6.9) 2 n n=1 where the coeﬃcients {an} are to be determined by the boundary conditions

∞ a0 X u(r, θ) = + an cos nθ = 0, for π − ε < θ ≤ π, (6.10) 2 r=1 n=1 ∞ ∂u(r, θ) X 1 = nan cos nθ = , for 0 ≤ θ < π − ε. (6.11) ∂r 2 r=1 n=1 114 Narrow Escape: The Circular Disk

We identify this problem with problem (5.4.4) in [184], where general functions appear on the right hand sides of equations (6.10) (6.11). Due to the invertibility of Abel’s integral operator, the equation

∞ Z π−ε a0 X 1 h1(t) dt + a cos nθ = cos θ √ , (6.12) 2 n 2 n=1 θ cos θ − cos t

deﬁnes h1(t) uniquely for 0 ≤ t < π − ε. The coeﬃcients are given by

Z π−ε Z π−ε 2 1 h1(t) dt an = cos nθ cos θ dθ √ π 0 2 θ cos θ − cos t Z π−ε Z t 1 1 1 cos n + 2 θ + cos n − 2 θ = h1(t) dt √ dθ. (6.13) π 0 0 cos θ − cos t

The integral √ Z u 1 2 cos n + 2 θ Pn(cos u) = √ dθ, (6.14) π 0 cos θ − cos u is Mehler’s integral representation of representation of the Legendre polyno- mial [1]. It follows that

1 Z π−ε an = √ h1(t)[Pn(cos t) + Pn−1(cos t)] dt, (6.15) 2 0 for n > 0, and

Z π−ε Z t 1 √ Z π−ε 2 cos 2 θ a0 = h1(t) dt √ dθ = 2 h1(t) dt. (6.16) π 0 0 cos θ − cos t 0

Integration of (6.11) gives

∞ X 1 a sin nθ = θ, for 0 ≤ θ < π − ε. (6.17) n 2 n=1

Changing the order of summation and integration yields

∞ Z π−ε 1 X 1 h (t)√ [P (cos t) + P (cos t)] sin nθ dt = θ. (6.18) 1 n n−1 2 0 2 n=1 6.2 Solution of a Mixed Boundary Value Problem 115

Using [184, eq.(2.6.31)],

∞ 1 1 X cos 2 θH(θ − t) √ [Pn(cos t) + Pn−1(cos t)] sin nθ = √ , (6.19) 2 n=1 cos t − cos θ we obtain

Z θ h1(t) dt θ √ = 1 , for 0 ≤ θ < π − ε. (6.20) 0 cos t − cos θ 2 cos 2 θ

The solution of the Abel-type integral equation (6.20) is given by

u Z t u sin 1 d 2 h1(t) = √ du. (6.21) π dt 0 cos u − cos t

Together with (6.16) this gives

√ u Z π−ε u sin 2 2 a0 = √ du. (6.22) π 0 cos u + cos ε

We expect the function u(r, θ), closely related to the MFPT, to be almost constant in the disk, except for a boundary layer near the absorbing arc. The value of this constant is a0, because all other terms of expansion (6.9) are oscillatory.

6.2.1 Small ε asymptotics

The results of the previous section are independent of the value of ε. Here we ﬁnd the asymptotic of a0 for ε 1. Substituting

rcos u + cos ε s = (6.23) 2 116 Narrow Escape: The Circular Disk

in the integral (6.22) yields

r ε arccos s2 + sin2 4 Z cos(ε/2) 2 a = ds 0 π r ε 0 s2 + sin2 2 r ε arcsin s2 + sin2 Z cos(ε/2) 1 4 Z cos(ε/2) 2 = 2 ds − ds q 2 ε π r 0 2 0 2 2 ε s + sin 2 s + sin 2 r ε arcsin s2 + sin2 ε ε 4 Z cos(ε/2) 2 = 2 log 1 + cos − 2 log sin − ds 2 2 π r ε 0 s2 + sin2 2 ε 4 Z 1 arcsin s = −2 log + 2 log 2 − ds + O(ε) 2 π 0 s ε = −2 log + O(ε), (6.24) 2

Z 1 arcsin s π because ds = log 2. The substitution (6.23) turns out to be 0 s 2 extremely useful in evaluating the integrals appearing here.

6.2.2 Expected lifetime

Now, that we have the asymptotic expansion of a0 (eq.(6.24)), the evaluation of expected lifetime (MFPT to the absorbing boundary ∂Ωa) becomes possible. Setting r = 0 in equations (6.7) and (6.9), we obtain the expression (6.2) for MFPT from the center of the disk. Averaging (6.8) with respect to a uniform initial distribution in Ω gives

1 Z 2π Z 1 Z 1 1 r3 Eτ = dθ v(r, θ)r dr = a0 + r − dr π 0 0 0 2 2 a 1 ε 1 = 0 + = − log + , (6.25) 2 8 2 8 6.2 Solution of a Mixed Boundary Value Problem 117 as asserted in eq.(6.3). The maximal value of the MFPT is attained at the point r = 1, θ = 0, ∂u which is antipodal to the center of the absorbing arc. At this point = 0, ∂θ as can be seen by diﬀerentiating expansion (6.9) term by term. Setting r = 1 and θ = 0, we ﬁnd that

∞ a0 X v = u(1, 0) = + a . (6.26) max 2 n n=1 The evaluation of the maximal exit time is not as straightforward as the previous evaluated MFPTs, because one needs to calculate the inﬁnite sum in (6.26). This calculation is done in Appendix C.1, where we ﬁnd (eq.(C.12)) 1 v = log + 2 log 2 + O(ε), max ε as asserted in equation (6.4).

6.2.3 Boundary layers 1 We see that the maximal exit time is only v − v = log 2 − = max center 4 .4431471806 ... longer than its value at the center of the disk. In other words, the variance along the radius θ = 0, 0 ≤ r ≤ 1 is very small. However, in the opposite direction θ = π, 0 ≤ r ≤ 1, we expect a much diﬀerent behavior. 1 In particular, the MFPT is decreasing from a value of v ≈ log at the center ε center of the disk to v(1, π) = 0 at the center of ∂Ωa. The calculation of the exit time

2 ∞ 1 − r a0 X v (r) ≡ v(r, θ = π) = + + a (−r)n, (6.27) ray 4 2 n n=1 is similar to that of√ the maximal exit time and is done in Appendix C.2. For ε 1 and 1 − r ε, we ﬁnd the asymptotic form (eq.(C.20)) ε 1 − r2 v (r) = − log + 2 log(1 − r) + − log(1 + r2) + q(r) + O(ε), (6.28) ray 2 4 where q(r) is a smooth function in the interval [0, 1] (eqs.(C.18)-(C.19)). Clearly, this asymptotic expansion does not hold all the way through to the 118 Narrow Escape: The Circular Disk

absorbing arc at r = 1, where the boundary condition requires vray(r = 1) = rε 0. Instead, the boundary condition is almost satisﬁed at r = 1 − 2

rε ε rε v 1 − = − log + 2 log + O(ε) = O(ε), (6.29) ray 2 2 2

In other words, the asymptotic series (6.28) is the outer expansion [10]. √ We proceed to construct the boundary layer for 1 − r ε. Setting δ = 1 − r, we have the identities

1 − r2 1 1 = δ − δ2, 4 2 4 ε 1 − 2r cos ε + r2 = 4 sin2 (1 − δ) + δ2. 2 The exact form of the MFPT along the ray, eq.(C.15), gives the expansion

δ a0δ vray(δ) = + ε (6.30) 2 4 sin 2 r ε arccos s2 + sin2 s2 ds δ Z cos(ε/2) 2 δ2 − + O . ε ε3/2 ε π sin 0 s2 + sin2 2 2 Evaluating the integral in eq.(6.30),

r ε arccos s2 + sin2 s2 ds Z cos(ε/2) 2 ε3/2 0 s2 + sin2 2 π h ε ε ε i = − log sin + cos − log 1 + cos + log 2 + O(ε), (6.31) 2 2 2 2 we obtain the boundary layer structure

δ δ2 v (δ) = + O δ, . (6.32) ray ε ε 6.2 Solution of a Mixed Boundary Value Problem 119

ε In particular, setting δ = −ε log yields 0 2 ε v (δ ) = − log + O(ε log2 ε), (6.33) ray 0 2 which is the value of the outer solution. We conclude that the width of the 1 boundary layer is O ε log . Furthermore, the ﬂux at the center of the ε hole is given by

∂vray ∂vray 1 fluxcenter = = − = − + O(1). (6.34) ∂r r=1 ∂δ δ=0 ε 6.2.4 Flux profile Next, we calculate the profile of the flux on the absorbing arc. Differentiating expansion (6.9) gives the flux as ∞ ∂v(r, θ) ∂u(r, θ) 1 1 X f(θ) = = − = − + nan cos nθ, (6.35) ∂r ∂r 2 2 r=1 r=1 n=1 for π − ε < θ ≤ π. Using equation (6.15) for the coefficients, we have

∞ 1 1 Z π−ε X f(θ) = − + √ h (t) dt n[P (cos t) + P (cos t)] cos nθ 2 1 n n−1 2 0 n=1 ∞ 1 1 d Z π−ε X = − + √ h (t) dt [P (cos t) + P (cos t)] sin nθ. 2 dθ 1 n n−1 2 0 n=1 Since θiπ − εit, equation (6.19) implies 1 d θ Z π−ε h (t) dt f(θ) = − + cos √ 1 . (6.36) 2 dθ 2 0 cos t − cos θ The evaluation of this integral is not immediate and is given in Appendix C.3. We ﬁnd that (eq.(C.37))

∞ 2 ! α2 1 X (2n+1(n + 1)!) (2nn!)2 f(α) = − √ − α2 − (1 − α2)n+1/2 2 ε (2n + 2)! (2n + 1)! ε 1 − α n=0 ∞ π X (2n)! (2n + 2)!(2n + 2) − − α2 (1 − α2)n + O(1), (6.37) 2ε (2nn!)2 (2n+1(n + 1)!)2 n=0 120 Narrow Escape: The Circular Disk

π − θ where α = , |α| < 1. The flux has a singular part, represented by the ε half-integer powers of (1 − α2), and a remaining regular part (the integer α2 powers.) The first term, − √ , is the most singular one, because it ε 1 − α2 becomes infinite as |α| → 1. In other words, the flux is infinitely large near the boundary of the hole. The splitting of the solution into singular and regular parts is common in the theory of elliptic boundary value problems in domains with corners (see e.g., [105, 106, 33]). The value of the flux at the center of the hole is to leading order

∞ 1 X π (2n)! (2nn!)2 1 f(0) = − − = − , (6.38) ε 2 (2nn!)2 (2n + 1)! ε n=0 in agreement with (6.34) (thanks Maple for calculating the infinite sum.) The size of the boundary layer is varying with θ proportionally to 1/f(θ). The singularity at the end points of the hole indicate that the layer shrinks there to zero. Therefore, the boundary layer is shaped√ as a small cap bounded by the absorbing arc and (more or less) the curve 1 − α2 (see Fig.6.2). In 1 particular, the MFPT on the reflecting boundary is O log , even when ε taken arbitrarily close to the absorbing boundary. The singularity of the flux near the endpoints indicates that the diffusive particle prefers to exit near the endpoints rather than through the center of the hole. The expansion (C.37) is useful in approximating the flux near the endpoints (α = ±1), where few terms are needed. However, it is slowly converg- ing near the center of the hole, where a power series in α2 should be used instead ∞ X 2n f(α) = fnα + O(1), (6.39) n=0 −1 where the coefficients fn are O(ε ). Equations (6.38) and (6.34) indicate 1 that f = − . All other coefficients can be found in a similar fashion. We 0 ε conclude that near the center (α 1) we have

1 α2 f(α) = − + O 1, . (6.40) ε ε 6.2 Solution of a Mixed Boundary Value Problem 121

Figure 6.2: The boundary layer, indicated by “BL”, is the area bounded by the absorbing boundary (dashed line) and the solid arc. Outside the 1 boundary layer the MFPT is O log . ε Chapter 7

Narrow Escape: Riemann Surfaces and Non-Smooth Domains

The contents of this chapter were submitted for publication [179]

We consider Brownian motion in a bounded domain Ω on a two-dimensional Riemannian manifold (Σ, g). We assume that the boundary ∂Ω is smooth and reflects the trajectories, except for a small absorbing arc ∂Ωa ⊂ ∂Ω. As ∂Ωa is shrunk to zero the expected time to absorption in ∂Ωa becomes infinite. The narrow escape problem consists in constructing an asymptotic expansion of the expected lifetime, denoted Eτ, as ε = |∂Ωa|g/|∂Ω|g → 0. We |Ω| 1 derive a leading order asymptotic approximation Eτ = g log + O(1) . Dπ ε The order 1 term can be evaluated for simply connected domains on a sphere by projecting stereographically on the complex plane and mapping conformally on a circular disk. It can also be evaluated for domains that can be mapped conformally onto an annulus. This term is needed in real life ap- 1 plications, such as trafficking of receptors on neuronal spines, because log ε is not necessarily large, even when ε is small. If the absorbing window is |Ω| 1 located at a corner of angle α, then Eτ = g log + O(1) , if near a Dα ε cusp, then Eτ grows algebraically, rather than logarithmically. Thus, in the domain bounded between two tangent circles, the expected lifetime is 7.1 Introduction 123

|Ω| 1 Eτ = + O(1) . (d−1 − 1)D ε

7.1 Introduction

In many applications it is necessary to find the mean first passage time (MFPT) of a Brownian particle to a small absorbing window in the otherwise reflecting boundary of a given bounded domain. This is the case, for example, in the permeation of ions through protein channels of cell membranes [73], and in the trafficking of AMPA receptors on nerve cell membranes [78], [14]. While the first example is three dimensional the second is two dimensional, which leads to very different results. In this chapter we consider the two dimensional case. In the first two parts of this series of chapters, we considered the narrow escape problem in three dimensions [177] and in the planar circular disk [178]. The leading order asymptotic behavior of the MFPT is different in the three and two dimensional cases; it is proportional to the relative size of the reflecting and absorbing boundaries in three dimensions, but in two dimensions it is proportional to the logarithm of this quotient. The difference in the orders of magnitude is the result of the different singularities of Neumann’s function for Laplace’s equation in the two cases. While the second term in the asymptotic expansion of the MFPT in three dimensions is much smaller then the first one, it is not necessarily so in two dimensions, because of the slow growth of the logarithmic function. It is necessary, therefore, to find the second term in the expansion in the two-dimensional case. This term was found for the case of a planar circular disk in [178], and can therefore be found for all simply connected domains in the plane that can be mapped conformally onto the disk. Similarly, it can be found for simply connected domains on two-dimensional Riemannian manifolds that can be mapped conformally on the planar disk. For example, the sphere with a circular cap cut off can be projected stereographically onto the disk, and so the second term for the narrow escape problem for such domains can be found. The specific mathematical problem can be formulated as follows. A Brow- nian particle diffuses freely in a bounded domain Ω on a two-dimensional Riemannian manifold (Σ, g). The boundary ∂Ω is reflecting, except for a small absorbing arc ∂Ωa. The ratio between the arclength of the absorbing 124 Narrow Escape: Riemann Surfaces and Non-Smooth Domains

boundary and the arclength of the entire boundary is a small parameter, |∂Ω | ε = a g 1. |∂Ω|g

The MFPT to ∂Ωa, denoted Eτ, becomes infinite as ε → 0. In this chapter we calculate the first term in the asymptotic expansion of Eτ for a general smooth bounded domain on a general two-dimensional Riemannian manifold. We find the second term for an annulus of two concentric circles, with a small hole located on its inner boundary. This result is generalized in a straightforward manner to domains that are conformally equivalent to the annulus. The calculation of the second term involves the solution of the mixed Dirichlet-Neumann problem for harmonic functions in Ω. While in the three dimensional case this is a classical problem in mechanics, diffusion, elasticity theory, hydrodynamics, and electrostatics [184, 49, 50], the two dimensional problem did not draw as much attention in the literature. First, we consider the problem of narrow escape on two dimensional manifolds, and derive the leading order asymptotic approximation |Ω| 1 Eτ = g log + O(1) for ε 1. (7.1) Dπ ε This generalizes the result of [78] from general smooth planar domains to general domains on general smooth two-dimensional Riemannian manifolds. The second term in the asymptotic expansion is found for the 2-sphere x2 + y2 + z2 = R2. The calculation is made possible by the stereographic projection that maps the Riemann sphere onto a circular disk, a problem that was solved in [178]. The boundary in this case is a spherical cap of central angle δ at the north pole, where ε is the ratio between the absorbing arc and the entire boundary circle. We find that the MFPT, averaged with respect to an initial uniform distribution, is given by |Ω| 1 1 1 Eτ = g log + 2 log + 3 log 2 − + O(ε, δ2 log δ, δ2 log ε) , (7.2) 2πD δ ε 2

2 where |Ω|g = 4πR is the surface area of the sphere. Note that there are two small parameters that control the behavior of the MFPT in this problem. The small ε contributes as equation (7.1) predicts, whereas the small δ parameter contributes half as much. 7.1 Introduction 125

The second case that we consider is that of narrow escape from an annulus, whose boundary is reflecting, except for a small absorbing arc on the inner circle. Specifically, the annulus is the domain R1 < r < R2, with all reflecting boundaries except for a small absorbing window located at the inner circle (see Fig. 7.1). The inversion w = 1/z transforms this case into that of R the absorbing boundary on the outer circle. Setting β = 1 < 1, the MFPT, R2 averaged with respect to a uniform initial distribution, can be written as

Eτ =

1 1 R2 1 1 (R2 − R2) log + log 2 + 2β2 + 2 log − R2 + O(ε, β4)R2. 2 1 ε 2 1 − β2 β 4 2 2

Also in this case we find two small parameters, the ε contribution belongs to a singular perturbation problem with a boundary layer solution and an almost constant outer solution with singular fluxes near the edges of the window, whereas the δ contribution is just the singularity of Green’s function at the origin–a problem with a regular flux. This result is generalized to a sphere with two antipodal circular caps removed. We find that for β 1 the maximum exit time is attained near the south pole, as expected. This result can be generalized to manifolds that can be mapped conformally onto the said domain. The asymptotic expansion of the MFPT to a non-smooth part of the boundary is different. We consider two types of singular boundary points: corners and cusps. If the absorbing arc is located at a corner of angle α, the MFPT is |Ω| 1 Eτ = g log + O(1) . (7.3) Dα ε For example, the MFPT from a rectangle with sides a and b to an absorbing window of size ε at the corner (α = π/2, see Figure 7.2), is

2|Ω| a 2 π b ε Eτ = log + log + + 2β2 + O , β4 , π ε π 6 a a where |Ω| = ab and β = e−πb/a. The calculation of the second order term turns out to be similar to that in the annulus case. The pre-logarithmic |Ω| factor g is the result of the diﬀerent singularity of the Neumann function Dα 126 Narrow Escape: Riemann Surfaces and Non-Smooth Domains

at the corner. It can be obtained by either the method of images, or by the conformal mapping z 7→ zπ/α that flattens the corner. In the vicinity of a cusp α → 0, therefore the asymptotic expansion (7.3) is invalid. We 1 find that near a cusp the MFPT grows algebraically fast as , where λ ελ is the order of the cusp. Note that the MFPT grows faster to infinity as the boundary is more singular. The change of behavior from a logarithmic growth to an algebraic one expresses the fact that entering a cusp is a rare Brownian event. For example, the MFPT from the domain bounded between two tangent circles to a small arc at the common point (see Figure 7.4) is |Ω| 1 Eτ = + O(1) , where d < 1 is the ratio of the radii. This (d−1 − 1)D ε result is obtained by mapping the cusped domain conformally onto the upper half plane. The singularity of the Neumann function is transformed as well. The leading order term of the asymptotics can be found for any domain that can be mapped conformally to the upper half plane. In three dimensions the class of isolated singularities of the boundary is much richer than in the plane. The results of [177] cannot be generalized in a straightforward way to windows located near a singular point or arc of the boundary. We postpone the investigation of the MFPT to windows at isolated singular points in three dimensions to a future paper. As a possible application of the present results, we mention the calculation of the diffusion coefficient from the statistics of the lifetime of a receptor in a corral on the surface of a neuronal spine [14].

7.2 Asymptotic Approximation to the MFPT on a Riemannian Manifold

We denote by x(t) the trajectory of a Brownian motion in a bounded domain Ω on a two-dimensional Riemannian manifold (Σ, g). For a domain Ω ⊂ Σ 1 with a smooth boundary ∂Ω (at least C ), we denote by |Ω|g the Riemannian surface area of Ω and by |∂Ω|g the arclength of its boundary, computed with respect to the metric g. The boundary ∂Ω is partitioned into an absorbing arc ∂Ωa and the remaining part ∂Ω − ∂Ωa is reflecting for the Brownian trajectories. We assume that the absorbing part is small, that is, |∂Ω | ε = a g 1, |∂Ω|g 7.2 Asymptotic Approximation to the MFPT on a Riemannian Manifold 127 however, Σ and Ω are independent of ε; only the partition of the boundary ∂Ω into absorbing and reflecting parts varies with ε. The first passage time τ of the Brownian motion from Ω to ∂Ωa has a finite mean and we define

u(x) = E[τ | x(0) = x].

The function u(x) satisﬁes the mixed Neumann-Dirichlet boundary value problem (see for example [127], [166])

D∆gu(x) = −1 for x ∈ Ω (7.4)

∂u(x) = 0 for x ∈ ∂Ω − ∂Ω (7.5) ∂n a

u(x) = 0 for x ∈ ∂Ωa, (7.6) where ∆g is the Laplace-Beltrami operator on Σ and D is the diﬀusion co- eﬃcient. Obviously, u(x) → ∞ as ε → 0, except for x in a boundary layer near ∂Ωa.

7.2.1 Expression of the MFPT using the Neumann function We consider the Neumann function defined on Σ by 1 ∆gN(x, y) = −δ(x − y) + , for x, y ∈ Ω (7.7) |Ω|g ∂N(x, y) = 0, for x ∈ ∂Ω, y ∈ Ω. ∂n The Neumann function N(x, y) is defined up to an additive constant and is symmetric [55]. The Neumann function exists for the domain Ω, because the compatibility condition is satisfied (i.e., both sides of eq.(7.7) integrate to 0 over Ω due to the boundary condition). The Neumann function N(x, y) is constructed by using a parametrix H(x, y) [6] ,

h(d(x, y)) H(x, y) = − log d(x, y), (7.8) 2π 128 Narrow Escape: Riemann Surfaces and Non-Smooth Domains

where d(x, y) is the Riemannian distance between x and y and h(·) is a regular function with compact support, equal to 1 in a neighborhood of y. As a consequence of the construction N(x, y) − H(x, y) is a regular function on Ω. To derive an integral representation of the solution u, we multiply eq.(7.4) by N(x, y), eq.(7.7) by u(x), integrate with respect to x over Ω, and use Green’s formula to obtain the identity I ∂u(x(S)) 1 Z N(x(S), ξ) dSg = − u(x) dVg + u(ξ) ∂Ω ∂n |Ω|g Ω Z − N(x, ξ) dVg. (7.9) Ω The integral 1 Z Cε = u(x) dVg (7.10) |Ω|g Ω is an additive constant and the ﬂux on the reﬂecting boundary vanishes, so we rewrite eq.(7.9) as Z Z ∂u(x(S)) u(ξ) = Cε + N(x, ξ) dVg + N(x(S), ξ) dSg, (7.11) Ω ∂Ωa ∂n

where S is the coordinate of a point on ∂Ωa, and dSg is arclength element on ∂Ωa associated with the metric g. Setting ∂u(x(S)) f(S) = , ∂n

and choosing ξ ∈ ∂Ωa in eq.(7.11), we obtain Z Z 0 = Cε + N(x, ξ) dVg + N(x(S), ξ)f(S) dSg. (7.12) Ω ∂Ωa The ﬁrst integral in eq.(7.11) is a constant (independent of ε), because due to the symmetry of N(x, y) eq.(7.7) gives the boundary value problem Z ∆ξ N(x, ξ) dVg = 0 for ξ ∈ Ω Ω ∂ Z N(x, ξ) dVg = 0 for ξ ∈ ∂Ω, ∂n(ξ) Ω 7.2 Asymptotic Approximation to the MFPT on a Riemannian Manifold 129

whose solution is any constant. Changing the deﬁnition of the constant Cε, equation (7.11) can be written as, Z u(ξ) = N(x(S), ξ)f(S) dSg + Cε, (7.13) ∂Ωa and both f(S) and Cε are determined by the absorbing condition (7.6) Z 0 = N(x(S), ξ)f(S) dSg + Cε for ξ ∈ ∂Ωa. (7.14) ∂Ωa

2 Equation (7.14) has been considered in [78] for a domain Σ ⊂ R as an integral equation for f(S) and Cε. Actually, the boundary coordinate S can be chosen as arclength on ∂Ωa, denoted s. Under the regularity assumptions of the boundary, the normal derivative f(s) is a regular function, but develops a singularity as ξ(s) approaches the corner boundary of ∂Ωa in ∂Ω [105]. Both can be determined from the representation (7.13), if all functions in eq.(7.14) and the boundary are analytic. In that case the solution has a series expansion in powers of arclength on Ωa. The method to compute Cε follows the same step as in [78].

7.2.2 Leading order asymptotics Under our assumptions, u(ξ) → ∞ as ε → 0 for any ﬁxed ξ ∈ Ω, so that eq.(7.10) implies that Cε → ∞ as well. It follows from eq.(7.14) that the integral in (7.14) decreases to −∞. An origin 0 ∈ ∂Ωa is ﬁxed and the boundary ∂Ω is parameterized by 1 1 (x(s), y(s)). We rescale s so that ∂Ω = (x(s), y(s)) : − 2 < s ≤ 2 and 1 1 1 1 x − 2 , y − 2 = x 2 , y 2 . We assume that the functions x(s) and y(s) are real analytic in the interval 2|s| < 1 and that the absorbing part of the boundary ∂Ωa is the arc

∂Ωa = {(x(s), y(s)) : |s| < ε} . The Neumann function can be written as 1 N(x, ξ) = − log d(x, ξ) + v (x, ξ), for x ∈ B (ξ), (7.15) 2π N δ where Bδ(ξ) is a geodesic ball of radius δ centered at ξ and vN (x; ξ) is a regular function. We consider a normal geodesic coordinate system (x, y) 130 Narrow Escape: Riemann Surfaces and Non-Smooth Domains

at the origin, such that one of the coordinates coincides with the tangent coordinate to ∂Ωa. We choose unit vectors e1, e2 as an orthogonal basis in the tangent plane at 0 so that for any vector ﬁeld X = x1e1 + x2e2, the metric tensor g can be written as

2 X kl 2 gij = δij + ε aij xkxl + o(ε ), (7.16) kl

where |xk| ≤ 1, because ε is small. It follows that for x, y inside the geodesic 2 ball or radius ε, centered at the origin, d(x, y) = dE(x, y) + O(ε ), where dE is the Euclidean metric. We can now use the computation given in the Euclidean case in [78]. To estimate the solution of equation (7.14), we recall that when both x and ξ are on the boundary, vN (x, ξ) becomes singular (see [55, p.247, eq.(7.46)]) and the singular part gains a factor of 2, due to the singularity of the “image charge”. Denoting byv ˜N the new regular part, equation (7.14) becomes

Z 0 0 log d(x(s), ξ(s )) 0 0 v˜N (x(s ); ξ(s)) − f(s ) S(ds ) = Cε, (7.17) |s0|<ε π

where S(ds0) is the induced measure element on the boundary, and x = (x(s), y(s)), ξ = (ξ(s), η(s)). Now, we expand the integral in eq.(7.17), as in [78],

log d(x(s), ξ(s0)) = log p(x(s0) − ξ(s))2 + (y(s0) − η(s))2 1 + O(ε2)

and

∞ ∞ X j 0 0 X 0 j 0 S(ds)f(s) = fjs ds, v˜N (x(s ); ξ(s))S(ds ) = vj(s )s ds (7.18) j=0 j=0

0 for |s| < ε, where vj(s ) are known coeﬃcients and fj are unknown coef- ﬁcients, to be determined from eq.(7.17). To expand the logarithmic term in the last integral in eq.(7.17), we recall that x(s0), y(s0), ξ(s), and η(s) are analytic functions of their arguments in the intervals |s| < ε and |s0| < ε, 7.2 Asymptotic Approximation to the MFPT on a Riemannian Manifold 131 respectively. Therefore

Z ε (s0)n log d(x(s), ξ(s0)) ds0 = (7.19) −ε Z ε (s0)n log p(x(s0) − ξ(s))2 + (y(s0) − η(s))2 1 + O(ε2) ds0 = −ε Z ε (s0)n log |s0 − s| 1 + O (s0 − s)2 1 + O(ε2) ds0. −ε

We keep in Taylor’s expansion of log {|s0 − s| (1 + O ((s0 − s)2))} only the leading term, because higher order terms contribute positive powers of ε to the series

∞ Z ε X 1 s2j log(s − s0)2 ds0 = 4ε (log ε − 1) + 2 . (7.20) (2j − 1)j ε2j−1 −ε j=1

For even n ≥ 0, we have

Z ε (s0)n log(s − s0)2 ds0 = −ε ∞ εn+1 εn+1 X εn−2j+1 4 log ε − − 2 s2j , (7.21) n + 1 (n + 1)2 j(n − 2j + 1) j=1 whereas for odd n, we have

∞ Z ε X s2j+1 εn−2j (s0)n log(s − s0)2 ds0 = −4 . (7.22) 2j + 1 n − 2j −ε j=1

Using the above expansion, we rewrite eq.(7.17) as

( ∞ ) Z ε −1 X 0 = log |s0 − s|2 1 + O (s0 − s)2 (1 + O(ε2)) + v (s0)sj × π j −ε j=0 ∞ X 0j 0 fjs ds + Cε, j=0 132 Narrow Escape: Riemann Surfaces and Non-Smooth Domains

and expand in powers of s. At the leading order, we obtain

X ε2p+1 ε2p+1 ε (log ε − 1) f + log ε − f = 0 2p + 1 (2p + 1)2 2p p Z ε π 0 0 v0(s ) ds + Cε. (7.23) 2 −ε

Equation (7.23) and

1 Z ε X ε2p+1 f(s) S(ds) = f 2 (2p + 1) 2p −ε p

determine the leading order term in the expansion of Cε. Indeed, integrating eq.(7.4) over the domain Ω, we see that the compatibility condition gives

Z ε f(s) S(ds) = −|Ω|g, (7.24) −ε

Z ε 0 0 and using the fact that v0(s )S(ds ) = O(ε), we ﬁnd that the leading −ε order expansion of Cε in eq.(7.23) is

|Ω| 1 C = g log + O(1) for ε 1. (7.25) ε π ε

If the diﬀusion coeﬃcient is D, eq.(7.13) gives the MFPT from a point x ∈ Ω, outside the boundary layer, as

|Ω| 1 E[τ | x] = u(x) = g log + O(1) for ε 1. (7.26) πD ε

7.3 The Annulus Problem

We consider a Brownian particle that is conﬁned in the annulus R1 < r < R2. The particle can exit the annulus through a narrow opening of the inner circle (see Fig.7.1). The MFPT v(x) satisﬁes 7.3 The Annulus Problem 133

Figure 7.1: An annulus R1 < r < R2. The particle is absorbed at an arc of length 2εR1 (dashed line) at the inner circle. The solid lines indicate reﬂecting boundaries. 134 Narrow Escape: Riemann Surfaces and Non-Smooth Domains

∆v = −1, for R1 < r < R2, (7.27)

∂v = 0, for r = R , ∂r 2 ∂v = 0, for r = R , |θ − π| > ε, ∂r 1

v = 0, for r = R1, |θ − π| < ε. R2 − r2 The function w = 1 is a solution of the Dirichlet problem for eq.(7.27) 4 in the exterior domain of the inner circle r > R1. More speciﬁcally, it satisﬁes the boundary value problem

∆w = −1, for R1 < r < R2,

∂w 1 = − R , for r = R , ∂r 2 1 1 ∂w 1 = − R , for r = R , ∂r 2 2 2

w = 0, for r = R1. (7.28) The function u = v − w satisﬁes

∆u = 0, for R1 < r < R2,

∂u 1 = R , for r = R , ∂r 2 2 2 ∂u 1 = R , for r = R , |θ − π| > ε, ∂r 2 1 1

u = 0, for r = R1, |θ − π| < ε. (7.29) Separation of variables produces the solution ∞ n n a0 X r R2 r u(r, θ) = + a + b cos nθ + α log , (7.30) 2 n R n r R n=1 2 1 7.3 The Annulus Problem 135

where an, bn and α are to be determined by the boundary conditions. Dif- ferentiating with respect to r yields ∞ " n−1 n−1# ∂u X an r bnR2 R2 α = n − cos nθ + . (7.31) ∂r R R r2 r r n=1 2 2

Setting r = R2 gives " ∞ # 1 1 X R = n (a − b ) cos nθ + α , (7.32) 2 2 R n n 2 n=1 1 therefore, a = b and α = R2, and we have n n 2 2 ∞ n n a0 X r R2 1 r u(r, θ) = + a + cos nθ + R2 log . (7.33) 2 n R r 2 2 R n=1 2 1

The boundary conditions at r = R1 become the dual series equations ∞ n n a0 X R2 R1 + a + cos nθ = 0, for |θ − π| < ε, 2 n R R n=1 1 2 ∞ " n+1 n−1# X R2 R1 R2 na − cos nθ = R2 − R2 , for |θ − π| > ε. n R R 2R 2 1 n=1 1 2 1 Setting " n+1 n−1# R1 R2 R1 cn = − an, for n ≥ 1, (7.34) R2 R1 R2 and c0 = a0 converts the dual series equations to ∞ c0 X cn + cos nθ = 0, for π − ε < θ < π, (7.35) 2 1 + H n=1 n ∞ X 1 nc cos nθ = (R2 − R2), for 0 < θ < π − ε, (7.36) n 2 2 1 n=1 2n 2β R1 where Hn = − 2n for n ≥ 1, and H0 = 0, with β = < 1. Note that 1 + β R2 2n Hn = O(β ) which tends to zero exponentially fast (much faster than the n−1 decay required for the Collins method [26, 27], see also [177]). 136 Narrow Escape: Riemann Surfaces and Non-Smooth Domains

The case Hn ≡ 0 was solved in [178]. We now try to ﬁnd the correction of that result due to the non vanishing Hn. As in [178] the equation ∞ c0 X cn + cos nθ = 2 1 + H n=1 n θ Z π−ε h (t) dt cos √ 1 for 0 < θ < π − ε 2 θ cos θ − cos t

deﬁnes the function h1(θ) uniquely for 0 < θ < π − ε, the coeﬃcients are given by Z π−ε 1 + Hn cn = √ h1(t)[Pn(cos t) + Pn−1(cos t)] dt, (7.37) 2 0 and √ Z π−ε c0 = 2 h1(t) dt. (7.38) 0 Integrating equation (7.36) gives

∞ X 1 c sin nθ = R2 − R2 θ, for 0 < θ < π − ε. (7.39) n 2 2 1 n=1 Substituting eq.(7.39) in equation (7.37), changing the order of summation and integration, while using [184, eq.(2.6.31)],

∞ 1 1 X cos 2 θH(θ − t) √ [Pn(cos t) + Pn−1(cos t)] sin nθ = √ , (7.40) 2 n=1 cos t − cos θ we obtain for 0 < θ < π − ε, Z θ h (t) Z π−ε (R2 − R2) θ √ 1 dt + K (θ, t)h (t) dt = 2 1 , (7.41) β 1 θ 0 cos t − cos θ 0 2 cos 2

where the kernel Kβ is ∞ 1 X K (θ, t) = H (P (cos t) + P (cos t)) sin nθ β √ θ n n n−1 2 cos n=1 2 √ θ = −2 2(1 + cos t) sin β2 + O(β4). (7.42) 2 7.3 The Annulus Problem 137

The infinite sum in eq.(7.42) is approximated by its first term, while using the first two Legendre polynomials P0(x) = 1,P1(x) = x. Using Abel’s inversion formula applied to equation (7.41), we find that u Z π−ε 2 2 Z t u sin ˜ R2 − R1 d 2 h1(t) − Kβ(t, s)h1(s) ds = √ du, (7.43) 0 π dt 0 cos u − cos t ˜ where the kernel Kβ is Z t ˜ 1 d Kβ(u, s) sin u Kβ(t, s) = − √ du (7.44) π dt 0 cos u − cos t √ u 2 2(1 + cos s) d Z t sin sin u = β2 √ 2 du + O(β4). π dt 0 cos u − cos t The substitution rcos u − cos t s = (7.45) 2 gives u Z t sin sin u π t √ 2 du = √ sin2 , (7.46) 0 cos u − cos t 2 2 therefore, s K˜ (t, s) = 2β2 cos2 sin t + O(β4). (7.47) β 2 Equation (7.43) is a Fredholm integral equation of the second kind for h1, of the form ˜ (I − Kβ)h = z, (7.48) where u R2 − R2 d Z t u sin z(t) = 2 1 √ 2 du. π dt 0 cos u − cos t Therefore, we h can be expanded as ˜ ˜ 2 h = z + Kβz + Kβz + ..., (7.49) √ 2 which converges in L . Since c0 = 2hh, 1i (eq.(7.38)), we find an asymptotic expansion of the form √ h ˜ i c0 = 2 hz, 1i + hKβz, 1i + ... . (7.50) 138 Narrow Escape: Riemann Surfaces and Non-Smooth Domains

The leading order term of this expansion was calculated in [178]. We now ˜ 2 estimate the error term hKβz, 1i, which is also the O(β ) correction. Inte- grating by parts and changing the order of integration yields u 2 2 Z π Z s u sin ˜ 2 R2 − R1 2 s d 2 Kβz(t) = 2β sin t cos ds √ du π 0 2 ds 0 cos u − cos s u R2 − R2 Z π Z s u sin du = β2 2 1 sin t sin s ds √ 2 π 0 0 cos u − cos s √ 2 2 2 = 2β (R2 − R1) sin t. (7.51) Therefore, √ Z π √ ˜ 2 2 2 2 2 2 hKβz, 1i = 2β (R2 − R1) sin t dt = 2 2β (R2 − R1). (7.52) 0 We conclude that 1 c = (R2 − R2) 2 log + 2 log 2 + 4β2 + O(ε, β4) . (7.53) 0 2 1 ε The MFPT averaged with respect to a uniform initial distribution is

4 c0 1 R2 R2 1 2 Eτ = + 2 2 log − R2 (7.54) 2 2 R2 − R1 R1 4 1 = (R2 − R2) log + log 2 + 2β2 2 1 ε 1 R2 1 1 + 2 log − R2 + O(ε, β4)R2. 2 1 − β2 β 4 2 2 Note that there are two diﬀerent logarithmic contributions to the MFPT. The “narrow escape” small parameter ε contributes |Ω| 1 g log , (7.55) π ε as expected from the general theory (equation (7.1)), whereas the parameter β contributes |Ω| 1 g log . (7.56) 2π β 7.4 Domains with Corners 139

These asymptotics differ by a factor 2, because they account for different singular behaviors. The asymptotic expansion (7.55) comes out from a singular perturbation problem with singular flux near the edges, boundary layer and an outer solution, whereas the asymptotics (7.56) is an immediate result of the singularity of the Neumann function, with a regular flux. The maximum exit time is attained at the antipode point of the center of the hole at the outer circle. Indeed, eq. (7.33) indicates that the maximum is attained for θ = 0. To determine its location along the cord R1 < r < R2, we note that Hn → 0 as β → 0. Therefore, in this limit, eqs. (7.35)-(7.36) are equivalent to the circular disk problem. The logarithmic term of eq. (7.33) 1 is monotonic increasing with r, and is O log for r = R . Therefore, for β 2 β 1 the maximum is attained at the outer circle r = R2, which is also the farthest point from the hole. Note that (7.54) is valid, with the obvious modifications, for any domain that is conformally equivalent to the annulus.

7.4 Domains with Corners

Consider a Brownian motion in a rectangle Ω = (0, a)×(0, b) of area ab. The boundary is reﬂecting except the small absorbing segment ∂Ωa = [a − ε, a] × {b} (see Fig. 7.2). The MFPT v(x, y) satisﬁes the boundary value problem

∆v = −1, (x, y) ∈ Ω,

v = 0, (x, y) ∈ ∂Ωa, ∂v = 0, (x, y) ∈ ∂Ω − ∂Ω . (7.57) ∂n a

b2 − y2 The function f = satisﬁes 2

∆f = −1, (x, y) ∈ Ω,

f = 0, (x, y) ∈ ∂Ωa, ∂f = 0, (x, y) ∈ {0} × [0, b] ∪ {a} × [0, b] ∪ [0, a] × {0}, ∂n ∂f = −b, (x, y) ∈ [0, a − ε] × {b}, (7.58) ∂n 140 Narrow Escape: Riemann Surfaces and Non-Smooth Domains

Figure 7.2: Rectangle of sizes a and b with a small absorbing segment of size ε at the corner. 7.4 Domains with Corners 141 therefore, the function u = v − f satisﬁes ∆u = 0, (x, y) ∈ Ω,

u = 0, (x, y) ∈ ∂Ωa, ∂u = 0, (x, y) ∈ {0} × [0, b] ∪ {a} × [0, b] ∪ [0, a] × {0}, ∂n ∂u = b, (x, y) ∈ [0, a − ε] × {b}. (7.59) ∂n A solution for u in the form of separation of variables is ∞ a0 X πny πnx u(x, y) = + a cosh cos , (7.60) 2 n a a n=1 where the coeﬃcients an are to be determined by the boundary conditions at y = b ∞ a0 X πnb πnx u(x, b) = + a cosh cos = 0, x ∈ (a − ε, a), 2 n a a n=1 ∞ ∂u π X πnb πnx (x, b) = na sinh cos = b, x ∈ (0, a − ε). (7.61) ∂y a n a a n=1 πnb Setting c = a sinh , we have n n a ∞ c0 X cn + cos nθ = 0, π − δ < θ < π, 2 1 + H n=1 n ∞ X ab nc cos nθ = , 0 < θ < π − δ, (7.62) n π n=1 πε πnb where δ = and H = tanh − 1, n ≥ 1. Note that H = O (β2n) a n a n πb for β = exp − < 1. The rectangle problem and annulus problem (eq. a (7.36)) are almost mathematically equivalent, and equation (7.53) gives the value of c0 2ab 1 c = 2 log + 2 log 2 + 4β2 + O(δ, β4) 0 π δ 4ab a 2 ε = log + log + 2β2 + O , β4 . (7.63) π ε π a 142 Narrow Escape: Riemann Surfaces and Non-Smooth Domains

Figure 7.3: A small opening near a corner of angle α.

The error term due to O(β4) is generally small. For example, in a square a = b and β = e−π so that β4 ≈ 3 × 10−6. The MFPT averaged with respect to a uniform initial distribution is c b2 2ab a 2 π b ε Eτ = 0 + = log + log + + 2β2 + O , β4 . (7.64) 2 3 π ε π 6 a a The leading order term of the MFPT is 2|Ω| a log , (7.65) π ε which is twice as large than (7.55). The general result (7.1) was proved for a domain with smooth boundary (at least C1). However, in the rectangle example, the small hole is located at the corner. The additional factor 2 is the result of the diﬀerent singularity of the Neumann function at the corner, which is 4 times larger than that of the Green function. At the corner there are 3 image charges — the number of images that one sees when standing near two perpendicular mirror plates. In general, for a small hole located at a corner of an opening angle α (see Fig. 7.3), the MFPT is to leading order |Ω| 1 Eτ = log + O(1) . (7.66) Dα ε This result is a consequence of the method of images for integer values of π π . For non-integer we use the complex mapping z 7→ zπ/α that ﬂattens α α 7.5 Domains with Cusps 143

1 the corner. The upper half plane Neumann function log z is mapped to π 1 log z and the analysis of Section 7.2 gives (7.66). α To see that the area factor |Ω| remains unchanged under the conformal mapping f :(x, y) 7→ (u(x, y), v(x, y)), we note that this factor is a consequence of the compatibility condition, that relates the area to the integral Z |Ω| ∆(x,y)w dx dy = − , Ω D where w(x, y) = E[τ| x(0) = x, y(0) = y] satisﬁes ∆(x,y)w = −1/D. The Laplacian transforms according to

2 2 ∆(x,y)w = (ux + uy)∆(u,v)w, by the Cauchy-Riemann equations and the Jacobian of the transformation is 2 2 J = ux + uy. Therefore, Z Z ∆(x,y)w dx dy = ∆(u,v)w du dv. Ω f(Ω) This means that the compatibility condition of Section 7.2 remains unchanged and gives the area of the original domain.

7.5 Domains with Cusps

Here we find the leading order term of the MFPT for small holes located near a cusp of the boundary. A cusp is a singular point of the boundary. As α = 0 at the cusp, one expects to find a different asymptotic expansion than (7.66). As an example, consider the Brownian motion inside the domain bounded between the circles (x−1/2)2 +y2 = 1/4 and (x−1/4)2 +y2 = 1/16 (see Figure 7.4). The conformal mapping z 7→ exp{πi(1/z − 1)} maps this domain onto the upper half plane. Therefore, the MFPT is to leading order

|Ω| 1 Eτ = + O(1) . (7.67) D ε

This result can also be obtained by mapping the cusped domain to the unit circle. The absorbing boundary is then transformed to an exponentially 144 Narrow Escape: Riemann Surfaces and Non-Smooth Domains

Figure 7.4: The point (0, 0) is a cusp point of the dotted domain bounded between the two circles. The small absorbing arc of length ε is located at the cusp point. 7.6 Diﬀusion on a 2-Sphere 145 small arc of length exp{−π/ε} + O(exp{−2π/ε}), and equation (7.67) is recovered. If the ratio between the two radii is d < 1, then the conformal map that maps the domain between the two circles to the upper half plane πi is exp (1/z − 1) (for d = 1/2 we arrive at the previous example), d−1 − 1 so the MFPT is to leading order

|Ω| 1 Eτ = + O(1) . (7.68) (d−1 − 1)D ε

1 The MFPT tends algebraically fast to infinity, much faster than the O log ε behavior near smooth or corner boundaries. The MFPT for a cusp is much larger because it is more difficult for the Brownian motion to enter the cusp than to enter a corner. The MFPT (7.68) can be written in terms of d instead of the area. Substituting |Ω| = πR2(1 − d2), we find

πR2d(1 + d) 1 Eτ = + O(1) , (7.69) D ε where R is the radius of the outer circle. Note that although the area of Ω is a monotonically decreasing function of d, the MFPT is a monotonically increasing function of d and tends to a finite limit as d → 1. Similarly, one can consider different types of cusps and find that the leading order term for the MFPT is proportional to 1/ελ, where λ is a parameter that describes the order of the cusp, and can be obtained by the same technique of conformal mapping.

7.6 Diﬀusion on a 2-Sphere

7.6.1 Small absorbing cap Consider a Brownian motion on the surface of a 2-sphere of radius R [141], described by the spherical coordinates (θ, φ)

x = R sin θ cos φ, y = R sin θ sin φ, z = R cos θ.

The particle is absorbed when it reaches a small spherical cap. We center the cap at the north pole, θ = 0. Furthermore, the FPT to hit the spherical 146 Narrow Escape: Riemann Surfaces and Non-Smooth Domains

cap is independent of the initial angle φ, due to rotational symmetry. Let v(θ) be the MFPT to hit the spherical cap. Then v satisﬁes

∆M v = −1, (7.70)

where ∆M is the Laplace-Beltrami operator [141] of the 2-sphere. This Laplace-Beltrami operator ∆M replaces the regular plane Laplacian, because the diﬀusion occurs on a manifold [141, and reference therein]. For a function v independent of the angle φ the Laplace-Beltrami operator is (see Appendix D.1) −2 00 0 ∆M v = R (v + cot θ v ) . (7.71) The MFPT also satisﬁes the boundary conditions

v0(π) = 0, v(δ) = 0, (7.72)

where δ is the opening angle of the spherical cap. The solution of the boundary value problem (7.71), (7.72) is given by sin(θ/2) v(θ) = 2R2 log . (7.73) sin(δ/2) Not surprisingly, the maximum of the MFPT is attained at the point θ = π with the value δ 1 vmax = v(π) = −2R2 log sin = 2R2 log + log 2 + O(δ2) . (7.74) 2 δ The MFPT, averaged with respect to a uniform initial distribution, is 1 Z π Eτ = v(θ) sin θ dθ δ 2 cos2 δ 2 log sin(δ/2) 1 = −2R2 + cos2(δ/2) 2 1 1 = 2R2 log + log 2 − + O(δ2 log δ) . (7.75) δ 2 Both the average MFPT and the maximum MFPT are |Ω| 1 τ = g log + O(1) , (7.76) 2π δ 7.6 Diﬀusion on a 2-Sphere 147

2 where |Ω|g = 4πR is the area of the 2-sphere. This asymptotic expansion is the same as for the planar problem of an absorbing circle in a disk. The result is two times smaller than the result (7.1) that holds when the absorbing boundary is a small window of a reflecting boundary. The factor two difference is explained by the aspect angle that the particle “sees”. The two problems also differ in that the “narrow escape” solution is almost constant and has a boundary layer near the window, with singular fluxes near the edges, whereas in the problem of puncture hole inside a domain the flux is regular and there is no boundary layer (the solution is simply obtained by solving the ODE).

7.6.2 Mapping of the Riemann sphere We present a diﬀerent approach for calculating the MFPT for the Brownian particle diﬀusing on a sphere. We may assume that the radius of the sphere is 1/2, and use the stereographic projection that maps the sphere into the plane [74]. The point Q = (ξ, η, ζ) on the sphere (often called the Riemann sphere) ξ2 + η2 + (ζ − 1/2)2 = (1/2)2 is projected to a plane point P = (x, y, 0) by the mapping

ξ η ζ x = , y = , r2 = x2 + y2 = , (7.77) 1 − ζ 1 − ζ 1 − ζ and conversely

x y r2 ξ = , η = , ζ = . (7.78) 1 + r2 1 + r2 1 + r2

The stereographic projection is conformal and therefore transforms harmonic functions on the sphere harmonic functions in the plane, and vice versa. How- ever, the stereographic projection is not an isometry. The Laplace-Beltrami 2 2 operator ∆M on the sphere is mapped onto the operator (1 + r ) ∆ in the plane (∆ is the Cartesian Laplacian). The decapitated sphere is mapped onto the interior of a circle of radius δ r = cot . (7.79) δ 2 148 Narrow Escape: Riemann Surfaces and Non-Smooth Domains

Therefore, the problem for the MFPT on the sphere is transformed into the planar Poisson radial problem 1 ∆V = − , for r < r , (7.80) (1 + r2)2 δ subject to the absorbing boundary condition

V (r = rδ) = 0, (7.81)

where V (r) = v(θ). The solution of this problem is

1 1 + r2 V (r) = log δ . (7.82) 4 1 + r2 Transforming back to the coordinates on the sphere, we get 1 sin(θ/2) v(θ) = log . (7.83) 2 sin(δ/2)

As the actual radius of the sphere is R rather than 1/2, multiplying eq.(7.83) by (2R)2, we ﬁnd that (7.83) is exactly (7.73).

7.6.3 Small cap with an absorbing arc Consider again a Brownian particle diﬀusing on a decapitated 2-sphere of radius 1/2. The boundary of the spherical cap is reﬂecting but for a small window that is absorbing (see Fig. 7.5). We calculate the mean time to absorption. Using the stereographic projection of the preceding subsection, we obtain the mixed boundary value problem 1 ∆v = − , for r < r , 0 ≤ φ < 2π, (1 + r2)2 δ

v(r, φ) = 0, for |φ − π| < ε,

r=rδ

∂v(r, φ) = 0, for |φ − π| > ε. (7.84) ∂r r=rδ 7.6 Diﬀusion on a 2-Sphere 149

Figure 7.5: A sphere of radius R without a spherical cap at the north pole of central angle δ. The particle can exit through an arc seen at angle 2ε. 150 Narrow Escape: Riemann Surfaces and Non-Smooth Domains

The function 1 1 + r2 w(r) = log δ 4 1 + r2 is the solution of the all absorbing boundary problem eq.(7.82), so the function u = v − w satisﬁes the mixed boundary value problem

∆u = 0, r < rδ, for 0 ≤ φ < 2π,

u(r, φ) = 0, for |φ − π| < ε,

r=rδ

∂u(r, φ) rδ = , for |φ − π| > ε. (7.85) ∂r 2(1 + r2) r=rδ δ

Scalingr ˜ = r/rδ, we find this mixed boundary value problem to be that of a planar disk [178], with the only difference that the constant 1/2 is now 2 rδ replaced by 2 . Therefore, the solution is given by 2(1 + rδ ) 2 2rδ h ε i a0 = − 2 log + O(ε) . (7.86) 1 + rδ 2 Transforming back to the spherical coordinate system, the MFPT is 1 sin θ/2 v(θ, φ) = log 2 sin δ/2 (7.87) ∞ n δ h ε i X cot (θ/2) − cos2 log + O(ε) + a cos nφ. 2 2 n cot (δ/2) n=1 The MFPT, averaged over uniformly distributed initial conditions on the decapitated sphere, is 1 log sin(δ/2) 1 δ 2 Eτ = − + + cos2 log + O(ε) . (7.88) 2 cos2(δ/2) 2 2 ε Scaling the radius R of the sphere into (7.88), we find that for small ε and δ the averaged MFPT is 1 1 1 Eτ = 2R2 log + 2 log + 3 log 2 − + O(ε, δ2 log δ, δ2 log ε) . (7.89) δ ε 2 7.6 Diffusion on a 2-Sphere 151

There are two diﬀerent contributions to the MFPT. The ratio ε between the absorbing arc and the entire boundary brings in a logarithmic contribution to the MFPT, which is to leading order

|Ω| 1 g log . π ε However, the central angle δ gives an additional logarithmic contribution, of the form |Ω| 1 g log . 2π δ The factor 2 diﬀerence in the asymptotic expansions is the same as encountered in the planar annulus problem. The MFPT for a particle initiated at the south pole θ = π is δ δ h ε i v(π) = −2R2 log sin − 4R2 cos2 log + O(ε) 2 2 2 1 1 = 2R2 log + 2 log + 3 log 2 + O(ε, δ2 log δ, δ2 log ε) .(7.90) δ ε

We also find the location (θ, φ) for which the MFPT is maximal. The sta- ∂v tionarity condition = 0 implies that φ = 0, as expected (the opposite ∂φ φ-direction to the center of the window). The infinite sum in equation (7.87) is O(1). Therefore, for δ 1, the MFPT is maximal near the south pole θ = π. However, for δ = O(1), the location of the maximal MFPT is more complex. Finally, we remark that the stereographic projection also leads to the determination of the MFPT for diffusion on a 2-sphere with a small hole as discussed above, and an all reflecting spherical cap at the south pole. In this case, the image for the stereographic projection is the annulus, a problem solved in Section 7.3. Chapter 8

Narrow Escape at Short Times

In Chapters 5-7 we calculated the mean first passage time (MFPT) to a narrow hole. In this Chapter we calculate the short time asymptotics of the survival probability. In particular, we calculate the initial location from which it is most difficult to exit at short times, that is, the initial location that maximizes the survival probability. We find that this location can be different than the location where the MFPT is maximal.

8.1 Absorbing Disk

First, we consider Brownian motion in a circular disk of radius R whose boundary is absorbing. We show the intuitively obvious result that at short times it is hardest to exit from the center of the disk. The survival probability is Z Pr{τ > t | x0 = x} = p(y, t | x) dy, (8.1) Ω where the transition pdf p(y, t | x) is the solution of the initial boundary value problem

∂p(y, t | x) = D∆yp(y, t | x) for |x|, |y| < R, ∂t

p(y, t | x) = 0 for |y| = R, |x| < R,

p(y, 0 | x) = δ(y − x), for |x|, |y| < R, (8.2) 8.1 Absorbing Disk 153 where D is the diﬀusion coeﬃcient. The pdf p(y, t | x) has a short time asymptotic expansion of the form [25, 168]

∞ X S2(x, y) p(y, t | x) = exp − i Z (x, y, t), (8.3) 4Dt i i=0

2 where the eikonals Si(x, y) satisfy the eikonal equation |∇ySi| = 1, whose characteristics are line segments, called rays. The pre-exponential factors have regular expansions in powers of t,

∞ X (n) n−1 Zi(x, y, t) = Zi (x, y)t , (8.4) n=0

(n) where Zi (x, y) satisfy transport equations. In particular, the ﬁrst eikonal is S0(x, y) = |x − y|, which is the length of the ray connecting x and y, 1 and Z (x, y, t) = . Therefore, the leading order short time asymptotics 0 4πt of the density p(y, t | x) is 1 |x − y|2 p(y, t | x) ∼ exp − . (8.5) 4πDt 4Dt Due to rotational symmetry, we can assume that the initial point is lying on the x-axis, x = (x, 0), x ≥ 0. Introducing polar coordinates with origin at x, we write y = (ξ, η), ξ = x + r cos θ and η = r sin θ. The integral (8.1) takes the form 1 Z 2π Z rmax(θ) r2 Pr{τ > t | x0 = x} = dθ exp − r dr 4πDt 0 0 4Dt 1 Z 2π r (θ)2 = 1 − exp − max dθ, (8.6) 2π 0 4Dt

p 2 2 2 where rmax(θ) = −x cos θ + R − x sin θ is the solution of the equation R2 = ξ2 + η2 = (x + r cos θ)2 + (r sin θ)2.

The integral (8.6) is of Laplace type with small parameter t. Therefore, the main contribution to this integral comes from the point of minimum of rmax,

min rmax(θ) = R − x. (8.7) 0≤θ<2π 154 Narrow Escape at Short Times

Hence, Z 2π r (θ)2 (R − x)2 exp − max dθ ∼ C exp − , (8.8) 0 4Dt 4Dt 00 where C > 0 is a constant that depends on rmax at the minimum. It is most difficult to exit when the survival probability is maximal, that is, when the exponential function (8.8) is smallest, which happens for x = 0. Therefore, at short times it is mostly difficult to exit from the center of the disk, and the survival probability is given by R2 Pr{τ > t | x = (0, 0)} ∼ 1 − exp − for t 1. (8.9) 0 4Dt The above calculation takes into account only the contribution of the first ray. Can the other rays of the short time expansion (8.3) alter the starting point from which it is hardest to exit at short times? To answer this question, 0 0 we consider the second eikonal S1(x, y) = |x − x | + |x − y|, which is the length of a ray the emanates from x, hits the boundary once at x0(x, y), and ends at y. The law of reflection [25] is that the angle of incidence equals the 1 angle of reflection. The transport function Z (x, y, t) = −Z (x, y, t) = − 1 0 4πt is chosen so as to satisfy the absorbing boundary condition at the boundary 0 0 0 point y = x . In such a case S1(x, x ) = S0(x, x ). However, the sum of the first two rays satisfies the boundary condition only on a part of the boundary. For example, the boundary condition is satisfied at y = (R, 0), but not at the antipodal point y = (−R, 0), where the error is exponentially small, and will be corrected by the next (third) eikonal. The integral of the second eikonal is evaluated using the Laplace method as well. Therefore, the entire contribution to the integral comes from the minimum of the second eikonal

min S (x, y) = R − x, (8.10) y 1 which is obtained at the boundary point y that connects the center of the disk and x. Thus Z 2 2 S1 (x, y (R − x) Z1 exp = −C exp − , (8.11) Ω 4Dt 4Dt with C > 0. We conclude that the contribution of the second eikonal to the expansion (8.6) is of the same order and the same sign as the exponentially 8.2 Narrow Escape from a Disk 155 small remainder of the ﬁrst eikonal. Therefore, the location of the point from which it is hardest to exit at short times is not altered, because the contributions of all higher order eikonals (three and above) are exponentially (R − x)2 smaller than exp − . 4Dt The survival probability given that the trajectory was initiated from the center of the disk is calculated explicitly, because the ﬁrst and second rays satisfy the boundary condition on the entire boundary

1 |y|2 (2R − |y|)2 p(y, t | (0, 0)) = exp − − exp − . (8.12) 4πt 4Dt 4Dt

Therefore, the survival probability is

1 Z R r2 (2R − r)2 Pr {τ > t | (0, 0)} = exp − − exp − r dr 2Dt 0 4Dt 4Dt (2R)2 = 1 − exp − 4Dt √ R 4πDt 2R R + erf √ − erf √ 2Dt 4Dt 4Dt R2 ∼ 1 − 2 exp − . (8.13) 4Dt

Comparing eqs.(8.9) and (8.13), we note that, indeed, the second eikonal contributes the same as the ﬁrst one.

8.2 Narrow Escape from a Disk

Next, we consider diffusion in a disk, whose boundary is reflecting, but for a small absorbing arc. We show again the intuitive result, that at short times it is hardest to exit from the point which is farthest from the hole, that is, from the antipodal point to the center of the hole. Clearly, if the entire boundary were reflecting, the diffusing particle could not exit, so its survival probability is

Pr{τ > t | x} = 1, for all t > 0, |x| < R. (8.14) 156 Narrow Escape at Short Times

The survival probability is also the integral of the density (eq. (8.1)), which is an infinite sum of rays (eq. (8.3)), therefore, the integral over all reflected rays is exactly 1. We also note that the contributions of the reflected rays are all positive, because they have to satisfy the no flux Neumann boundary condition. However, in the case at hand, a small part of the boundary is absorbing. Therefore, the contribution of rays that hit the absorbing boundary an odd number of times is negative, whereas the contribution of rays that hit the boundary an even number of times is positive. Therefore, to leading order,

Z ( 2 ) Si(x,y)(x, y) Pr{τ > t | x} ∼ 1 − 2 Zi(x,y)(x, y, t) exp dy, (8.15) Ω 4Dt

where i(x, y) is the index of the minimal ray that emanates from x, hits the absorbing boundary once (and can hit the reflecting part of the boundary any number of times), and finally gets to y. This ray was already counted with a positive (instead of a negative) sign in the first term 1, and therefore it is subtracted twice in eq.(8.15). Evaluating the integral in eq.(8.15) by the Laplace method, shows that the main contribution to the integral comes 2 from y which minimizes Si(x,y))(x, y). Clearly, the shortest ray is obtained when y is in the absorbing part of the boundary. Therefore,

ρ(x)2 Pr{τ > t | x} ∼ 1 − C exp − , (8.16) 4Dt

where ρ(x) is the distance of x from the absorbing boundary, and C > 0 is the Laplace method constant. It follows that the point x from which it is hardest to exit is the farthest point from the absorbing boundary. In a disk, it is the antipodal point to the center of the hole. We note that the above considerations hold also for a general convex domain, because all points can be seen from all points.

8.3 Narrow Escape in an Annulus

Consider an annulus between two concentric circles of radii R1 R2, centered at O, with all reflecting boundary, except for a small absorbing arc at the outer circle (see Fig. 7.1). We show that the point from which it is hardest to exit at short times is the antipodal point to the center of the 8.3 Narrow Escape in an Annulus 157 absorbing hole in the inner circle. Although this point is not the farthest point from the hole (the antipodal point of the hole on the outer circle is the one, almost twice as much in distance), it is most difficult to exit from there. This result is somewhat counterintuitive, because the antipodal point on the outer circle is not only the farthest point, but also the point from which the MFPT is maximal (see Chapter 7). The analysis of the previous section shows that the survival probability Pr{τ > t | x} is asymptotically determined by the shortest ray that connect the point x to the absorbing√ boundary that hits the reflecting boundary any number of times. If 2R1 < R2, then the length of the shortest eikonal that connects√ the center of the hole and its antipodal point on the outer circle is 2 2R2 (see Fig. 8.1). However, the shortest ray that connects the center of the hole and its antipodal point at the inner circle turns out to be longer. Therefore, it is more difficult to exit from the antipodal point on the inner circle, rather than the antipodal point on the outer circle. The calculation of the length of the ray is geometric, and is similar to the calculations of trajectories of billiard balls in the field of quantum chaos (see, e.g. [19]). Note that a trajectory of a billiard ball inside a disk is tangent to an inscribed circle. Therefore, the ray that emanated from the center of the hole and finally hits its antipodal point on the inner circle must hit the inner circle before it hits the outer circle (for otherwise it will never hit the inner circle). Specifically, consider a ray emanating from a point A1 on the outer circle and reflected once in the inner circle, at a point B1, to a point A2 on the outer circle (see Fig. 8.2). The ray is reflected from the inner circle for the n-th time at Bn to the point An on the outer circle. We set α = ∠A1OB1, so ∠A1OBn = (2n − 1)α. The condition of hitting the antipodal point at the inner circle is (2n − 1)α = π mod 2π, (8.17) π R with the constraint 0 ≤ α ≤ − arcsin 1 (see Fig. 8.3). For β = 2 R2 R R R 1 1, we have arcsin 1 ≈ 1 1. Therefore, the minimal so- R2 R2 R2 π lution is obtained with α = and n = 2. The length of this ray is 3 p 2 2 R1 3 R2 − R2R1 + R1 = 3R2 1 + O (see Fig. 8.4). This ray is R2 √ longer than the ray that hits the outer circle, because 3 > 2 2. Note that the two rays are equal in length, if 9 (1 − β + β2) = 8, which holds for 158 Narrow Escape at Short Times

√ 1 5 π π β = − = .12732200 .... Also 1.047 ... = < −arcsin β = 1.443 ... , 0 2 6 3 2 0 so this ray is permissible. For β > β0 it is once again the hardest to exit from the antipodal point on the outer boundary. 8.3 Narrow Escape in an Annulus 159

Figure 8.1: The length of the shortest ray that connects√ the center of the hole and its antipodal point on the outer circle is 2 2R2. 160 Narrow Escape at Short Times

Figure 8.2: The geometry of a billiard ball in an annulus: the accumulative angle after n hits of the inner circle is (2n − 1)α mod 2π. 8.3 Narrow Escape in an Annulus 161

Figure 8.3: A billiard ball would hit the antipodal point, only if it hits π the inner circle before it hits the outer circle. Therefore α ≤ α = − max 2 R arcsin 1 . R2 162 Narrow Escape at Short Times

Figure 8.4: The shortest billiard trajectory that connects the center of the hole and its antipodal point on the inner circle is of length p 2 2 3 R2 − R2R1 + R1. Part II

Non-Equilibrium Statistical Mechanics of Interacting Particles 164

The most common continuum model of ion permeation is the coupled Poisson-Nernst-Planck (PNP) system of equations [43], also known as the drift-diffusion equation in the physics of semiconductors [170]. In this model, ions are described as point charges, while both short range (e.g., size effects) and long range (electric forces) interactions are neglected. Thus, phenomena such as blocking and selectivity are not expected to be recovered by this model. Recently, an alternative model, derived at the molecular level, was shown to lead to a slightly different equation [167], the conditional-PNP equations, where conditioned densities (pair correlation function) and potentials replace the unconditioned quantities of the PNP equations. In Chapter 9 we outline the infinite hierarchy of equations for the correlation functions in systems out of equilibrium. However, there are two missing ingredients to make this hierarchy well posed. The first is a boundary condition for the pair correlation function, derived from the stochastic differential equations that describe the physical microscopic model. The second is a closure relation that relates high order correlation functions to correlation functions of lower order. Even when both ingredients are at hand, solving the C-PNP system numerically in a finite or infinite domain is computationally prohibitive, because it is a problem in high dimensions (each particle is three-dimensional). The numerical challenge exceeds the scope of this dissertation. The diffusion of interacting particles in confined geometries is dominated by the pair correlation function of a non-equilibrium system. In Chapter 10, we derive the Kirkwood superposition approximation to a closure relation for systems at equilibrium in the thermodynamic limit, using a variational formulation. We define the entropy of the triplet correlation function and show that the Kirkwood closure brings the entropy to its maximum. This approach leads to a new interpretation of the Kirkwood closure relation, usually explained by probabilistic considerations of dependence and independence of particles. The Kirkwood closure is generalized to systems of finite volume at equilibrium by computing the pair correlation function in finite domains. Closure relations for high order correlation functions are also found by the variational approach. In particular, maximizing the entropy of quadruplets leads to the high order closure used in the BGY2 (Born-Green- Yvon) equations, which are a pair of integral equations for the triplet and pair correlation functions. A new direction will be the generalization of this approach to non-equilibrium systems, such as ions crossing a channel. One of the fundamental problems in the theory of ionic solutions is the question of how far the electrostatic field extends in a disordered system of 165 charges. This question is important if we were to use closure relations in a charged system, such as Kirkwood’s superposition approximation. We develop a new theory for this problem, based on a statistical approach and on large deviations theory. In Chapter 11 we define the concept of a totally disordered and electroneutral infinite system of charges and evaluate the probability distribution of their electric potentials and fields. We show that in one dimension the electric field is always small, of the order of the field of a single charge. If the system is electroneutral, the spatial variations in potential do not exceed those that can be produced by a single charge. In two and three dimensions, the electric field in similarly disordered electroneutral systems is usually small, with small variations. Interestingly, in two and three dimensional systems, the electric potential is usually very large, even though the electric field is not, because large amounts of energy are needed to put together a typical disordered configuration of charges in two and three dimensions, but not in one dimension. If the system is locally electroneutral—as well as globally electroneutral—the potential is usually small in all dimensions. An additional problem concerning the electric field in an ionic solution is that of the compatibility of the electric current carried by the diffusing ions and Maxwell’s equations. We resolve this problem by applying the Ramo-Shockley theorem, which leads to the intuitively obvious result that the current carried by the ionic flux is identical to the electric current measured by the electrodes. Finally, in Chapter 12, we combine the results of the previous chapters to derive boundary conditions for the pair correlation function, both for infinite systems in steady state and for finite systems carrying a quasi steady state diffusion current. Chapter 9

Ionic Diffusion Through Confined Geometries: From Langevin Equations to Partial Differential Equations

The contents of this chapter were published in [138]

Ionic diffusion through and near small domains is of considerable importance in molecular biophysics in applications such as permeation through protein channels and diffusion in a domain near the charged active sites of macromolecules. The validity of standard continuum-type descriptions of the motion of interacting ions in domains of nanoscale geometry and charge distribution in and near a channel are domain is questionable. The description in terms of density of only a few ions in such a domain may be overstretch- ing continuum mechanics beyond its underlying assumptions. In particular, non-equilibrium diffusion of interacting particles has not been adequately captured by continuum theories to this day. More specifically, although the standard machinery of equilibrium statistical mechanics contains microscopic details it is not applicable, because these our system is usually not in equilibrium. Concentration gradients and the presence of an external applied potential drive a non-vanishing stationary current through the system, rendering it a non-equilibrium system. A molecular level study of a stochastic molecular model for the diffusive motion of interacting particles in an external field of force and a derivation of effective partial differential equations 9.1 Introduction 167 and their boundary conditions that describe the stationary non-equilibrium system began in [167]. The interactions can include electrostatic, Lennard- Jones, and other pairwise forces. The main result of [167] is the derivation of a diffusion equation for the single ion density, coupled to the joint density of two ions, but not for the pair or higher order densities. Also the boundary conditions that these densities satisfy were not derived so far. The analysis of this chapter yields a new type of Poisson-Nernst-Planck equations, that involve conditional and unconditional charge densities and potentials. The conditional charge densities are the non-equilibrium analogs of the well-studied pair correlation functions of equilibrium statistical physics. Our proposed theory is an extension of equilibrium statistical mechanics of simple fluids to stationary non-equilibrium problems. The proposed system of equations differs from the standard Poisson-Nernst-Planck system in two important aspects. First, the force term depends on conditional densities and thus on the finite size of ions, and second, it contains the dielectric boundary force on a discrete ion near dielectric interfaces. Recently, various authors (see citations below) have shown that both of these terms are important for diffusion through confined geometries, in the context of ion channels. The problems of deriving the missing boundary conditions and decoupling the equations for higher order densities are studied in later chapters of this part.

9.1 Introduction

With ongoing advances in technology and experimental techniques, the physical systems that are either studied or designed become smaller and smaller, nowadays reaching the nanometer and nanosecond length and time scales, respectively. In this chapter we focus on a particular type of systems, containing a nanoscale (nearly picoscale) pore connected to two large reservoirs of electrolyte solutions. Two examples of such systems, which are of great importance in molecular biophysics and biotechnological applications, are ionic permeation through protein channels [43], [73] and through carbon nanotubes [91]. These nanoscale systems exhibit a range of phenomena not encountered in larger macroscopic systems. For example, due to the confined geometry of the pore through which the fluid flows, ions typically cannot pass by each other, leading to single filing phenomena, complex relations between unidirectional currents, and other nonlinear phenomena in mixtures [73]. The confined 168 From Langevin Equations to Partial Differential Equations

geometry enhances the importance of individual ion-ion interactions, which are thought to be the cause of the high selectivity of these nanopores to specific ionic species [58]. The function of these systems depends not only on the nanoscale atomic details and geometry of the pore, but also on experimentally controlled macroscopic variables, such as the applied electrostatic potential and the surrounding bath concentrations. Therefore, the function of these systems involves many different time and length scales, from the femto-second and Angströmscales for the motion of single water molecules till the microsec- ond time scale at which current measurements are made and micro- to milli- meter distances at which measuring devices are placed. One final important feature worth mentioning is that these systems almost always function away from equilibrium, as does almost all biology. These nanopores separate regions of different concentrations and an applied voltage is usually present across them. Therefore, these systems can be viewed as nanodevices, with well defined input and output relations that are the purpose of the device. Many of these biological systems are of interest because they are highly sensitive, with atomic scale details controlling macroscopic flows. Mutations or modifications to the atomic structure of the nanopore lead to significant changes in its characteristics. Thus, the theoretical analysis of such systems necessarily resides at the interface between continuum and discrete atomic molecular physics, presenting new challenges in non-equilibrium statistical physics. On the one hand, due to the importance of confined geometry, discrete ion-ion interactions and non-uniform dielectric coefficient, standard continuum equations of electrolyte solutions, such as Poisson-Boltzmann (PB), Poisson-Fokker-Planck, or Poisson-Nernst-Planck (PNP) are not valid [167, 132, 28]. On the other hand, due to the wide range of time and length scales involved — mixing nanoscale and continuum — direct molecular dynamics simulations are impractical for the study of the macroscopic function of these systems. Regretfully, even coarse grained Brownian Dynamics (BD) simulations are not always the appropriate tool to study such systems. For example, it is not possible to study the effects of a 10µM concentration of calcium ions in a 100 mM Na+Cl− electrolyte in a simulation that contains only a few hundred ions, even though the biological effects of such trace concentrations are often of overwhelming importance [2]. Therefore, there is a need for a hierarchy of models each valid in its own scale, and connections between them. While the standard PNP system may not be valid in confined geometries, 9.1 Introduction 169 it is important to note the advantages of a continuum description of nanoscale systems. The computational complexity of continuum descriptions, that typically involve the solution of a system of coupled partial differential equations, is often orders of magnitude lower than that of corresponding BD or MD simulations. Another significant advantage of continuum descriptions over BD and MD simulations is that they easily accommodate non-equilibrium boundary conditions for the macroscopic electrostatic potential and ionic concentrations. Simulations in the chemical tradition have difficulty with such conditions (see however [89]). The goal, then, is to derive continuum equations for non-equilibrium systems that include molecular details, absent in the PNP theory. We note that for the corresponding equilibrium problem, the theory of equilibrium statistical mechanics of simple fluids incorporates particle-particle interactions by the Bogolyubov-Born-Green-Kirkwood-Yvon (BBGKY) hierarchy [7], [12]. This hierarchy, coupled with a closure relation, has proven successful in the calculation of a number of important macroscopic properties of electrolytes from their underlying molecular interactions [158]. The main problem discussed in this chapter is thus how to generalize the theory of equilibrium statistical mechanics of simple fluids, which is based on the equilibrium Boltzmann distribution, to stationary non-equilibrium problems. To this end, we note that one of the key missing elements in the equilibrium theory is the dynamics of the moving particles. The Boltz- mann distribution defines the probability of configurations but does not include a description of how individual particles move in the system. In non- equilibrium problems there is usually a net flow of particles. It is not possible to compute net fluxes from equilibrium theories. In this chapter, following [167], we present a molecular model of permeation, based on diffusive motion of ions, and propose a mathematical averaging procedure that results in a hierarchy of Poisson and Nernst-Planck type equations containing conditional and unconditional charge densities. The proposed conditional system, called C-PNP, includes molecular details such as excluded volume effects and the dielectric force on a discrete ion that are absent in the standard PNP system. Recently, various authors have shown that both of these terms are important for diffusion through confined geometries, in the context of ion channels [58, 133, 136, 30, 119, 59]. The C-PNP system, along with a closure relation and boundary conditions, provides a theoretical framework for the study of non-equilibrium diffusing systems. The C-PNP system reduces to the BBGKY hierarchy when equilibrium boundary conditions are given, thus generalizing equilibrium statistical mechanics to stationary non-equilibrium 170 From Langevin Equations to Partial Differential Equations

problems. Application of C-PNP to ion channels can be expected to predict blocking and possibly selectivity. This chapter is organized as follows. In Section 9.2 we describe the standard Poisson-Boltzmann and Poisson-Nernst-Planck theories and discuss their limitations in confined geometries. Then, in Section 9.3, we briefly review the theory of equilibrium statistical mechanics of simple fluids, and its limited applicability to non-equilibrium systems. The main results of this chapter are described in Sections 9.4 and 9.5, where we formulate a non- equilibrium stochastic molecular model based on trajectories and present a mathematical derivation of the corresponding continuum equations. We conclude in Section 9.6 with summary and discussion.

9.2 Standard Continuum Treatments and their Failure in Conﬁned Systems

We consider the setup shown in Figure 2.1. A rigid nanopore connects two reservoirs of electrolyte solutions. The system is kept in a stationary non- equilibrium condition by an external feedback mechanism that maintains average concentrations cL and cR in the left and right reservoirs, respectively, and an average applied voltage V across them. The problem at hand is to compute the stationary ionic current through the nanopore as a function of the parameters of the system: the geometry, dielectric coefficients, and fixed charge distribution of the pore, the atomic characteristics of the ions (e.g., radii, ion-ion interaction forces, and their diffusion coefficients), and as a function of the experimentally controlled macroscopic variables, cL, cR and V . Throughout this chapter we consider modeling approaches at the level of the primitive model [7, 72], that do not consider explicitly the individual molecules of the solvent, but rather treat them as a continuous structureless dielectric, and as a viscous and noisy medium. Some of the limitations of this approach are discussed in the summary section. For simplicity, we consider an ionic solution with only two ionic species, with positive and negative charges, such as Na+Cl−, although this assumption can easily be relaxed. Two of the most common mean-field continuum theories of ionic solutions are the equilibrium Poisson-Boltzmann (PB) and the non-equilibrium Poisson-Nernst-Planck (PNP) theories [7], [12]. Both PB and PNP assume 9.2 Standard Continuum Treatments 171 a constitutive relation of the form ezp J p(x) = −Dp ∇cp(x) + cp(x) ∇ψ(x) kBT between the local averaged particle flux of the positive ionic species at location x, called J p(x), their concentration cp(x), and a potential of mean field ψ(x). In this equation kB is Boltzmann’s constant, T is temperature, Dp is the diffusion coefficient of the positive ions and zp is their valence. A similar equation for the flux of the negative ions is also assumed. The equilibrium PB theory assumes that the local flux vanishes everywhere,

J p(x) = J n(x) = 0 (9.1) while the PNP theory assumes conservation of ﬂux, e.g. the Nernst-Planck equation

∇ · J p(x) = ∇ · J n(x) = 0. (9.2) The simplicity of these two theories lies in their crude approximation for the potential of the mean field. Both PB and PNP assume that the potential ψ(x) satisfies Poisson’s equation with the average concentrations cp(x) and cn(x), ∇ · [ε(x)∇ψ(x)] = −e zpcp(x) − zncn(x) + ρfixed(x) (9.3) where ε(x) is the location dependent dielectric coefficient and ρfixed is the concentration of fixed charges in the system. Stated differently, the electrostatic field exerted on an ion at location x is approximated by the gradient of the mean electrostatic potential. The PB theory is often the starting point of modern electrochemistry, with the famous Debye-Hückel theory of ionic shielding [12], [13]. The PNP theory has also found numerous applications in the study of electrolyte transport, plasma physics and in modeling of semiconductor devices [12, 170]. Both of these theories represent the ions as a continuous charge distribution in an ambient mean potential field, defined by the Poisson equation (9.3). Both theories neglect effects due to the finite size of ions and due to the discrete nature of their charge. The limitations of the PB approach, even in bulk electrolyte solutions with moderate to high concentrations, are well known [12], [72], [13]. Recently, the failure of PB and PNP in confined geometries has also been shown both in simulations [132], [28] and in theoretical treatments [167]. 172 From Langevin Equations to Partial Differential Equations

9.3 Equilibrium Statistical Mechanics of Sim- ple Fluids

The preceding discussion indicates that the problem at hand is how to include molecular details in non-equilibrium continuum type equations. It is important to note that molecular details can be included in continuum models with satisfactory results. For example, the theory of equilibrium statistical mechanics of simple fluids yields continuum type equations with molecular detail, and has proven successful in the computation of many macroscopic quantities [12], [158], some of which (e.g., activity of ions) are difficult to determine in MD or BD simulations. We now briefly describe the key elements in equilibrium theory and its limitations for non-equilibrium systems. For simplicity we consider the canonical (NVT) ensemble formulation. The classical description of the statistical physics of interacting particles (at the level of the primitive model) starts with a finite system of N particles in a finite volume V at a fixed temperature T with the N particles at locations x1, ..., xN . It is usually assumed that the potential of the forces acting on the ions, U(x1, x2, ..., xN ), is explicitly known, and is a sum of pair radial interactions, X U(x1, x2, ..., xN ) = Ui,j(|xi − xj|), i

with the role of forces at the boundary and electrostatic interactions with boundary charges being often ignored. The conﬁgurational partition function for this system is deﬁned as

Z Z N −βU Y QN (β) = ··· e dxi, N V i=1

where β = 1/kBT . The main assumption of the theory is that the probability of a given conﬁguration of the N particles follows the Boltzmann distribution,

e−βU(x1,...xN ) p(x1, x2, ..., xN ) = . (9.4) QN Since the number of particles in a given system is usually of the order of Avo- gadro’s number, one must consider a reduced description of the system; thus 9.3 Equilibrium Statistical Mechanics of Simple Fluids 173 sometimes the marginal densities of only one or two particles are considered. By definition, the average physical density at location x1 is given by N Z Z Y ρ(x1) = N ··· p(x1, ..., xN ) dxi. (9.5) N−1 V i=2 In equilibrium statistical mechanics, it is often customary to consider the limit of very large systems, namely N, |V | → ∞, such that N/|V | = ρ. In this limit, it is possible to show that ρ(x) satisfies the following equation, known as the first BBGKY equation, Z 1 ∇ρ(x) + ρ(x) ∇xU1,2(x − y)ρ(y | x) dy = 0, (9.6) kT where ρ(y | x) describes the conditional density of particles at y, given the presence of a particle at x. By definition, ρ(x, y) ρ(y | x) = ρ(x) where ρ(x, y) is the joint physical concentration of a pair of particles. For homogeneous infinite systems ρ(x) is uniform, due to symmetry, and thus equal to the bulk concentration ρ. The microscopic structure of the solution is described by the pair correlation function g2(x, y), which is the non-dimensional version of ρ(x, y), given by ρ(x , x ) g (x , x ) = 1 2 (9.7) 2 1 2 ρ2 N Z Z Y = lim N(N − 1) ··· p(x1, x2, ..., xN ) dxi. N,V →∞ N−2 V i=3 The pair correlation function is important for the determination of many thermodynamic properties of an equilibrium system, such as the free energy, pressure, chemical potential and compressibility of electrolytes, to name just a few [12], [158]. By differentiating (9.7) with respect to x1, it is possible to show that g2 satisfies the second BBGKY equation, which depends on the higher order triplet correlation function g3, Z kBT ∇x1 g2 + g2∇x1 U1,2 + ρ ∇x1 U1,3 |x1 − x3| g3(x1, x2, x3) dx3 = 0. (9.8) 174 From Langevin Equations to Partial Differential Equations

Equations (9.6) and (9.8) correspond to a system with only one species of interacting particles. In the case of two species of interacting particles —say positive and negative ions — there are two singlet density functions, p n g1 and g1 , for the positive and negative species, respectively. There are α,β also four pair correlation functions, g2 , where α, β are one of the four possible combinations (α, β) ∈ (p, p), (p, n), (n, p), (n, n) , though obvi- p,n n,p ously g2 (x, y) = g2 (y, x). All of these quantities satisfy first and second BBGKY equations similar to (9.6) and (9.8). The BBGKY equations constitute an infinite hierarchy of forwardly coupled integro-differential equations. Further approximations are needed to solve the problem. For example, approximate solutions for the single and pair densities are often computed using closure relations, such as HNC or MSA, relating higher order correlation functions to lower order correlation functions, and the relevant thermodynamical quantities are calculated from the resulting densities [12], [158].

9.3.1 Equilibrium vs. Non-equilibrium Statistical Me- chanics In contrast to the simple equilibrium system described in the previous section, the biophysical systems we consider are in steady state, but not equilibrium. Our system is connected to an energy source, either a natural one, as in living organisms, or an artificial one, that mimics the natural one, but also allows experimental control of parameters. Usually the nanopore carries a steady net flux of particles (e.g., either into or out of a biological cell). A steady state system carrying a constant flow cannot be described by the equilibrium theory, since the Boltzmann distribution assumes a symmetrical velocity distribution with zero mean and thus zero net flux. We construct below a non-equilibrium analogue of the Boltzmann distribution and use it to calculate fluxes. It is instructive to note that the first BBGKY equation (9.6) can be rewritten as J(x) = 0, (9.9) where J(x) is the average flux at location x, f¯(x) J(x) = −D ∇ρ(x) − ρ(x) , kBT 9.4 Non-Equilibrium Statistical Mechanics 175 with D the diffusion coefficient and f¯(x) the average force on a particle at x, given by Z ¯ f(x) = − ∇xU1,2(x − y)ρ(y | x) dy.

Upon comparison of equation (9.9) with equations (9.1) and (9.2) for the PB and PNP ﬂuxes, it is tempting to write the following equation for the non-equilibrium case, ∇ · J(x) = 0. This result, however plausible, cannot be derived from the assumptions of equilibrium statistical mechanics, because those assumptions imply J(x) = 0. We present below a derivation of this non-equilibrium equation from the dynamics of diﬀusing ions.

9.4 Non-Equilibrium Statistical Mechanics

9.4.1 A Trajectory Based Approach Our point of departure for the statistical description of non-equilibrium systems of diffusing particles is a probability measure defined on their trajectories in phase space, rather than on the statistics of points in configuration space, which is the starting point of traditional equilibrium statistical mechanics. First, we introduce some notation. We consider a finite domain Ω, that consists of the two macroscopic reservoirs and the connecting rigid nanopore (see Figure 2.1). Its boundary ∂Ω is composed of reflecting boundaries ∂ΩR h and the feedback boundaries ∂ΩF . We assume that there are N ions of species h (h =Ca++, Na+,Cl−,... ) in Ω, which are numbered at time t = 0, P h h N = N, and we follow their trajectories at all times t > 0. The coordinates of a point are x = (x, y, z), while the location and velocity coordinates h h of the j-th ion of species h at time t are xj (t) and vj (t), respectively. Ac- cording to our assumptions, an ion that reaches ∂ΩF is instantly re-injected by the feedback mechanism at one or another part of the boundary, so that its individual identity is preserved, and consequently the total number of ions inside Ω is fixed at all times. The feedback mechanism serves as the energy source for the system, keeping it in stationary non-equilibrium by keeping average concentrations cL and cR in the left and right reservoirs and a constant applied voltage V across the system. In experimental situations, 176 From Langevin Equations to Partial Differential Equations

electronic devices and chemical apparatus provide the energy and feedback. In biological systems, metabolic machinery and the active transport systems it fuels provide the energy and feedback. The evolution of the joint probability density function (pdf) of all the ions and water molecules in this system can be described by the Liouville equation. However, at the level of the primitive model, we follow the evolution of the pdf of only the ions in the system, which is a lower dimensional projection of the former. Since the motion of ions in solution is strongly overdamped, on time scales larger than the relaxation time of the solution, memory effects due to the thermal motion of the solvent can be neglected [7], and the joint motion of only the ions can be described as diffusion with interactions. Therefore, our starting point is a memoryless system of coupled Langevin equations for the different ion species h = Ca++, Na+, Cl−, etc., j = 1,...,N h,

f h q x¨h + γh xh x˙ h = j + 2εhγh xh w˙ h, (9.10) j j j M h j j where a dot on a variable indicates differentiation with respect to time, γh(xh) is the location dependent friction coefficient per unit mass, M h is h h the effective mass of an ion of species h and ε = kBT/M . The force on h the j-th ion of species h is f j and includes all ion-ion interactions. It thus h depends on the locations of all ions. The functions w˙ j are, by assumption, independent standard Gaussian white noises. Thus, the effects of the solvent are modeled as a source of noise and friction for ionic motion and as an averaged dielectric coefficient for the ion-ion interactions. We discuss the limitations of these approximations in Section 9.6.

9.4.2 The Fokker-Planck Equation

We deﬁne by pN (x1,..., xN , v1,..., vN ) the stationary probability density function (pdf) of the system of all N ions. Since the coupled motion of all ions is governed by the Langevin system (9.10) with independent noise terms, the stationary pdf pN satisﬁes the multi-dimensional stationary Fokker-Planck equation (FPE) [166]

N h X X h 0 = Lj pN , (9.11) h j=1 9.5 The C-PNP System 177

h where Lj is the Fokker-Planck operator acting on the phase space coordinates of the j-th ion of species h. It is given by

h ! h h h h f j Lj pN = ∇vh · γ (xj )vj − pN + j M h

h h h h ∆ h ε γ (x )pN − v · ∇ h pN , vj j j xj where the operators ∇v and ∆v denote the gradient and the Laplacian with respect to the variable v, respectively. Equation (9.11) is defined in the 3N N 3N dimensional region (x1,..., xN ) ∈ Ω and (v1,..., vN ) ∈ R . The solution pN of the stationary FPE (9.11) is the non-equilibrium analog of the Boltzmann distribution. It is the stationary transition probability density function of the 6N-dimensional trajectory of the system in phase space and thus it is also the probability density function of the particle configurations in phase space. Obviously, since the FPE is defined in a finite region, a unique solution is determined only after specification of appropriate boundary conditions. If no-flux boundary conditions are given, the FPE (9.11) can be solved explicitly and the solution is the Boltzmann distribution. Thus, our formulation of a non-equilibrium system reduces to the equilibrium theory in this special case. It is thus clear that the boundary conditions are what drive the system out of equilibrium. The boundary conditions for the FPE (9.11) need to be determined from the action of the feedback mechanism at the boundaries and will be described in Chapter 12. Finally, we note that a time dependent FPE, similar to (9.11), was suggested [7], [40] as the starting point for the study of the transport characteristics of bulk electrolytes, by an analysis of the decay into equilibrium of transient infinite non-equilibrium electrolyte systems.

9.5 The C-PNP System

With the interpretation of the stationary joint transition pdf of the phase- space trajectories, pN (x1,..., v1,...), as the probability density of conﬁgu- rations of all particles in phase space, we can follow the steps taken in the theory of equilibrium statistical mechanics, and study the single and pair densities. An equation similar to (9.5), for ch(x), the time-averaged steady 178 From Langevin Equations to Partial Diﬀerential Equations

h state physical concentration of ions of species h at location x1 , is given by Z h h h h h Y h0 Y h0 c (x ) = N pN (x ,..., v ,...) dx dv . 1 N−1 3N 1 1 i i Ω × R (i,h0)6=(1,h) i,h0 (9.12) In the equilibrium case, equation (9.4) gives an explicit expression for pN , so that some of the integrations in (9.12) can be performed explicitly. Moreover, in deriving the BBGKY equation, the limit N, |V | → ∞ is taken. In the non-equilibrium case, the system remains finite and pN is not known explicitly. What is known is only that pN satisfies the FPE equation (9.11) in a finite domain. However, by integration of this equation over all particle coordinates but one, the following Nernst-Planck type equation for the concentration ch(x) can be derived [167],

0 = −∇x · J h(x), (9.13) where J h(x) is the ﬂux density of type h ions, given by

" h # f¯ (x) J h(x) = −Dh(x) ∇ch(x) − ch(x) , (9.14) kBT

h h h where D (x) = kBT/M γ (x) is the local diﬀusion coeﬃcient of species h. The quantity f¯ h(x) in (9.14) is the average force on a single ion of type h. It is given by Z f¯ h(x) = f hp (x˜h | xh = x) dx˜h, (9.15) ΩN−1 1 N−1 1 1 1

h h where x˜1 is the vector of all N − 1 particle coordinates except x1 and h h pN−1(x˜1 | x1 = x) is the conditional probability density of the N − 1 remaining ions given that the ﬁrst ion of species h is located at x. In the case of charged ions in solution, the ion-ion interaction forces are pair-wise additive, and thus the force on the ﬁrst ion of species h can be written as

h h h X h,h0 h0 h f 1 = f ed(x1 ) + f (xi , x1 ), (9.16) (i,h0)6=(1,h)

0 where f h,h is the ion-ion interaction force that an ion of type h0 acts on an ion of type h. It includes both Coulombic interactions as well as short 9.5 The C-PNP System 179 range interactions, such as excluded volume or Lennard-Jones forces. The h force f ed contains both the effects of an applied external field as well as the dielectric self-force near dielectric boundaries [167, 133, 4]. Interactions between charges in the system and boundary charges, including both fixed and induced charges, may determine the entire behavior of the system, e.g., to stop the flow of ions through an open pore. These interactions change the energy of the system also in the equilibrium case. As shown in [167], with the specific form (9.16) for the force on a single ion, equation (9.15) for the average force simplifies to

¯ h h ¯ h h ¯h f (x) = f ed(x) + f SR(x) − z e∇yφ (y|x) y=x where Z ¯ h X h,h0 h0|h f SR(x) = f SR (y, x)c (y|x) dy, (9.17) h0 Ω is the average short range force on a type h ion, zh is the valence of type h ions and φ¯h(y|x) is the conditional electrostatic potential at y given a type h ion at x. It satisﬁes the (conditional) Poisson equation

h X h0 h0|h ∇y · ε(y)∇yφ¯ (y|x) = −e z c (y|x), (9.18) h0 where ε(y) is the dielectric coeﬃcient at y. In both eq. (9.17) and (9.18), ch0|h(y, x) is the conditional concentration of h0 ions at y given an h-type ion at x. In terms of unconditional quantities, it is given by

h,h0 0 c (x, y) ch |h (y|x) = . ch(x) To summarize, the density ch(x) satisfies a Nernst-Planck type equation (9.13), with an average force f¯ h that is the sum of a dielectric self force, an averaged short range force (9.17) and an averaged electrostatic force. The latter is a solution of a conditional Poisson equation that depends on conditional densities, in contrast to the unconditional densities in the standard PNP formulation, equation (9.3). We call this resulting system of conditional Poisson and Nernst-Planck equations C-PNP. The NP equation (9.13) is defined in the finite domain Ω. Therefore, in addition to the determination of the averaged force f¯ h, boundary conditions 180 From Langevin Equations to Partial Differential Equations

on ∂Ω must be specified in order to determine the unique solution for ch(x). h Obviously, on ∂ΩR, c (x) satisfies no flux boundary conditions,

J h(x) · ν = 0. x ∈ ∂ΩR

In addition, on ∂ΩF , according to our assumptions, the average concen- h h trations c (x) are maintained at ﬁxed known values cF (x) by the feedback mechanism. Therefore, regardless of the exact method by which the feedback mechanism maintains these average concentrations,

h h c (x) = cF (x), for x ∈ ∂ΩF . In case the domain consists of the entire space, separated by an impermeable membrane with a small hole, the reﬂecting boundary ∂ΩR are the two sides of the membrane, and the conditions ate inﬁnity are

h h c (x) → cR for |x| → ∞, z > 0 (9.19) h h c (x) → cL for |x| → ∞, z < 0.

9.5.1 The C-PNP Hierarchy An important difference between the PNP and the C-PNP systems is that the C-PNP system is not closed, because, as seen from (9.17) and (9.18), the averaged force in the Nernst-Planck equation (9.13) depends on conditional higher order concentrations. Specifically, we consider the equation for the pair concentrations. Employing similar methods to those of [167], we obtain that ch,h0 (x, y) satisfies the six-dimensional Nernst-Planck equation

h,h0 h,h0 ∇ h · J x, y + ∇ h0 · J 0 x, y = 0, (9.20) x xh y yh where " ¯ h,h0 # h,h0 h h,h0 f (x, y) h,h0 Jxh (x, y) = −D (x) ∇xc (x, y) − c (x, y) kBT

0 and f¯ h,h (x, y) is the average force on an ion of species h located at x, 0 h,h0 given an ion of species h located at y. The second ﬂux J 0 is given by a yh 9.5 The C-PNP System 181 similar expression. For the case of pairwise additive forces, this force can be simpliﬁed to

¯ h,h0 h h,h0 ¯ h,h0 f (x, y) = f ed(x) + f (x, y) + f SR (x, y) (9.21) 0 −ezh∇zφ¯h,h (z|x, y) z=x

0 ¯ h,h ¯h,h0 where f SR and φ are the higher order analogs of (9.17) and (9.18), which depend on the third order conditional concentrations ch00|h,h0 . Equation (9.20) is the non-equilibrium analog of the second BBGKY equation (9.8), and as 0 in equilibrium, determination of the forces f¯ h,h (x, y) requires knowledge of the triplet densities ch,h0,h00 (x, y, z). Similarly, it is possible to write an equation for the triplet density, whose average forces depend on fourth order conditional densities. We arrive this way at an infinite hierarchy of conditional Poisson and Nernst-Planck type equations, all defined in finite domains. The resulting equations are very similar to those used in the study of macroscopic bulk dynamical properties of electrolytes [7], where the time dependence of similar equations is considered in infinite domains. In these studies, closure relations between the triplet and pair densities, similar to those of equilibrium statistical mechanics, are employed in order to compute the average forces [38]. Closure relations require further physics or approximations, or both. In our case, however, since we are concerned with a finite system in non- equilibrium, a closure relation between the triplet and the pair densities is not enough to close the system. Specifically, the Smoluchowski type equation (9.20) is defined in the finite domain (x, y) ∈ Ω × Ω. Therefore, to uniquely determine its solution, boundary conditions have to be prescribed on the domain boundaries, e.g. for (x, y) ∈ ∂Ω×Ω and (x, y) ∈ Ω×∂Ω. Only after these boundary conditions are specified, the pair concentration ch,h0 (x, y) is 0 completely determined, provided the forces f¯ h,h (x, y) are known. As in the case of the single ion densities, the boundary conditions for the pair densities should also be determined by the action of the feedback mechanism. The derivation of boundary conditions for the pair concentrations, as well as for higher order densities, requires a more detailed description of the feedback mechanism, developed in Chapter 12. 182 From Langevin Equations to Partial Differential Equations

9.5.2 PNP revisited The simplest possible closure is

ch0|h(y|x) = ch0 (y), (9.22)

which assumes independence of ions and thus neglects ion-ion ﬁnite size effects. Therefore, it is also necessary to neglect all short range forces in this approximation, if one wishes to be consistent. This closure recovers the

(unconditional) PNP system, but with an additional force term, f ed, the dielectric boundary force on a single ion near dielectric interfaces. This term represents the forces on a single ion by surface charges induced by the ion itself at dielectric interfaces [133, 4]. This force term has a crucial importance in the determination of the permeation characteristics of the gramicidin channel, see [133]. The above analysis of this closure clarifies the assumptions and validity of PB and PNP. These two theories are obtained as approximations of the exact BBGKY and C-PNP systems by use of the simple closure (9.22) that neglects both the discreteness of charge and the finite size of ions. Therefore, in any system where discreteness of charge and the finite size of particles are important, such as near dielectric interfaces or in confined regions, the validity of these theories is questionable.

9.6 Summary and Discussion

The function of biological systems such as ion channels typically involves the atomic control of macroscopic flows. These channels are nanoscale non- equilibrium systems connecting large electrolyte reservoirs, so their description involves many different length and time scales. Yet, the majority of published work on channels typically considers only a single level of modeling, which is valid or computable on limited time and length scales. In order to study the function of such systems, connections between theories at different levels are essential. In this chapter we presented a connection between the level of Brownian dynamics (BD) and of continuum theories. This connection is shown graphically as the diagonal red arrow in figure 9.1. Our analysis shows an equivalence between BD simulations and the C- PNP hierarchy. This equivalence can be used to validate specific closure relations as well as BD computer codes by comparing the results of the two 9.6 Summary and Discussion 183

Figure 9.1: Typical time scales of various processes for permeation through a protein channel 184 From Langevin Equations to Partial Diﬀerential Equations

computations. The resulting C-PNP hierarchy is not closed as it involves conditional and unconditional charge densities. We find that it is the boundary conditions that drive the system out of equilibrium. When equilibrium boundary conditions are imposed the equilibrium BBGKY hierarchy is recovered from the C-PNP equations. While the single ion densities satisfy simple concentration and no flux boundary conditions, the corresponding conditions for the higher order densities are not obvious and require further analysis (see Chapter 12). The two main force terms that appear in the C-PNP system and are absent in standard PNP are the dielectric boundary force and the effects due to the finite size of ions. Both of these terms have been shown to be important in the context of narrow protein channels [58, 133, 136, 30, 119, 59]. However, to the best of our knowledge, the validity of various closure relations has not yet been considered in confined and highly non-homogeneous systems. The derivation presented in this chapter is at the level of the primitive model and thus relies on a few assumptions concerning the properties of the implicit solvent. One of these assumptions is that the noise terms of different ions are independent. This assumption may not hold in very narrow regions occupied by one or more ions and only a few water molecules, where ions and water may move in a highly coordinated fashion. The analysis of this configuration requires separate analysis. Another assumption concerns the representation of the solvent as an effective dielectric constant. Given the substantial frequency dependence [8], and possible location dependence of induced charge, higher resolution methods must be used, along with exper- imentation, to evaluate this representation and its parameters. Specifically, the description of induced charge by a dielectric coefficient (with its inher- ent assumption of a linear invariant relation between induced charge and local electrical field) must be validated, and the frequency/time and location dependence of this dielectric coefficient evaluated. Such high resolution calculations are not trivial because they themselves must be shown to represent the electric field accurately over the relevant length and time scales. We note that in the context of proteins, the dielectric coefficient for dielectric boundary forces may not be the same as the one coefficient for charge-charge interactions [169]. Finally, we note that in this chapter we assumed that the nanopore is rigid. If this is not the case, then the dynamics of the fluctuations of the nanopore structure from its average configuration should be coupled to the Langevin equations for the motion of all the ions as described in [167]. This 9.6 Summary and Discussion 185 would result in a conditional Poisson equation with a conditional averaged structure of the nanopore, that depends on the location of the mobile ion inside it. It seems likely that gating can be described by such a coupled system of equations. Chapter 10

Maximum Entropy Formulation of the Kirkwood Superposition Approximation

The contents of this chapter were published in [174]

Using a variational formulation, we derive the Kirkwood superposition approximation for systems at equilibrium in the thermodynamic limit. We define the entropy of the triplet correlation function and show that the Kirk- wood closure brings the entropy to its maximal value. This approach leads to a different interpretation for the Kirkwood closure relation, usually explained by probabilistic considerations of dependence and independence of particles. The Kirkwood closure is generalized to finite volume systems at equilibrium by computing the pair correlation function in finite domains. Closure relations for high order correlation functions are also found using a variational approach. In particular, maximizing the entropy of quadruplets leads to the high order closure g1234 = g123g124g134g234/[g12g13g14g23g24g34] used in the BGY2 (Born-Green-Yvon 2) equations which are a pair of integral equations for the triplet and pair correlation functions.

10.1 Introduction

The pair correlation function is one of the cornerstones in the theory of simple liquids [12, 34, 7, 72, 67, 158]. Many thermodynamic properties of the fluid 10.1 Introduction 187 can be derived from the pair function. There are two main approaches to finding the pair function. The first approach is based on the Ornstein-Zernike integral equation and a closure relation for the direct and indirect correlation functions. Many closure relations fall into this category, such as the Percus- Yevick approximation (PY), the hypernetted chain approximation (HNC) and the mean spherical approximation (MSA). The second approach relies on the Bogoliubov-Born-Green-Kirkwood-Yvon (BBGKY) hierarchy, which relates the n-th correlation function with the n + 1-th correlation function, and assumes a closure relation that connects them. The Kirkwood superposition, that approximates the triplet correlation function by the product of the three pair correlation functions (n = 2)

g3(x1, x2, x3) = g2(x1, x2)g2(x1, x3)g2(x2, x3),

(see also eq. (10.42)), was the first suggested closure of this kind [99, 100]. Approximating the quadruplet correlation function by the triplet correlation function (n = 3) using the Fisher-Kopeliovich closure [53] eq. (10.43) turned out to yield a much better approximation for the pair correlation function than the Kirkwood superposition approximation (SA), as noted by Ree et al. [113, 151]. However, truncating the BBGKY hierarchy at a higher level (n ≥ 4) is computationally impractical at the moment. Both approaches and the underlying closure relations have their own advantages and disadvantages. Their success is often evaluated in comparisons with either molecular dynamics (MD) or Monte-Carlo (MC) simulations. All closures succeed in the limit of (very) low density. However, when the density is increased they eventually fail, sooner or later. Choosing the “best” closure relation is an art in itself. Obviously, each choice of closure relation results in a different approximate solution for the pair function. The BBGKY equation (with or without the SA) has a great advantage over any other approach. For dp hard spheres it predicts a point where = 0, and hence it predicts solid- dρ ification. Neither the PY or HNC theories can do this and so one expects the BBGKY approach to be superior at high densities, which is particularly important in applications to protein and channel biology [44]. Rice et al. [152, 153] improved the Kirkwood SA for systems of hard- spheres and Lennard-Jones 6-12 potential. Meeron [130] and Salpeter [160] proposed a formal expression for the triplet correlation function in the form

g3(x1, x2, x3) = g2(x1, x2)g2(x1, x3)g2(x2, x3) exp[τ(x1, x2, x3, ρ)], (10.1) 188 Maximum Entropy Formulation of the Kirkwood Superposition Approximation

P∞ n where τ(x1, x2, x3, ρ) ≡ n=1 ρ δn+3(x1, x2, x3). The coefficients δn+3(x1, x2, x3) consist of simple 123-irreducible diagrams. Rowlinson [159] evaluated analytically the term δ4 for a system of hard spheres. Rice [152, 153] evaluated the ρδ4 term δ5 numerically and showed that the Padéapproximation τ ≈ 1 − ρδ5 gives a better fit to the computer simulation of the hard-sphere system than the Kirkwood SA. However, Ree proved that this improvement to the Kirk- wood SA is inferior to the high order closure relation of Refs [113, 151], because the latter takes into account all the diagrams of Refs [152, 153] and more. Hu et al. [81] showed that the integral closure relations (PY, HNC and MSA) can be obtained from the Kirkwood SA using another approximation. In this chapter we derive the Kirkwood SA from a variational (Euler- Lagrange) formulation. We define an entropy functional for the triplet correlation function, and show that it produces the Kirkwood closure in the thermodynamic limit of number of particles and volume taken to infinity, while keeping the number density constant, i.e. N,V → ∞, N/V = ρ. The triplet entropy functional is different from the physical entropy of the full N-particle system. Maximizing the triplet entropy functional, given the constraint that the pair correlation function is the marginal of the triplet correlation function, yields the SA. This maximum entropy formulation leads to a different closure for a finite volume system which we write as an integral equation. In the thermodynamic limit, the solution of this integral equation reduces to the Kirkwood SA. We describe an iterative procedure for solving the BBGKY equation with this generalized entropy closure. Using this entropy closure instead of the Kirkwood SA produces different pair correlation functions. The entropy closure can be used to calculate the pair function in confined geometries. Specifically, we can deduce the resulting pair function near the domain walls (boundaries), and compare it to MC or MD simulations, and to the results of the original Kirkwood closure. The present attempt to find the pair correlation function using a variational formulation is not new. Richardson [154] used a maximum entropy (minimum Helmholtz free energy) argument to obtain an equation for the pair correlation function. That variational approach consists in maximizing the entropy while assuming an approximate relation (closure) between the pair and triplet function, and resulted in an integral equation which differs from the BGY equation. On the other hand, the variational approach applied in this chapter assumes that the BGY equation is exact, and maximizes 10.2 Maximum Entropy 189 the triplet entropy under the constraint that the pair correlation function is the marginal of the triplet function. This approach leads to a closure for the BGY equation. The resulting closure coincides with the Kirkwood SA for the radial function. Mathematically, the approach of Richardson [154] consists in solving a variational problem without imposing constraints, while the current proposed variational formulation includes constraints and is therefore solved by the method of Lagrange multipliers. The definition of the n-particle entropy (3 ≤ n ≤ N) is similar to the definition of the triplet entropy functional. The N-particle entropy coincides with the physical entropy and its maximization yields the Boltzmann distribution, as expected. High order closure relations are obtained by maximizing the n-particle entropy. In particular, maximizing the quadruplets entropy leads to the Fisher-Kopeliovich closure [53] and the corresponding BGY2 equation [113, 151], that has been shown to give a very good fit to experimental data.

10.2 Maximum Entropy

3 We consider a ﬁnite domain Ω ⊂ R . Suppose p2(x1, x2) is a known symmetric probability distribution function (pdf)

Z p2(x1, x2) dx1 dx2 = 1 Ω×Ω

p2(x1, x2) = p2(x2, x1) (10.2)

p2(x1, x2) ≥ 0,

where p2(x1, x2) represents the joint pdf of finding two particles at locations x1 and x2, as usually defined in the statistical mechanics of fluids. This means a relationship between p2(x1, x2) and p3(x1, x2, x3) can be expressed 190 Maximum Entropy Formulation of the Kirkwood Superposition Approximation

in the form of the constraints

Z φ1(p3) = p3(x1, x2, x3) dx1 − p2(x2, x3) = 0, Ω Z φ2(p3) = p3(x1, x2, x3) dx2 − p2(x1, x3) = 0, (10.3) Ω Z φ3(p3) = p3(x1, x2, x3) dx3 − p2(x1, x2) = 0. Ω

An approximate closure relation can be found by solving the optimization problem of maximizing the triplet entropy functional

Z H(p3) = − p3(x1, x2, x3) ln p3(x1, x2, x3) dx1 dx2 dx3, (10.4) Ω×Ω×Ω

under the constraints (10.3). Note that H(p3) diﬀers from the physical entropy of the full N-particle system

Z H ≡ H(pN ) = − pN (x1, x2,..., xN ) ln pN (x1, x2,..., xN ) dx1 dx2 ··· dxN , Ω

where pN (x1, x2,..., xN ) is the Boltzmann distribution. The motivation for maximizing the triplet entropy in finding a closure relation, is that the Boltzmann distribution brings the physical entropy to a maximum. Thus, maximizing H(p3) instead of H introduces errors and is expected to be an approximation to the closure problem. This issue will be further discussed in Section 10.4. The notation ”triplet entropy” for H(p3) of equation (10.4) is in agreement with the definition of the Shannon entropy of three random variables in information theory (see eg. Ref [31]). This variational problem can be solved by the method of Lagrange multipliers. Thus, we define Lagrange multipliers λ1(x2, x3), λ2(x1, x3), λ3(x1, x2) 10.2 Maximum Entropy 191 and the functional

F (p3, λ1, λ2, λ3) = H + λ1φ1 + λ2φ2 + λ3φ3

Z = − p3(x1, x2, x3) ln p3(x1, x2, x3) dx1 dx2 dx3 Ω×Ω×Ω Z +λ1(x2, x3) p3(x1, x2, x3) dx1 − p2(x2, x3) Ω Z +λ2(x1, x3) p3(x1, x2, x3) dx2 − p2(x1, x3) Ω Z +λ3(x1, x2) p3(x1, x2, x3) dx3 − p2(x1, x2) . Ω The Euler-Lagrange equation is

− ln p3(x1, x2, x3) − 1 + λ1(x2, x3) + λ2(x1, x3) + λ3(x1, x2) = 0, (10.5) or equivalently

p3(x1, x2, x3) = γ1(x2, x3)γ2(x1, x3)γ3(x1, x2), (10.6) where

λ1(x2,x3)−1/3 γ1(x2, x3) = e ,

λ2(x1,x3)−1/3 γ2(x1, x3) = e ,

λ3(x1,x2)−1/3 γ3(x1, x2) = e .

Clearly, γi ≥ 0 (i = 1, 2, 3), and therefore p3 ≥ 0. Moreover, γ1 = γ2 = γ3 because p2 is symmetric. Setting γ = γ1 = γ2 = γ3 we ﬁnd that

p3(x1, x2, x3) = γ(x1, x2)γ(x2, x3)γ(x1, x3). (10.7)

The constraint that p2 is the marginal of p3 gives an equation for γ, Z p2(x1, x2) = γ(x1, x2) γ(x1, x3)γ(x2, x3) dx3. (10.8) Ω 192 Maximum Entropy Formulation of the Kirkwood Superposition Approximation

The symmetry of p2 (eq.(10.2)) implies that of γ,

γ(x1, x2) = γ(x2, x1).

If equation (10.8) has a unique solution, then p2 determines γ. However, the pdf p2 is unknown. We know that p2 satisﬁes the BBGKY equation

0 = fex(x1)p2(x1, x2) + f(x2, x1)p2(x1, x2) − kBT ∇x1 p2(x1, x2) Z +(N − 2) f(x3, x1)p3(x1, x2, x3) dx3, Ω

where f(x2, x1) is the force exerted on a particle located at x1 by another particle located at x2; and fex(x1) is an external force ﬁeld acting on a particle located at x1. Substituting the maximum entropy closure (10.7) into the BBGKY equation (10.9), together with the integral equation (10.8), produces a system of integral equations for p2 and γ,

0 = fex(x1)p2(x1, x2) + f(x2, x1)p2(x1, x2) − kBT ∇x1 p2(x1, x2) Z +(N − 2) f(x3, x1)γ(x1, x2)γ(x2, x3)γ(x1, x3) dx3 Ω Z p2(x1, x2) = γ(x1, x2) γ(x1, x3)γ(x2, x3) dx3. (10.9) Ω

To determine the pair correlation function p2, we need to solve the system (10.9).

10.3 Two Examples

The system (10.9) can be solved or simpliﬁed in the case that the particles interact only with the external ﬁeld and in the thermodynamic limit. 10.3 Two Examples 193

10.3.1 Non-interacting particles in an external ﬁeld Non-interacting particles in an external ﬁeld are described by f ≡ 0 and the system (10.9) is reduced to

0 = fex(x1)p2(x1, x2) − kBT ∇x1 p2(x1, x2) (10.10) Z p2(x1, x2) = γ(x1, x2) γ(x1, x3)γ(x2, x3) dx3 (10.11) Ω Equation (10.10) can be integrated to yield

−U (x1)/kB T p2(x1, x2) = e ex h(x2), (10.12) where fex(x1) = −∇x1 Uex(x1), and h(x2) is an arbitrary function (the integration constant). The symmetry condition (10.2) gives

−1 −[U (x1)+U (x2)]/kB T p2(x1, x2) = C e ex ex , (10.13) Z 2 where C = e−Uex(x)/kB T dx is a normalization constant. As expected, Ω we ﬁnd that x1, x2 are independent random variables, p2(x1, x2) = p1(x1)p1(x2). In this case, the solution to (10.11) is given by p p p γ(x1, x2) = p2(x1, x2) = p1(x1) p1(x2). (10.14)

Indeed, Z γ(x1, x2) γ(x1, x3)γ(x2, x3) dx3 = Ω p Z p p = p1(x1)p1(x2) p1(x1)p1(x3) p1(x2)p1(x3) dx3 Ω Z = p1(x1)p1(x2) p1(x3) dx3 = p2(x1, x2). Ω

10.3.2 The Thermodynamic Limit The thermodynamic limit is described by the following limiting process in which the domain is gradually enlarged to be all space, Ω → R3, keeping 194 Maximum Entropy Formulation of the Kirkwood Superposition Approximation

the ratio between the number of particles, N, and the volume of the domain, V = |Ω|, constant. The ratio N/V = ρ is the number density. In this limiting process, all the pdfs tend to zero, because the volume of the domain tends to infinity, and the pdfs are normalized with respect to that volume. It is therefore more convenient to work with number densities, such as ρ, which are normalized with respect to a fixed volume, e.g., 1 cm3, and do not vanish in the limiting process. First, we consider a bounded domain Ω ⊂ R3. The previous example of non-interacting particles suggests the definition

γ(x1, x2) δ(x1, x1) = p , (10.15) p1(x1)p1(x2) which transforms equation (10.11) into Z p2(x1, x2) = δ(x1, x2) p1(x3)δ(x1, x3)δ(x2, x3) dx3. (10.16) p1(x1)p1(x2) Ω We rewrite equation (10.16) as

(Ω) p2 (x1, x2) (Ω) (Ω) = (10.17) p1 (x1)p1 (x2) Z (Ω) (Ω) (Ω) (Ω) δ (x1, x2) p1 (x3)δ (x1, x3)δ (x2, x3) dx3, Ω

(Ω) (Ω) (Ω) where p2 (x1, x2) = p2(x1, x2), δ (x1, x2) = δ(x1, x2), p1 (x1) = p1(x1), to emphasize their dependency on the speciﬁc domain Ω. We set

(Ω) p2 (x1, x2) g2(x1, x2) = lim . (10.18) Ω→ 3 (Ω) (Ω) R p1 (x1)p1 (x2) For example, if the two particles become independent when they are separated,

(Ω) (Ω) (Ω) p2 (x1, x2) = p1 (x1)p1 (x2) (1 + o(1)) , for |x1 − x2| 1, (10.19)

then lim|x2|→∞ g2(x1, x2) = 1. Next, we show that under the assumption (10.19)

(Ω) lim δ (x1, x2) = g2(x1, x2). 3 Ω→R 10.4 Minimum Helmholtz Free Energy 195

Indeed, Z Z p2(x1, x3) p2(x2, x3) p1(x3) dx3 = p1(x3) (1 + o(1)) dx3 Ω p1(x1)p1(x3) p1(x2)p1(x3) Ω

= 1 + o(1). (10.20) Taking the limit Ω → R3, the o(1) term vanishes, and equation (10.17) gives

δ(x1, x2) = g2(x1, x2), (10.21) as asserted. We interpret equation (10.21) as the Kirkwood SA. Equations (10.7) and (10.15) imply that the triplet pdf satisﬁes (Ω) p3 (x1, x2, x3) (Ω) (Ω) (Ω) (Ω) (Ω) (Ω) = δ (x1, x2)δ (x1, x3)δ (x2, x3). (10.22) p1 (x1)p1 (x2)p1 (x3) Taking the limit Ω → R3 and using equation (10.21), we obtain (Ω) p3 (x1, x2, x3) g3(x1, x2, x3) = lim Ω→ 3 (Ω) (Ω) (Ω) R p1 (x1)p1 (x2)p1 (x3)

= g2(x1, x2)g2(x1, x3)g2(x2, x3), (10.23) which is the Kirkwood closure relation for the triplet correlation function. Usually, the motivation for using the Kirkwood SA is a probabilistic consid- eration of dependency and independency of particles [99] (see also Section 10.5.) Here we ﬁnd another interpretation for the Kirkwood closure. The Kirkwood closure is the (only) closure relation that brings the entropy of triplets of particles to its maximum value. The principle of maximum entropy is a well known principle in statistical mechanics, in testing statistical hypotheses [114, 115] and beyond [155, 18]. The maximum entropy principle is also applicable to systems out of equilibrium [35]; therefore Kirkwood’s SA can be generalized to systems out of equilibrium. In the next section we give further motivation for its use.

10.4 Minimum Helmholtz Free Energy

Elementary textbooks in statistical mechanics mention that the Boltzmann distribution 1 −U(x1,...,xN )/kB T pN (x1,..., xN ) = e , (10.24) ZN 196 Maximum Entropy Formulation of the Kirkwood Superposition Approximation

brings the Helmholtz free energy

F (p) = U(p) − kBTH(p) (10.25) Z = U(x1,..., xN )p(x1,..., xN ) dx1 ··· dxN ΩN Z +kBT p(x1,..., xN ) ln p(x1,..., xN ) dx1 ··· dxN , ΩN to its minimum under the normalization constraint Z p(x1,..., xN ) dx1 ··· dxN = 1. (10.26) ΩN For a pairwise additive potential together with an external ﬁeld force

N X X U(x1,..., xN ) = U(xi, xj) + Uex(xj), (10.27) 1≤i

the potential energy term U(p) of the Helmholtz free energy (10.25) takes the simple form

N ! Z X X U(p) = U(xi, xj) + Uex(xj) p(x1,..., xN ) dx1 ··· dxN N Ω 1≤i

are the marginal densities. If the pdf p2(x1, x2) is assumed to be known, as in Section 10.2, then the energy term of the Helmholtz free energy U(p) is also known. Therefore, minimizing the Helmholtz free energy, under the assumption that the pdf p2(x1, x2) is known, is equivalent to maximizing the entropy, since U(p) is constant during the minimizing process. 10.5 Probabilistic Interpretation of the Kirkwood Closure 197

10.5 Probabilistic Interpretation of the Kirk- wood Closure

The Kirkwood SA (10.23) was the first closure relation to be suggested [99] and tested [100] in the theory of simple liquids. This fact might be explained by its simplicity and its intuitive origin. In this section we give a probabilistic interpretation of the Kirkwood SA, and find its generalization for closure relations of higher orders of the BBGKY hierarchy. The problem at level n (n ≥ 2) is to find an approximation for the n + 1-particle pdf in terms of the n-particle pdf. For example, the Kirkwood SA (10.23) closes the hierarchy at level n = 2. High order closures are much more accurate, fitting better the experimental and simulated (MC or MD) data [113, 151]. However, the computational complexity increases drastically with n, making the truncation at level n ≥ 4 impractical at the moment. First, consider the case n = 2. We assume that particles become independent as they are separated (10.19), although we are certainly aware that long range forces such as the electric field can produce strong correlations over all of even a very large domain, as already noted by Kirkwood in his original paper [99]. In order to make the exact equality (10.19) into an approximation, we assume that there exists a distance d > 0 such that

p2(x1, x2) = p1(x1)p1(x2), (10.28) for |x1 − x2| > d. Three interchangeable particles can be in four different configurations with respect to the distance d, depending on the number of intersections (see Figure 10.1). In all configurations but configuration (d), where all three particles intersect, there are at least two particles that do not intersect. Since the particles are interchangeable we may assume that |x1 − x3| > d. Applying Bayes’ law we have

p3(x1, x2, x3) = p3(x3|x1, x2)p2(x1, x2). (10.29) By the independence assumption (10.28) we have

p3(x3|x1, x2) = p2(x3|x2). (10.30) Therefore,

p2(x2, x3) p3(x1, x2, x3) = p2(x3|x2)p2(x1, x2) = p2(x1, x2). (10.31) p1(x2) 198 Maximum Entropy Formulation of the Kirkwood Superposition Approximation

Figure 10.1: Four conﬁgurations of three particles. (a) no intersections (b) one intersection (c) two intersections (d) three intersections. 10.5 Probabilistic Interpretation of the Kirkwood Closure 199

p (x , x ) Multiplying by 1 = 2 1 3 we obtain p1(x1)p1(x3)

p2(x1, x2)p2(x2, x3)p2(x1, x3) p3(x1, x2, x3) = , (10.32) p1(x1)p1(x2)p1(x3) which is the Kirkwood SA. We see that the Kirkwood closure is a good approximation when at least two particles are suﬃciently far apart and independent. However, it fails when all three particles are close to each other. Next, we ﬁnd the n-level Kirkwood closure relation for the n + 1-particle pdf in terms of the n-particle pdf, using probabilistic considerations.

Proposition: The n-level Kirkwood closure relation is given by

n−1 Y Y (−1)n−1−k pn(x1, x2,..., xn) = pk(xi1 , xi2 ,..., xik ) .

k=1 1≤i1

Proof: We have already seen that the approximation holds for n = 2, 3. Assuming, by induction, that at least two particles are suﬃciently far apart and independent, we can assume without loss of generality, that particles 1 and n are far apart, and independent, |x1 − xn| > d. Using Bayes’ law, we ﬁnd that

pn(x1, x2,..., xn) = pn−1(x1, x2,..., xn−1)pn(xn|x1, x2,..., xn−1). Because particles 1 and n are suﬃciently far apart it follows that

pn−1(x2, x3,..., xn) pn(xn|x1, x2,..., xn−1) = pn−1(xn|x2, x3,..., xn−1) = . pn−2(x2, x3,..., xn−1) Hence,

pn−1(x1, x2,..., xn−1)pn−1(x2, x3,..., xn) pn(x1, x2,..., xn) = . (10.34) pn−2(x2, x3,..., xn−1) It follows from the induction assumption that for every j = 2, 3, . . . , n − 1, with particles 1 and n far apart, we have

1 = pn−1(x1,..., xj−1, xj+1,..., xn) (10.35) n−2 Y Y (−1)n−1−k × pk(xi1 , xi2 ,..., xik ) .

k=1 1≤i1

Multiplying eqs.(10.34) and (10.35) (for all j = 2, 3, . . . , n−1) ends the proof.

Corollary: For n = 4 Kirkwood’s formula becomes

p4(x1, x2, x3, x4) = p1(x1)p1(x2)p1(x3)p1(x4) × (10.36) p (x , x , x )p (x , x , x )p (x , x , x )p (x , x , x ) 3 1 2 3 3 1 2 4 3 1 3 4 3 2 3 4 . p2(x1, x2)p2(x1, x3)p2(x1, x4)p2(x2, x3)p2(x2, x4)p2(x3, x4)

In the case Ω = R3, we deﬁne

pn(x1, x2,..., xn) gn(x1, x2,..., xn) = lim n . (10.37) 3 Q Ω→R j=1 p1(xj)

Qn Dividing equation (10.33) by j=1 p1(xj) gives

n−1−k n−1 !(−1) pn(x1, x2,..., xn) Y Y pk(xi , xi ,..., xi ) = 1 2 k , Qn Qk j=1 p1(xj) p1(xi ) k=1 1≤i1

n−1 X n − 1 (−1)n−1−k = 1. (10.39) k − 1 k=1

Note that the k = 1 terms in the product of equation (10.38) cancel out, so the product may begin from k = 2

n−1−k n−1 !(−1) pn(x1, x2,..., xn) Y Y pk(xi , xi ,..., xi ) = 1 2 k . Qn Qk j=1 p1(xj) p1(xi ) k=2 1≤i1

n−1 Y Y (−1)n−1−k gn(x1, x2,..., xn) = gk(xi1 , xi2 ,..., xik ) . (10.41)

k=2 1≤i1

Examples: 10.6 High Level Entropy Closure 201

• n = 3 (Kirkwood SA)

g3(x1, x2, x3) = g2(x1, x2)g2(x1, x3)g2(x2, x3). (10.42)

• n = 4 (Fisher-Kopeliovich [53])

g4(x1, x2, x3, x4) = (10.43)

g (x , x , x )g (x , x , x )g (x , x , x )g (x , x , x ) 3 1 2 3 3 1 2 4 3 1 3 4 3 2 3 4 . g2(x1, x2)g2(x1, x3)g2(x1, x4)g2(x2, x3)g2(x2, x4)g2(x3, x4) 10.6 High Level Entropy Closure

In this section we use the maximum entropy principle to derive the n-level closure relation, and compare the resulting closure relation with the n-level probabilistic Kirkwood closure of Section 10.5. The problem at level n (n ≥ 2) is to ﬁnd an approximation for the n + 1-particle pdf in terms of the n- particle pdf. For example, the Kirkwood SA (10.23) closes the hierarchy at level n = 2. We use the principle of maximum entropy to obtain the closure relation. As in the derivations of Section 10.2, we assume that the n-particle pdf pn(x1, x2,..., xn) is known, and we search for the n + 1-particle pdf pn+1(x1, x2,..., xn+1) that maximizes the entropy

H =

Z − pn+1(x1, x2,..., xn+1) ln pn+1(x1, x2,..., xn+1) dx1 dx2 ··· dxn+1, Ωn+1 with the n + 1 constraints that the pn are the marginals of pn+1

pn(x1, x2,..., xn) =

Z pn+1(x1,..., xj,..., xn+1) dxj, j = 1, 2, . . . , n + 1. Ω

Since p2 is the marginal of pn (n ≥ 2), it follows that p2 is also known. Therefore, for a pairwise additive potential, maximizing the Helmholtz free energy of the n + 1-particle system is equivalent to minimizing the entropy of the n + 1-particle system. Introducing the Lagrange multipliers 202 Maximum Entropy Formulation of the Kirkwood Superposition Approximation

λj(x1,..., xj−1, xj+1,..., xn+1), j = 1, 2, . . . , n+1, the Euler-Lagrange equation gives n+1 X − ln pn+1 − 1 + λj = 0. (10.44) j=1 Since the n particles are interchangeable,

n+1 Y pn+1(x1,..., xn+1) = γ(x1,..., xj−1, xj+1,..., xn+1). (10.45) j=1

Integration with respect to xn+1 yields

pn(x1,..., xn) = (10.46)

n Z Y γ(x1,..., xn) γ(x1,..., xj−1, xj+1,..., xn+1) dxn+1. Ω j=1 Solving the non-linear integral equation (10.46) for γ and substituting in equation (10.45) gives the n-level closure relation of the n-level BBGKY hierarchy equation for pn

n X 0 = fex(x1)pn(x1, x2,..., xn) + f(xj, x1)pn(x1, x2,..., xn) j=2

−kBT ∇x1 pn(x1, x2,..., xn) (10.47) Z +(N − n) f(xn+1, x1)pn+1(x1, x2,..., xn+1) dxn+1. Ω 10.6.1 The thermodynamic limit We have seen in Section 10.3 that the maximum entropy principle yields the Kirkwood SA (n = 2) in the thermodynamic limit. In this section we show that in the thermodynamic limit, the maximum entropy principle results in the probabilistic Kirkwood closure (10.33) for all levels n ≥ 2. First, we consider a bounded domain Ω ⊂ R3. We set (Ω) (Ω) pn (x1, x2,..., xn) hn (x1, x2,..., xn) = , Qn−1 Q p(Ω)(x , x ,..., x )(−1)n−1−k k=1 1≤i1

(Ω) (Ω) γ (x1, x2,..., xn) δ (x1, x2,..., xn) = . n−1 (Ω) (−1)n−1−k n−k Q Q p (x , x ,..., x ) n−k+1 k=1 1≤i1

(Ω) (Ω) hn (x1, x2,..., xn) = δ (x1, x2,..., xn) × (10.50)

Z n Y (Ω) F (x1, x2,..., xn, xn+1) δ (x1,..., xj−1, xj+1,..., xn+1) dxn+1, Ω j=1 where

F (x1, x2,..., xn, xn+1) n−2 Y Y (Ω) (−1)n−1−k(n−k−1) = pk (xi1 , xi2 ,..., xik ) k=1 1≤i1

For min1≤j≤n |xn+1 − xj| 1, the n + 1-th particle becomes independent of particles 1, 2, . . . , n, and we have

(Ω) pk (xi1 , xi2 ,..., xik−1 , xn+1) =

(Ω) (Ω) p1 (xn+1)pk−1(xi1 , xi2 ,..., xik−1 ) (1 + o(1)) ,

for all k = 1, 2, . . . , n and all sets of indices 1 ≤ i1 < i2 < . . . < ik−1 ≤ n. 204 Maximum Entropy Formulation of the Kirkwood Superposition Approximation

−1 Therefore, to leading order in (min1≤j≤n |xn+1 − xj|)

F (x1, x2,..., xn, xn+1) =

n−2 Y Y (Ω) (−1)n−1−k(n−k−1) pk (xi1 , xi2 ,..., xik ) × k=1 1≤i1

k=1 1≤i1

n−1    X n  (−1)n−1−k(n − k)  k − 1  h (Ω) ik=1  = p1 (xn+1)

(Ω) = p1 (xn+1), (10.52)

where we have used the combinatorial identity

n−1 X n (−1)n−1−k(n − k) = 1. (10.53) k − 1 k=1 We conclude that Z (Ω) (Ω) (Ω) hn (x1, x2,..., xn) = δ (x1, x2,..., xn) p1 (xn+1) (1 + o(1)) Ω n Y (Ω) × δ (x1,..., xj−1, xj+1,..., xn+1) dxn+1. j=1

Therefore,

(Ω) (Ω) lim δ (x1, x2,..., xn) = lim h (x1, x2,..., xn) (10.54) 3 3 n Ω→R Ω→R 10.6 High Level Entropy Closure 205

Substituting eq. (10.49) in eq. (10.45), using the relations (10.48) and (10.54), and the deﬁnition (10.37), we ﬁnd that n Y Y (−1)n−k gn+1(x1,..., xn+1) = gk(xi1 , xi2 ,..., xik ) .

k=2 1≤i1

10.6.2 Closure at the highest level n = N − 1 Although the probabilistic Kirkwood closure and the maximum entropy closure agree in the case of systems in the entire space, they differ in the case of confined systems with a finite number of particles N. It appears that the maximum entropy closure (10.45) is exact in a confined system when applied at the highest level n = N − 1, whereas the probabilistic Kirkwood closure (10.33) is not exact. In other words, the maximum entropy closure relation yields the Boltzmann distribution (10.24). What appeared at first to be an approximation, turns out to be the exact result. Indeed, setting 1 1/N γ(x1, x2,..., xN−1) = × (10.56) ZN

( " N−1 # ) 1 X 1 X exp − U(xi, xj) + Uex(xj) /kBT , N − 2 N − 1 1≤i

N Y 1 γ(x ,..., x , x ,..., x ) = e−U(x1,...,xN )/kB T , (10.57) 1 j−1 j N Z j=1 N which is the Boltzmann distribution. The Boltzmann distribution obviously satisﬁes the BBGKY equation (10.47). Therefore, we have found a solution to BBGKY equation which satisﬁes the closure relation (10.45). This solution coincides with the Boltzmann distribution, and so we conclude that it is the exact solution. For the N − 1-particle pdf pN−1 we have 1 Z −U(x1,...,xN )/kB T pN−1(x1,..., xN−1) = e dxN , (10.58) ZN Ω 206 Maximum Entropy Formulation of the Kirkwood Superposition Approximation

which is both the exact and “approximate” result. The probabilistic Kirkwood approximation (10.33) gives however a different result. If it were to be exact, then the resulting N − 1-particle pdf would have been given by equation (10.58). Therefore, all lower level pdf’s pn (n ≤ N − 1) would be (multiple) integrals of the Boltzmann distribution. Therefore, by the closure (10.33), pN should be a product of integrals of the Boltzmann distribution, which is a contradiction to the known form of the Boltzmann distribution (10.24). Although the maximum entropy closure is exact at the highest order n = N −1, it is not exact at lower orders. The observation that the maximum entropy closure is exact at the highest level, while the probabilistic Kirkwood closure is not, may indicate that the maximum entropy closure may ﬁt the experimental data better than the probabilistic Kirkwood closure does in conﬁned systems, even when used at lower levels (n = 2, 3).

10.7 Conﬁned Systems

Systems in bounded domains are particularly important, because only bounded domains can include the spatially nonuniform boundary conditions needed to describe devices, with spatially distinct inputs, outputs, and (sometimes) power supplies. A large fraction of electrochemistry involves such devices as batteries or concentration cells. A large fraction of molecular biology involves such devices as transport proteins that move ions across otherwise impermeable membranes [45, 46]. In the general case, where Ω ⊂ R3 is a bounded domain, there is no known analytic solution to the system (10.9). We propose to solve this system by the following iterative scheme:

(0) 1. Initial guess γ (x1, x2). Set i = 0.

(i) 2. Solve the non-homogeneous linear equation for p2 (x1, x2)

(i) (i) (i) kBT ∇x1 p2 (x1, x2) − f(x2, x1)p2 (x1, x2) − fex(x1)p2 (x1, x2) Z (i) (i) (i) = (N − 2) f(x3, x1)γ (x1, x2)γ (x2, x3)γ (x1, x3) dx3. Ω 10.7 Conﬁned Systems 207

3. Solve the non-linear system for γ(i+1) Z (i) (i+1) (i+1) (i+1) p2 (x1, x2) = γ (x1, x2) γ (x2, x3)γ (x1, x3) dx3. Ω (10.59)

4. i ← i + 1. Return to step 2, until convergence is achieved.

The analysis of the preceding section indicates that a good initial guess might be (0) p γ (x1, x2) = p1(x1)p1(x2) g2(x1, x2), (10.60) where g2(x1, x2) is the solution to the BBGKY equation with the Kirkwood SA in the entire space R3. This solution can be found rather easily using the inherited symmetries of the problem. For example, it is well known that if fex = 0, then g2(x1, x2) = g2(|x1 − x2|), and the problem for g2 becomes one dimensional. Step 2 requires the solution of a linear partial diﬀerential equation in a bounded region. This equation can be written in a gradient form as

(U(x1,x2)+Uex(x1))/kB T ∇x1 e p2(x1, x2) =

N − 2 Z (U(x1,x2)+Uex(x1))/kB T − e ∇x1 U(x1, x3)p3(x1, x2, x3) dx3. kBT Ω

The identity ∇x1 × ∇x1 u(x1) = 0, for all u, imposes a solvability condition for γ. Indeed, taking the curl of the last equation, together with the closure (10.7) results in

0 = ∇x1 ×

Z (U(x1,x2)+Uex(x1))/kB T e ∇x1 U(x1, x3)γ(x1, x2)γ(x1, x3)γ(x2, x3) dx3 . Ω In step 3 we solve a non-linear integral equation. We suggest solving the non-linear equation (10.8) by a Newton-Raphson iterative scheme. Let (n) (n) 2 2 γ (x1, x2) be the n-th iteration. Deﬁne the operator Γ :Ω → Ω as follows Z Γ(n)u(x, z) = γ(n)(x, y)u(z, y) dy. (10.61) Ω 208 Maximum Entropy Formulation of the Kirkwood Superposition Approximation

Let the operator S :Ω2 → Ω2 be the symmetrization operator

Su(x, y) = u(y, x). (10.62)

The Newton-Raphson iteration scheme suggests

(n+1) (n) γ = γ + ∆(x1, x2), (10.63)

where ∆(x1, x2) satisﬁes the linear integral equation

(n) (n) p2 (x1, x2) p2(x1, x2) − p2 (x1, x2) = (n) ∆(x1, x2) γ (x1, x2)

(n) (n) +γ (x1, x2)Γ ∆(x1, x2)

(n) (n) +γ (x1, x2)SΓ ∆(x1, x2), where Z (n) (n) (n) (n) p2 (x1, x2) = γ (x1, x2) γ (x1, x3)γ (x2, x3) dx3. (10.64) Ω We may write the iteration equivalently as !−1 p(n) γ(n+1) = γ(n) + 2 + γ(n)Γ(n) + γ(n)SΓ(n) p − p(n) . (10.65) γ(n) 2 2

The steps in the algorithm are iterated until convergence is achieved. We have yet to test our generalized Kirkwood closure in practice. The resulting pair correlation function should be compared with MD or MC simulations of particles in a conﬁned region. The observation of Subsection 10.6.2 and the generality of the maximum entropy principle (minimum Helmholtz free energy) may indicate that it will outperform the regular Kirkwood SA in bounded domains. The diﬀerence between the results of the two closure methods should be seen near the boundary walls of the domain.

10.8 Mixtures

The maximum entropy principle can also be used to ﬁnd closure relations of mixtures, both in conﬁned domains and in the entire space. Suppose a 10.8 Mixtures 209

mixture of S ≥ 2 species, with Nα (α = 1, 2,...,S) particles of each species. PS Let N = α=1 Nα be the total number of particles of all specie. There are S2 2-particle pdfs,

αβ p2 (x1, x2) α, β ∈ {1, 2,...,S},

αβ βα that exhibit the symmetry p2 (x1, x2) = p2 (x2, x1). In this section we brieﬂy discuss how to ﬁnd the closure relation in the mixture problem. In the maximum entropy approach, one is searching for S3 3-particle pdfs αβγ p3 (x1, x2, x3), α, β, γ ∈ {1, 2,...,S}, that bring the entropy S Z X Nα Nβ Nγ H = − pαβγ(x , x , x ) ln pαβγ(x , x , x ) dx dx dx N N N 3 1 2 3 3 1 2 3 1 2 3 α,β,γ=0 Ω (10.66) to maximum, with the 3S3 marginal constraints Z αβ αβγ p2 (x1, x2) = p3 (x1, x2, x3) dx3, (10.67) Ω Z αγ αβγ p2 (x1, x3) = p3 (x1, x2, x3) dx2, Ω Z βγ αβγ p2 (x2, x3) = p3 (x1, x2, x3) dx1. Ω This variational problem is solved using the Euler-Lagrange formulation similar to the derivation done in Section 10.2. In the case of a system in the entire space, the methods of subsections 10.3.2 and 10.6.1 show that the mixture entropy closure coincides with the probabilistic Kirkwood closure. Both subsections suggest that the triplets correlation functions are related to the pair correlation function through

αβγ αβ αγ βγ g3 (x1, x2, x3) = g2 (x1, x2)g2 (x1, x3)g2 (x2, x3), (10.68) for α, β, γ ∈ {1, 2,...,S}. Closures of higher orders can be obtained in a similar manner. In conﬁned systems, the Euler-Lagrange formulation leads to integral equations of the form (10.8). Note that since the entropy (10.66) depends N on the particle fraction α , we expect the resulting conﬁned system pair N correlation will also depend on the particle fraction. 210 Maximum Entropy Formulation of the Kirkwood Superposition Approximation

10.9 Discussion and Summary

We have used the maximum entropy principle to derive a closure relation for the BBGKY hierarchy. It is possible to consider functionals over distributions other than the entropy that can yield different (known) closures. In fact, a somewhat similar approach is used in density functional theory (DFT). In the DFT problem setup, a functional of the pair correlation function (e.g., the Helmholtz free energy) is maximized. The function that brings the given functional to its maximal value is the resulting pair correlation function. Using this method, one can recover some of the Ornstein-Zernike integral closures that relate the direct and indirect correlation functions, such as the PY closure, for instance. Our approach differs from that of the DFT in that we optimize over relations between the probability correlation (density) functions of successive orders, rather than over relations between the direct and indirect correlation functions. In this chapter we have used the maximum entropy principle to derive a closure relation for the BBGKY hierarchy. This approach to the closure problem appears to be new. We proved that for systems in the entire space, the maximum entropy closure relation coincides with the probabilistic Kirk- wood SA for all orders of the hierarchy. In finite systems the maximum entropy closure differs from Kirkwood’s SA. In particular, when applied to the highest level of the hierarchy, the maximum entropy closure is exact, whereas the probabilistic Kirkwood approximation is not. Besides the advantage of generality, the maximum entropy closures are expected to perform better than the Kirkwood SA, even in low order approximations. We expect the differences between the pair correlation functions predicted by the two methods to be significant especially near the domain boundaries. The maximum entropy closure may be applicable to non-equilibrium systems, and in particular to systems with spatially inhomogeneous boundary conditions. That implementation of the maximum entropy closure will be the subject of a separate paper. Chapter 11

Attenuation of the Electric Potential and Field in Disordered Systems

The contents of this chapter were published in [176]

We study the electric potential and field produced by disordered distributions of charge to see why clumps of charge do not produce large potentials or fields. The question is answered by evaluating the probability distribution of the electric potential and field in a totally disordered system that is overall electroneutral. An infinite system of point charges is called totally disordered if the locations of the points and the values of the charges are random. It is called electroneutral if the mean charge is zero. In one dimension, we show that the electric field is always small, of the order of the field of a single charge, and the spatial variations in potential are what can be produced by a single charge. In two and three dimensions, the electric field in similarly disordered electroneutral systems is usually small, with small variations. Interestingly, in two and three dimensional systems, the electric potential is usually very large, even though the electric field is not: large amounts of energy are needed to put together a typical disordered configuration of charges in two and three dimensions, but not in one dimension. If the system is locally electroneutral—as well as globally electroneutral—the potential is usually small in all dimensions. The properties considered here arise from the superposition of electric fields of quasi-static distributions of charge, as in nonmetallic solids or ionic solutions. These properties are found 212 Attenuation of the Electric Potential and Field in Disordered Systems

in distributions of charge far from equilibrium.

11.1 Introduction

There is no danger of electric shock when handling a powder of salt or when dipping a finger in a salt solution, although these systems have huge numbers of positive and negative charges. It seems intuitively obvious that the alternating arrangement of charge in crystalline Na+Cl− should produce electric fields that add almost to zero; it also seems obvious that Na+ and Cl− ions will move in solution to minimize their equilibrium free energy and produce small electrical potentials. But what about random arrangements of charge that occur in a random quasi-static arrangement of charge such as a snapshot of the location of ions in a solution? Tiny imbalances in charge distribution produce large potentials, so why doesn’t a random distribution of charge produce large potentials, particularly if the distribution is not at thermodynamic equilibrium? Indeed, some arrangements of charge produce arbitrarily large potentials, but as we shall see, these distributions occur rarely enough that the mean and variance of stochastic distributions are usually finite and small. More specifically, we determine the conditions under which stochastic distributions of fixed charge produce small fields. The quasi-static arrangements of charge can represent the fixed charge in amorphous non-metallic solids or snapshots of charge arrangement of ions in solution, due to their random (Brownian) motion. Our analysis does not apply to quantum systems [17], and in particular it fails if electrons move in delocalized orbitals, as in metals. Note that the random arrangements of charge considered here do not necessarily minimize free energy. We consider the field and potential in overall electroneutral random configurations of infinitely many point charges. An infinite system of point charges is called totally disordered if the locations of the points and the charges are random, and it is called overall electroneutral if the mean charge is zero. The configurations of charge may be static or quasi-static, that is, time dependent, but varying sufficiently slowly to avoid electromagnetic phenomena: the electric potential is described by Coulomb’s law alone. In one dimensional systems of this type, the potential is usually finite—even though the system usually contains an infinite number of positive and negative charges. Even if the system is disordered and spatially random, charges of the same sign do not clump together often enough to produce large fields 11.1 Introduction 213 or potentials, in one dimensional systems. Our approach is stochastic. We ask how disordered can a random electroneutral system be, yet still have a small field or potential. We find the answer by evaluating the probability distribution of the electric potential and field of a disordered system of charges. We find that the electric field in a totally disordered one dimensional system is small whether the system is locally electroneutral or not. The potential behaves differently; it can be arbitrarily large in a one dimensional system, but it is usually small in electroneutral systems. In two or three dimensional disordered systems, the electric field is not necessarily small. We show that in such systems that are also electroneutral the field is usually small. The potential, however, is usually large, even if the system is electroneutral. Both potential and field are small, if the system is locally—as well as globally—electroneutral (see definition below) in one, two and three dimensions. We consider several types of random arrays of charges: (a) A lattice with random distances between two nearest charges; (b) A lattice (of random or periodic structure) with a random distribution of positive and negative charges (charge ±1). Charges in the lattice need not alternate between positive and negative, nor need they be periodically distributed; (c) A lattice (of random or periodic structure) with random charge strengths. Not all charges are ±1, but they are chosen from a set q1, q2, . . . , qn with probabilities p1, p2, . . . , pn, respectively, such that

n X qipi = 0. (11.1) i=1

Equation (11.1) is our deﬁnition of electroneutrality in an inﬁnite system. We use renewal theory [97], perturbation theory [10], and saddle point approximation [90] to calculate the electric potential of one dimensional systems of charges and show that it is usually small. That is to say, the probability is small that the potential takes on large values. Thus, randomly distributed particles produce small potentials even in disordered systems in one dimension, if the system is electroneutral. The analysis of one dimensional systems requires the calculation of the probability density function (pdf) of weighted independent identically distributed (i.i.d.) sums of random variables. This pdf looks like the normal distribution near its center, but the tail distribution has the double exponential decay of the log-Weibull distribution [156]. 214 Attenuation of the Electric Potential and Field in Disordered Systems

We conclude that the electric potential of totally disordered electroneutral one dimensional systems is necessarily small, comparable to that of a single charge. Later in the chapter, we define local electroneutrality precisely and show that two and three dimensional systems with local electroneutrality usually have small potentials, because the potential of a locally neutral system of charges decays like the potential of a point dipole, as 1/r2. We show that the potential of typical totally disordered arrays of charges in two and three dimensions is infinite even if the system is electroneutral. Historically, little attention seems to have been paid to quasi-static random arrangements of charge, although much attention has been paid to the equilibrium arrangements of mobile charge. In systems of mobile charges, such as liquids and ionic solutions, the decay of the electric potential may even be exponential, after the mobile charges assume their equilibrium distribution. The early theory of Debye-Hückel [7] shows a nearly exponential decay (with distance from a given particle) of the average electric potential at equilibrium, originally found by solving the linearized Poisson-Boltzmann equation. In classical physics, perfect screening of multipoles (of all orders) occurs in both homogeneous and inhomogeneous systems at equilibrium in the thermodynamic limit, when boundary conditions at infinity are chosen to have no effect [129] and there is no flux of any species. This type of screening in electrolytic solutions is produced by the equilibrium configuration of the mobile charges [72, 125], which typically takes 100 psec to establish (compared to the 10−16 time scale of most atomic motions) [8]. Many other systems are screened by mobile charges after they assume their equilibrium configuration of lowest free energy [23], such as ionic solutions, metals and semiconductors. The spatial decay of potential in ionic solutions determines many of the properties of ionic solutions and is a striking example of screening or shielding. “Sum rules” of statistical mechanics [72, 125] describe these properties. These rules depend on the system assuming an equilibrium distribution, which can only happen if the charges are mobile. We consider finite and infinite systems of charges which may or may not be mobile and which are not necessarily at equilibrium. We show that the potential of a finite disordered locally electroneutral system is attenuated to the potential of a single typical charge, whether the potential is evaluated inside or outside a finite system or in an infinite system. We note that the behavior of the electric potential and field outside the line or plane of 11.2 A One-Dimensional Ionic Lattice 215 the lattice can be analyzed in a straightforward manner by the methods developed below.

11.2 A One-Dimensional Ionic Lattice

Consider a semi-inﬁnite array of alternating electric charges ±q with a distance d between neighboring charges. The electric potential Φ at a point P , located at a distance R from and to the left of the ﬁrst charge (see Fig. 11.1) is given by

q 1 1 1 1 Φ = − + − + ··· 4πε0 R R + d R + 2d R + 3d q 1 1 1 = 1 − + − + ··· 4πε0R 1 + a 1 + 2a 1 + 3a ∞ q X (−1)n = , (11.2) 4πε R 1 + na 0 n=0 where a = d/R is a dimensionless parameter. The series (11.2) is condition- ally convergent, so it can be summed to any value by changing the order of summation [108]. The order of summation reflects the order of construction of the system; different orders may lead to different potential energies of the system. However, the infinite series that determines the electric field

∞ q X (−1)n E = 4πε R2 (1 + na)2 0 n=0 is absolutely convergent, so the field does not depend on the order of summation of its defining series. Thus, all potentials differ from each other by a constant, which presumably reflects the different ways the charge distribution could be constructed, while having the same electric field. From here on, we consider the ordering in equation (11.2). Setting R = d (a = 1) we find the potential at a vacant lattice point (to avoid infinite potentials) due to charges located at both directions of the infinite lattice is

∞ q X (−1)n−1 q 2Φ(R = d) = 2 = · 2 log 2. 4πε d n 4πε d 0 n=1 0 216 Attenuation of the Electric Potential and Field in Disordered Systems

Figure 11.1: A semi inﬁnite lattice of alternating charges with a distance d between neighboring charges. The point P is located at a distance R from and to the left of the ﬁrst charge.

The constant 2 log 2 is known as the Madelung constant of a one dimensional lattice [101]. Next we find the asymptotic behavior of the potential Φ away from the semi-infinite lattice, that is for R d, or equivalently a 1. The following analysis is independent of the order of summation of the series (11.2). Clearly, the infinite sum in eq. (11.2) converges, because it is an alternating sum with a decaying general term. We expand the potential for a 1 (away from the lattice) in the asymptotic form

q 1 2 Φ = V0 + aV1 + a V2 + ··· . (11.3) 4πε0 R The eﬀect of the ﬁrst charge can be separated from all the others,

q 1 q 1 2 Φ = − V0 +aV ˜ 1 +a ˜ V2 + ··· , (11.4) 4πε0 R 4πε0 R + d where d a a˜ = = . R + d 1 + a Comparing eqs.(11.3) and (11.4) we obtain " # 1 a a 2 V + aV + a2V + ··· = 1 − V + V + V + ... . 0 1 2 1 + a 0 1 + a 1 1 + a 2

The coefficients V0,V1,... are found by equating the coefficients of like powers of a. In particular, we find that V0 = 1/2,V1 = 1/4,V2 = 0, so the potential 11.2 A One-Dimensional Ionic Lattice 217 has the asymptotic form

q 1 1 1 Φ = + a + O a3 . (11.5) 4πε0 R 2 4

All coeﬃcients Vn can easily be computed in a similar fashion. This result also determines the rate at which the potential far away reaches its limiting 1 q 1 value, . The divergent series for x = 1 has the value V0 = 2 if 2 4πε0R interpreted as a limit using the Abel sum [108]

∞ X 1 1 1 − 1 + 1 − 1 + 1 − 1 + ... = lim (−1)nxn = lim = . x→1− x→1− 1 + x 2 n=0

We note that the asymptotic expansion (11.5) can also be found directly from the diﬀerential equation that the sum

∞ X (−1)n y(x) = xn 1 + na n=0 satisﬁes [131] 1 axy0 + y = , (11.6) 1 + x with initial condition y(0) = 1. The asymptotic form of y(x) can easily be found by standard methods [10]. In particular,

∞ X (−1)n lim y(x) = . x→1− 1 + na n=0

The physical interpretation of the asymptotic expansion (11.5) is that the electric potential away from an infinite lattice of charged particles is about the same as if half a single charge were located at the origin. The spatial arrangement of the lattice attenuates the effect of its charge. The potential near the lattice is determined by a few of the nearest charges and the contribution of the remaining charges reduces to that of a half charge placed at a distance R d. Obviously, as R → 0 the potential becomes infinite, approaching the potential produced by just the nearest charge. 218 Attenuation of the Electric Potential and Field in Disordered Systems

11.3 One Dimensional Random Ionic Lattice

We turn now to solids in which the charges are distributed randomly in several different ways. First, consider a semi-infinite lattice of electric charges, in which the sign of each charge is determined randomly by a flip of a fair coin. That is, the charges that are located at the lattice points Xn (n = 0, 1, 2,...) are independent Bernoulli random variables that take the values ±1 with probability 1/2. The electric potential of this random lattice is given by

∞ q X Xn Φ = . (11.7) 4πε R 1 + na 0 n=0 Some discussion of the nature of convergence of the series (11.7) is needed at this point. The convergence of the sum of variances means that the partial sums converge in L2 with respect to the probability measure, so the sum (11.7) exists as a random variable Φ ∈ L2, whose variance is the sum of the variances. Now, the Cauchy-Schwarz inequality implies that Φ ∈ L1, so hΦi = 0. Note that (11.7) also converges with probability 1 [15]. We use fair coin tossing to maintain the condition of global electroneutrality, though arbitrary long runs of positive or negative charges occur in this distribution. Thus some realizations of the sequence Xn have runs (‘clumps’) of substantial net charge and potential. The standard deviation of the net charge in a region gives some feel for the size of the clumps. The√ standard deviation in the net charge of a region containing N charges is q N. For large values of N, substantial regions are not charge neutral. The condition of local charge neutrality (defined later) is violated for many of the realizations of charge in this distribution. Note that a particular set of Xn can produce an infinite potential, despite our general conclusions. If, for example, Xn = 1 for all n, the electric poten- ∞ X 1 tial becomes infinite (see eq. (11.7)), because = ∞. Nonetheless, 1 + na n=0 the L2 convergence of (11.7) implies that the probability that (11.7) is infinite is 0. In other words, even though the potential is infinite for a particular set of Xn, the potential is finite with probability 1. This is a striking example of the attenuation of the electric field, even without mobile charge. The attenuation of the potential produced by some ‘clumpy’ configurations of charges occurs even though there is no correlation in position, and there is no motion whatsoever. 11.3 One Dimensional Random Ionic Lattice 219

The electric ﬁeld, given by

∞ q X Xn E = − , 4πε R2 (1 + na)2 0 n=0 remains ﬁnite for all realizations of Xn, because the sum

∞ q X 1 S = 4πε R2 (1 + na)2 0 n=0 converges. The electric field is bounded (above and below) by S and so there is zero probability that the function is outside the interval (−S,S). The pdf of the electric field is compactly supported, even when all charges are positive (or negative). The electric field—unlike the potential—is attenuated even if the net charge of the system is not zero, taken as a whole. The standard ( ∞ )1/2 q X 1 deviation of the field is , which is of the order of 4πε R2 (1 + na)4 0 n=0 the field of a single charge at a distance R.

11.3.1 Moments The expected value of Φ is hΦi = 0, as mentioned above. The variance of Φ is given by 2 ∞ q X 1 V ar (Φ) = . (11.8) 4πε R (1 + na)2 0 n=0 A vacant lattice point in an inﬁnite (not semi-inﬁnite) lattice corresponds to R = d for both the charges to the right and to the left. It follows that the variance of the potential there is twice that given in (11.8) with a = 1, that is, 2 ∞ 2 q X 1 q π2 V ar (Φ) = 2 = 2 , (11.9) 4πε d n2 4πε d 6 0 n=1 0 so that the standard deviation is q π σΦ = √ . (11.10) 4πε0d 3 220 Attenuation of the Electric Potential and Field in Disordered Systems

√ As expected, the constant π/ 3 is larger than the Madelung constant 2 log 2 of the periodic lattice, because the potential of the disordered system is larger than that of the ordered one. Away from the semi inﬁnite lattice, i.e., for a 1, we can approximate the variance (11.8) by the Euler-Maclaurin formula, which replaces the sum by an integral,

q 2 Z ∞ 1 1 V ar (Φ) = 2 dx + + O(a) 4πε0R 0 (1 + ax) 2 q 2 1 1 = + + O(a) , (11.11) 4πε0R a 2 so the standard deviation is q σφ|R = √ (1 + O(a)) . (11.12) 4πε0 dR 1 1 The decay law of √ is more gradual than the decay law of a single R R charge.

11.3.2 The electrical potential as a weighted i.i.d. sum P The potential (11.7) is a weighted sum of the form anXn, where Xn are i.i.d. random variables. The distribution of potential is generally not normal. P∞ −n For example, consider the weighted sum n=1 2 Xn, where Xn are the same Bernoulli random variables. This weighted sum represents the uniform distribution in the interval [−1, 1]. It is, in fact equivalent to the binary representation of real numbers in the interval. Not only does this distribution not look like the Gaussian distribution for small deviations, it does not look at all Gaussian for large deviations. In fact, this distribution has compact support. It is zero outside a ﬁnite interval, without the tails of the better endowed Gaussian. Other unusual limit distributions can be easily obtained P∞ −n from sums of the form (11.7). For example, the weighted sum n=1 3 Xn is equivalent to the uniform distribution on the Cantor “middle thirds” set [195] in [−1, 1], whose Lebesgue measure (length) is 0. Note that the sum ∞ X Xn (1 + na)1+ε n=0 11.3 One Dimensional Random Ionic Lattice 221 has compact support for every ε > 0, because the series

∞ X 1 (1 + na)1+ε n=0 converges for every ε > 0. In our case ε = 0, so that the limit distribution does not necessarily have compact support. Nonetheless, we expect that the probability distribution function of the potential will have tails that decay steeply, even steeper than those of the normal distribution.

11.3.3 Large and small potentials. The saddle point approximation The existence of the first moment of the sum (11.7) depends on its tail distribution, which we calculate below by the saddle point method [90]. That is, we calculate the chance of finding a pinch of (noncrystalline) salt with a very large potential. For a potential Φ defined in equation (11.7), we denote q −1 the pdf of Φ by f(x). The Fourier transform fˆ(k) of this pdf is 4πε0R given by the infinite product

∞ Y k fˆ(k) = cos , (11.13) 1 + na n=0 which is an entire function in the complex plane, because the general term is 1 + O(n−2). The inverse Fourier transform recovers the pdf

1 Z ∞ f(x) = fˆ(k)eikx dk, (11.14) 2π −∞ which we want to evaluate asymptotically for large x. Setting

∞ X k g(k, x) = log cos + ikx, (11.15) 1 + na n=0 we write 1 Z ∞ f(x) = exp{g(k, x)} dk. (11.16) 2π −∞ 222 Attenuation of the Electric Potential and Field in Disordered Systems

d The saddle point is the point k for which g(k, x) = 0. Differentiating dk equation (11.15) with respect to k, we find that k ∞ tan d X 1 + na g(k, x) = − + ix. (11.17) dk 1 + na n=0 We look for a root of the derivative on the imaginary axis, and substitute k = is. The vanishing derivative condition of the saddle point method is then s ∞ tanh X 1 + na x = . (11.18) 1 + na n=0 The infinite sum on the right hand side represents a monotone increasing function of s in the interval 0 < s < ∞, so equation (11.18) has exactly one solution for every x. Near the saddle point k = is, we approximate g(k) by its Taylor expansion up to the order 1 d2 g(k) ≈ g(is) + g(is)(k − is)2, (11.19) 2 dk2 to find the leading order term of the full asymptotic expansion (derivatives of higher order of the Taylor expansion can be used to find all terms of the asymptotic expansion [21]). We use the Cauchy integral formula to calculate our Fourier integral (11.16) on the line parallel to the real k axis through k = is (see Fig. 11.2) 1 Z ∞ 1 f(x) ≈ eg(is) exp g00(is)(k − is)2 dk 2π −∞ 2 1 Z ∞ z2 eg(is) = eg(is) exp g00(is) dz = . (11.20) p 00 2π −∞ 2 −2πg (is) Equation (11.18) has no analytic solution, so we construct asymptotic approximations for large and small values of s separately.

11.3.4 Tail asymptotics Throughout this subsection we assume that a is small and s is large and we ﬁnd the tail asymptotics of the pdf away from the system (for a 1). For 11.3 One Dimensional Random Ionic Lattice 223

Figure 11.2: The integration contour passes through the saddle point k = is in the complex plane. 224 Attenuation of the Electric Potential and Field in Disordered Systems

s 1 the Euler-Maclaurin sum formula gives s tanh Z ∞ 1 + ax 1 x = dx + tanh s + O(a). (11.21) 0 1 + ax 2 s Substituting z = , we obtain 1 + ax 1 Z s tanh z 1 x = dz + tanh(s) + O(a). (11.22) a 0 z 2 Writing Z s tanh z Z 1 tanh z Z s tanh z − 1 Z s dz dz = dz + dz + 0 z 0 z 1 z 1 z Z 1 tanh z Z ∞ tanh z − 1 = log s + dz + dz + O(e−2s), 0 z 1 z we obtain (11.22) in the form a ax = log s + C + + O(a2, e−2s), (11.23) 2 where the constant C is given by Z 1 tanh z Z ∞ tanh z − 1 C = dz + dz. (11.24) 0 z 1 z Exponentiation of equation (11.23) gives the location of the saddle point asymptotically for small a and large s as s = eax−C−a/2+O(a2,e−2s). (11.25) The saddle point approximation (11.20) requires the evaluation of g and its second derivative at k = is. The Euler-Maclaurin sum formula gives ∞ X s g(is) = log cosh − sx 1 + na n=0 s Z s log cosh z 1 = 2 dz + log cosh s − sx + O(as) a 0 z 2 s Z 1 log cosh z Z s dz Z ∞ log cosh z − z 1 = 2 dz + + 2 dz + O a 0 z 1 z 1 z s s log 2 + − − sx + O(as). 2 2 11.3 One Dimensional Random Ionic Lattice 225

Using equations (11.23) and (11.24), we ﬁnd s log 2 1 g(is) = C − + O a, , as , (11.26) 1 a 2 a where Z 1 log cosh z Z ∞ log cosh z − z Z 1 tanh z Z ∞ tanh z − 1 C1 = 2 dz+ 2 dz− dz− dz, 0 z 1 z 0 z 1 z (11.27) and integration by parts shows that C1 = −1. It follows that s log 2 1 g(is) = − − + O a, , as . (11.28) a 2 a The second derivative of g is evaluated in a similar fashion 2 s 2 ∞ 1 − tanh d X 1 + na g(k) = − dk2 (1 + na)2 k=is n=0 1 Z s 1 = − 1 − tanh2 z dz − 1 − tanh2 s + O(ase−2s) as 0 2 tanh s = − + O(ase−2s, e−2s) as 1 1 = − + O(as, 1, )e−2s. (11.29) as as Substitution of (11.28), (11.29), and (11.25) into the saddle point approximation (11.20) gives √ a 1 (ax−C−a/2) − 1 eax−C−a/2 f(x) ≈ √ e 2 e a , (11.30) 2 π where the constant C = .8187801402 ··· is given by equation (11.24). There- fore, the small a and large s approximation to the tail of the pdf of Φ is given by √ 4πε R a f (x) ∼ 0 √ × (11.31) Φ q 2 π 1 4πε d 1 4πε d exp 0 x − C − a/2 − exp 0 x − C − a/2 , x → ∞. 2 q a q 226 Attenuation of the Electric Potential and Field in Disordered Systems

It follows from equation (11.31) that the pdf decays to zero as a double exponential as x → ∞, which implies that all moments exist. This decay is similar to the extreme value or the log-Weibull (Gumbel) distributions [156]. The compact support of the distributions of convergent series is replaced here with a steep decay. Note also that the decay becomes steeper further away from the system, as expected, because the pre-exponential factor of the inner exponent is 1/a = R/d. For small x the pdf can be approximated by a zero mean Gaussian with variance V ar (Φ), which for small a is

r ( 2) 4πε Rd Rd 4πε x f (x) ∼ 0 exp − 0 , x → 0. (11.32) Φ q 2π 2 q

Near its center, the distribution√ looks like a Gaussian with a standard deviation that decays like 1/ R, in agreement with equation (11.12). We conclude that the pdf looks normal near its center, but, far away from there, it decays to zero much more steeply, rather like a cutoﬀ. This conclusion is the answer to the question posed in subsection 11.3.2 about the normality of weighted sums of i.i.d. random variables. The non Gaussian tails of the distribution are characteristic of large deviations [90].

11.4 Random Distances

Consider a one-dimensional system of alternating charges without the restric- tion of equal distance between successive charges. In particular, we assume a renewal model, in which the distances between two neighboring charges are non-negative i.i.d random variables with pdf f(l) and finite expectation value Z ∞ d = lf(l) dl < ∞. 0 The potential of this random system is also a random variable. We show below that away from the system the mean value of the potential V¯ has the asymptotic form q 1 V¯ = + O(a) , (11.33) 4πε0R 2 where a = d/R. Equation (11.33) defines the attenuation produced by the configuration of charges. The mean potential of the system is produced by 11.5 Dimensions Higher than One 227

(in effect) half a charge. We note that the value 1/2 is exactly the same for both random and non-random systems of alternating charges (eq.(11.5)). We first note that Z ∞ Pr{V (R) = V } = f(l) Pr {V (R + l) = V ∗ − V } dl, (11.34) 0 q where V ∗ = . To find the mean value, we multiply (11.34) by V and 4πε0R integrate (note that 0 ≤ V ≤ V ∗), and then change the order of integration Z V ∗ Z ∞ V¯ (R) = V dV f(l) Pr {V (R + l) = V ∗ − V } dl 0 0 Z ∞ Z V ∗ = f(l) dl V Pr {V (R + l) = V ∗ − V } dV 0 0 Z ∞ Z V ∗ = V ∗ − f(l) dl V Pr{V (R + l) = V } dV 0 0 Z ∞ = V ∗ − f(l)V¯ (R + l) dl. (11.35) 0 We look for an asymptotic expansion of the form ¯ q ¯ ¯ 2 ¯ V (R) = V0 + aV1 + a V2 + ... . (11.36) 4πε0R ¯ Substituting this asymptotic expansion into (11.35) gives V0 = 1/2 for the O(1) term, because Z ∞ R 1 − a ≤ f(l) dl ≤ 1. (11.37) 0 R + l 1 The first inequality is due to the inequality ≥ 1 − x. Hence (11.33) 1 + x follows.

11.5 Dimensions Higher than One

11.5.1 The condition of global electroneutrality In dimensions higher than one, global electroneutrality is enough to dramatically attenuate the electric ﬁeld, but it is not enough to produce a small potential, as shown below. 228 Attenuation of the Electric Potential and Field in Disordered Systems

Consider the electric potential at a vacant site of random charges located at the points of a 2D square lattice

X Xnm Φ = √ . (11.38) n2 + m2 (n,m)6=(0,0) The variance of Φ is X 1 V ar(Φ) = = ∞. (11.39) n2 + m2 (n,m)6=(0,0) The infinite value of the variance means that arbitrarily large potentials can occur with high probability. That is, the electric potential is not attenuated. The divergence of the variance of the potential of three dimensional systems is even steeper. Therefore, attenuation of the potential of totally disordered systems can occur in two or three dimensional systems only if some correlation is introduced into the distribution of the locations of the charges. If, for example, the signs of all charges alternate, as in a real Na+Cl− crystal, the distribution of potential will be dramatically different, and greatly attenuated, compared to a two or three dimensional system in which many charges of one sign are clumped together. The condition of global electroneutrality is enough to ensure the dramatic attenuation of the electric field. Indeed, consider a 3D cubic lattice of random charges. The z-component of the electric field at a vacant lattice point is n Xnml cos √ X n2 + m2 + l2 E = . (11.40) z n2 + m2 + l2 (n,m,l)6=(0,0,0)

The variance of Ez is finite, n cos2 √ X n2 + m2 + l2 V ar(E ) = < ∞, z (n2 + m2 + l2)2 (n,m,l)6=(0,0,0) because convergence is determined by the integral Z π Z ∞ 2 1 2 2π cos θ sin θ dθ 4 r dr < ∞. 0 d r The large potential means that much work has to be done to create the given spatial configuration of the charges, however, the resulting field remains usually small. 11.5 Dimensions Higher than One 229

11.5.2 The condition of local electroneutrality Here we show that the condition of local electroneutrality implies the attenuation of the potential in two and three dimensions. For example, the potential of a two or three dimensional lattice of extended dipoles is ﬁnite with probability 1, if the orientation of dipoles is distributed independently, identically, and uniformly on the unit sphere (see Fig. 11.3). Paraphrasing [88, p.136], we say that a (net) charge distribution ρ(x) has local charge neutrality if the (net) charge inside a sphere of radius R falls with increasing R faster than any power, that is, for any x Z lim Rn ρ(y) dy = 0 for all n > 0. (11.41) R→∞ |x−y|

∞ l 1 X X 1 Ylm(θ, φ) Φ0,0,0(x) = qlm l+1 , (11.42) 4πε0 2l + 1 r l=0 m=−l where qlm are the multipole moments. In particular, the zeroth order multipole moment is 1 Z q00 = √ ρ(y) dy = 0, (11.43) 4π by the condition of local electroneutrality (11.41): the far potential due to a 2 single lattice point decays as 1/r (or steeper). The coeﬃcients qlm assigned to each lattice point are randomized as in the previous sections so their mean value vanishes, meaning that there is no preferred orientation in space. (Compare the example of dipoles which do not have a preferred orientation.) The mean value of the potential of the entire lattice is then hΦi = 0. The variance is given by X V ar(Φ) = V ar(Φijk), (11.44) ijk 230 Attenuation of the Electric Potential and Field in Disordered Systems

Figure 11.3: Two dimensional lattice of dipoles of randomly chosen orienta- tions produce attenuation due to the condition of local electroneutrality. 11.5 Dimensions Higher than One 231

where Φijk is the potential of the charge at lattice point (i, j, k). The potential decays as 1/r2 (or steeper); therefore the variance decays as 1/r4 = 1/(i2 + j2 + k2)2 (or steeper). The convergence of the inﬁnite sum (11.44) is determined by the convergence of the integral Z 1 Z ∞ 1 4π 4 dV = 4π 2 dr = < ∞. (11.45) r>d r d r d Thus, the variance of the potential is ﬁnite and we have shown that local electroneutrality produces a dramatic attenuation of potential. As above, the potential away from a charge is usually of the order of the potential of a single charge.

11.5.3 The liquid state Screening in the liquid state involves a least three phenomena. (1) The movement of charge to a distribution of minimal free energy. (2) The properties of a static charge distribution with minimal free energy. (3) The properties of any charge distribution. If the charge correlation function ρ(x) minimizes free energy, and is at equilibrium, as in ionic solutions, the far field potential is strongly screened. However, the relaxation into such a state takes time, typically psec to nsec in an ionic solution under biological conditions (see measurements reported in [8], and theory summarized in [107]). As long as local charge neutrality exists during the relaxation period, the potential changes from attenuated (as described above) to exponentially screened, as equilibrium is reached. In fact, the spread of potential in ionic solutions has the curious property that it is much less shielded at short times than at long times; potentials on the (sub) femtosecond time scale of atomic dynamics spread macroscopic distances while potentials on long time scales spread only atomic distances. Specifically, potentials on a time scale greater than nano or microseconds spread a few Debye lengths, only a nanometer or so under biological conditions, although potentials on a femtosecond time scale can spread arbitrarily far depending on the configuration of dielectrics at boundaries that govern the violations of local electroneutrality. To make this verbal analysis of fast phenomena rigorous, the potentials and fields should be computed from Maxwell’s equations, not Coulomb’s law. Non-equilibrium fluctuations may violate local charge neutrality, therefore field fluctuations can be large. For example, in systems which are not lo- 232 Attenuation of the Electric Potential and Field in Disordered Systems

cally electroneutral, potential can spread a long way, as in the telegraph [57], Kelvin’s transatlantic cable, or the axons of nerve cells [87]. In such systems, d.c. potential spreads arbitrarily far—kilometers in telegraphs; thousands of kilometers in the transatlantic cable; centimeters in a squid nerve ﬁlled with salt water—even if an abundance of ions (≈ 1023) are present. Local electroneutrality is violated in such systems (at the insulating boundary which separates the inside and outside of the cable, e.g., the cell membrane) and that violation allows large far ﬁeld potentials.

11.6 Summary and Discussion

Global electroneutrality ensures the dramatic attenuation of the electric potential and field of a one dimensional system of charges. Even if local electroneutrality is violated, and the local net charge is not zero, the potential remains finite in these one dimensional systems, even in a random lattice that includes arbitrarily long strings of equal charges. We have shown that the distribution of the weighted sum of i.i.d. random variables that define the one-dimensional electric potential is almost normal near its center, but has very steep double exponentially decaying tails. The distances between neighboring charges can also be random, without changing the attenuation effect. In higher dimensions, global electroneutrality is sufficient to dramatically attenuate the electric field, but not the potential. However, local electroneutrality ensures a small potential in two and three dimensions, so the electric potential and field is short range in one, two, and three dimensions, if the systems are locally electroneutral. Chapter 12

Boundary Conditions and a Closure Relation for the Pair Correlation Function in Non-Equilibrium Diﬀusion

12.1 Inﬁnite system in Steady-State

Consider the following physical setup (Fig.12.1). The space is divided by an infinite long impermeable membrane located at z = 0. Particles of several species can traverse the membrane only through a narrow channel. The interaction between particles is described by short ranged forces, such as the Lennard-Jones 6-12 interaction, whereas long ranged interactions are ignored. h Far away from the membrane the concentrations are maintained fixed: cL is h the concentration of the h species at z → −∞ and cR is the concentration of the h species at z → ∞. The total number of particles in this system is infinite. This system is carrying net flux of particles due to the concentration gradient. In the case of non-interacting point particles the concentration profile and flux were calculated explicitly in [98, 50]. This is a mixed boundary value problem that is similar to the well known electrified disk problem of electrostatics [88]. Mixed boundary value problems occur in many field of applications (see [184] for different applications and methods of solving such problems). For example, in Chapter 5 we calculated the time it takes a non Boundary Conditions and a Closure Relation for the Pair Correlation Function in 234 Non-Equilibrium Diffusion

Figure 12.1: The physical setup: the space is divided by an infinite membrane. Particles can permeate only through the channel. The concentrations to the left and to the right of the membrane are different. 12.1 Infinite system in Steady-State 235 interacting diffusing particle to get to the pore. The predicted flux density 1 is singular at the edges of the pore, where it blows up like , where pa2 − ρ2 a is the radius of the pore, and ρ is the distance from the center of the pore. Though the flux density is singular, the total flux through the pore is finite, as this is an integrable singularity. Away from the channel, the flux density decreases to zero as does the density of the field lines of a dipole in electrostatics. The concentrations tend to their values at infinity as fast as (a/z)2. This is the rate at which the system tends to equilibrium, which is maintained by fixed concentrations there. Non-interacting point particles can pass through each other, so important phenomena are lost, such as blocking (especially when the LJ radius is of the order of a) and selectivity [73]. These are expected to be effects of the pair correlation function, which is the joint pdf of two particles (see Chapter 9). The pair function is the basis to the theory of simple liquids [12, 7, 34, 72, 67, 158] of equilibrium statistical mechanics, from which thermodynamic properties of matter can be calculated. There are two main approaches to finding the pair function. The first approach is based on the Ornstein-Zernike integral equation and a closure relation for the direct and indirect correlation functions. Many closure relations fall into this category, such as the Percus- Yevick approximation (PY), the hypernetted chain approximation (HNC), and the mean spherical approximation (MSA). The second approach relies on the Bogoliubov-Born-Green-Kirkwood-Yvon (BBGKY) hierarchy, which relates the n-th correlation function with the n + 1-th correlation function, and assumes a closure relation that connects them. However, the starting point of all approaches is the assumption that the system is in equilibrium. Certainly, this is not the case at hand, where a non vanishing steady state flux is flowing through the system. In [167] an equation for the singlet function is derived in terms of the pair correlation function for systems out of equilibrium, i.e. carrying steady state flux. The motion of each particle was described by a separate Langevin equation, all coupled by the mutual forces. Integrating the full Fokker-Planck equation in the Smoluchowski overdamped limit resulted a partial differential equation (PDE) for the singlet function in terms of the pair correlation function. Using similar integrations, we find in Chapter 9 a PDE for the pair function in terms of the triplet function 9.20, and in general, an entire hierarchy of PDEs. However, two ingredients, that are necessary for the well- posedness of the pair function problem, are still missing. First, the PDE is Boundary Conditions and a Closure Relation for the Pair Correlation Function in 236 Non-Equilibrium Diffusion

not well posed, unless (correct) boundary conditions are specified. Second, a closure relation that relates the pair function and the triplet function is needed. In this chapter we supply these two missing ingredients. The boundary conditions for the pair function, which satisfies the equations (9.20)-(9.21), are to be specified when one particle is somewhere inside the domain, while the second particle is at the boundary, which can be either the reflecting membrane, or infinity. At the reflecting wall we impose a no flux condition for the two particle flux density (Neumann type boundary condition), h,h0 Jxh (x, y) = 0, x ∈ ∂ΩR, y ∈ Ω. (12.1) At infinity, the two particles become independent, therefore the pair function satisfies

h,h0 h h0 c (x, y) = cLc (y), |x| → ∞, z < 0, y ∈ Ω, (12.2) h,h0 h h0 c (x, y) = cRc (y), |x| → ∞, z > 0, y ∈ Ω. Thus the singlet and the pair densities ch(x), ch,h0 (x, y) satisfy the nonlin- early coupled system (9.13)-(9.19), (9.20)-(9.21), (12.1)-(12.2). We choose Kirkwood’s superposition approximation, because it is grounded in solid probabilistic considerations, which are independent of the state of equilibrium or non-equilibrium of the systems, as shown in Chapter 10. Therefore, it is appropriate to use the Kirkwood closure in systems carrying steady state flux. Other closure relations that have been successful in calculating pair function in equilibrium system are not guaranteed to give correct 0 results for systems out of equilibrium. The force term f¯ h,h (x, y) is now defined in terms of the pair function, thus rendering (9.20) a nonlinear partial differential equation. We assume that the nonlinear problem (9.13)-(9.19), (9.20)-(9.21), (12.1)-(12.2) is well posed. Still, this leaves us with the computational problem of solving the above problem, numerically or otherwise. The computational complexity of such high dimensional problem seems to be impractical at present. The rotational symmetry of the problem (the distance from the wall and the distance from the center of the pore suffice to describe the position of a single particle) reduces the dimensionality from six to five, which still requires heavy computational resources (even the two dimensional analogous problem requires solving a four dimensional PDE, 2D for each particle.) This remains a numerical challenge that hopefully will be solved someday. 12.2 Quasi Steady State: The Neumann Problem in Finite Domains 237

12.2 Quasi Steady State: The Neumann Prob- lem in Finite Domains

In this section we consider a finite system with all boundaries reflecting (see Figure 2.1). In other words, there are no control mechanisms to maintain fixed concentrations and voltages. In the case of all reflecting boundary condition, no spurious boundary layers appear in the solution of the Fokker- Planck equation. Moreover, the Smoluchowski overdamped approximation is valid in the entire domain. In this model, time evolution is governed by the Nernst-Planck equation and no feedback mechanisms (of the kind considered in Chapter 2) are needed. In this setup, which describes actual channel current measurements made with polarizing (e.g., platinum) electrodes, there is no flux in the steady state. However, the long transient is the quasi steady state that Hodgkin, Huxley, and Katz actually observed on the (biological) time scale of 1 − 10 msecs [76, 77]. The eigenfunction expansion of the solution of the Smoluchowski equation [167]

Nh ∂pN (x˜, t) X X = − ∇xh · Jxh (x˜, t), (12.3) ∂t j j h j=1 where J h (x˜, t) is the 3-dimensional probability ﬂux vector of the j-th par- xj ticle of species h, given by

h f j (x˜) kBT J h (x˜, t) = pN (x˜, t) − ∇ h pN (x˜, t), (12.4) xj h h h h h h xj M γ (xj ) M γ (xj ) results in

∞ X −λnt pN (x˜, t) = ane φn(x˜), (12.5) n=0 where λ0 < λ1 < . . . < λn < . . . are the eigenvalues of the linear elliptic operator

Nh X X n ∇ h · J h (x˜) = λnφn(x˜), (12.6) xj xj h j=1 Boundary Conditions and a Closure Relation for the Pair Correlation Function in 238 Non-Equilibrium Diﬀusion

with Neumann (no-ﬂux) boundary conditions

n J h (x˜) · ν(x˜) = 0, x˜ ∈ ∂Ω, (12.7) xj where

h n f j (x˜) kBT J h (x˜) = φn(x˜) − ∇ h φn(x˜). (12.8) xj h h h h h h xj M γ (xj ) M γ (xj )

Clearly, the first eigenvalue is λ0 = 0, which corresponds to the equilibrium 0 −U(x˜ )/kB T solution J h (x˜) = 0 for all x˜, j, h, or equivalently, φ0(x˜) = e xj is the equilibrium Boltzmann distribution. This means, that the system equilibrates after sufficiently long time (t 1/λ1). As mentioned above, the long transient (1/λ2 t 1/λ1) describes measurements on the biological time scale. In this scale, a quasi steady flux flows through the channel. For such time scales, we can approximate pN (x˜, t) by only the first two eigenfunctions, as the other eigenfunctions decay exponentially faster,

−U(x˜ )/kB T −λ1t pN (x˜, t) ≈ a0e + a1φ1(x˜)e . (12.9)

Indeed, for non-interacting particles, it can be shown ([62, 32] and Chapter 5) that the second eigenvalue λ1 is much smaller than all other eigenvalues (λ1 aD D λ ), as λ ∝ , while λ ∝ , where a is the radius of the channel 2 1 |Ω| 2 |Ω|2/3 hole, D is the diffusion coefficient and |Ω| is the volume of the chamber. We conjecture that λ1 λ2 also for the case of interacting particles. From the calculation of the eigenfunction φ1(x˜), one can find the pair correlation function of the quasi steady state. The resulting pair correlation function depends on the choice of the closure relation used in the calculation, as is the case in equilibrium systems [34].

12.3 The Electrical Current and the Ramo- Shockley Function

We consider the diffusion of interacting ions in an electrolytic solution in a finite domain that consists of two large baths connected by one or more narrow channels, and includes two or more metal electrodes maintained at fixed voltages, imposing a controlled voltage difference across the domain. 12.3 The Electrical Current and the Ramo-Shockley Function 239

In addition, an external circuitry is connected to one of the electrodes and measures the current flowing through it, similar to a typical experimental voltage clamp experiment. The problem at hand is to determine the electric current flowing through this electrode due to the diffusion currents of the different ionic species in the system. Obviously, the total current through the electrode is the sum of the contributions of the diffusion currents of all ionic species. Note that in this formulation, we neglect magnetic fields, radiation terms, convection terms, water flow, and electrolysis. The Ramo-Shockley theorem [146, 172] relates the microscopic motion of mobile charges in the finite domain to the electric current measured at any given electrode. For a single moving charge q at location x with velocity v, the instantaneous current at the j-th electrode is given by

Ij = qv · ∇uj(x) (12.10) where uj is the solution of capacitor problem

∇ · [ε(x)∇uj] = 0 (12.11) with the boundary conditions

uj = 1 (12.12) ∂Ωj

uj = 0 (i 6= j), (12.13) ∂Ωi where ∂Ωj is the boundary of the j-th electrode. In addition, on the insulating parts of the boundary of the domain, the conditions on uj are that ∂u ∂u ε j − ε j = 0, 1 ∂n 2 ∂n where derivatives are taken in the normal direction to the boundary, and ε1, ε2 are the dielectric coeﬃcients on the two sides of the interface. In the case of many particles, due to superposition, the total current recorded at the j-th electrode is given by X Ij = qivi · uj(xi) i Boundary Conditions and a Closure Relation for the Pair Correlation Function in 240 Non-Equilibrium Diﬀusion

12.4 The Connection between the Diﬀusion Current and the Electric Current

We now relate the diﬀusion current carried by the mobile ions inside the domain to the electrical current measured by the Amp´ere meter connected to the electrode. Recall that pN (x˜, v˜, t) denotes the probability density function of all particles of all species at time t. Using the Ramo-Shockley theorem (12.10), the macroscopic current due to the motion of all N h particles of species h through the j-th electrode is given by Z h h h h h Ij = N q v1 · ∇uj(x1 )pN (x˜, v˜, t) dx˜ dv˜, (12.14)

where we compute the contribution to the current due only to the ﬁrst particle of species h, and multiply by N h, because all particles of species h are interchangeable. The total current measured at electrode j is the sum over all species, X h Ij = Ij . (12.15) h Note that in equation (12.14), we can easily integrate over all coordinates, except that of the ﬁrst particle, which gives the marginal density Z h h h h h h h h h Ij = N q v1 · ∇uj(x1 )p(x1 , v1 , t) dx1 dv1 . (12.16)

As shown in [167], by integrating the full Fokker-Planck equation (9.11) over all particle coordinates but those of the ﬁrst particle of species h, we obtain a reduced Fokker-Planck equation, that can be written in the conservation law form ∂ p(xh, vh, t) + ∇x · J h (xh, vh, t) + ∇v · J h (xh, vh, t) = 0, (12.17) ∂t 1 1 x 1 1 v 1 1

h h with Jx and Jv given by

h h h h Jx = v1 p(x1 , v1 , t) (12.18) f¯ h J h = −(γhvh − 1 )p(xh, vh, t) − ∇Dhp(xh, vh, t). v 1 M h 1 1 1 1 12.4 The Connection between the Diﬀusion Current and the Electric Current 241

Therefore, combining equations (12.18) and (12.16), and suppressing the index j of the electrode, the electric current due to species h is Z h h h h I = N q ∇u(x) · Jx(x, v, t) dx dv. (12.19)

12.4.1 The Steady State We ﬁrst consider the current measured in the steady state, in which ∂p/∂t = 0 in equation (12.17). In this case we can write Z h h h h h h I = N q ∇xu(x) · Jx + u∇x · Jx + u∇vJv dxdv Z h h h h = N q ∇x · uJx + u∇vJv dxdv. (12.20)

The second term in (12.20) vanishes, because integration with respect to v 3 extends over R . Integration with respect to v in the first term yields the h total probability flux Jx(x) at location x. The divergence theorem converts the integral with respect to x into Z h h h h I = N q ∇ · u(x)Jx(x) dx Ω Z h h h = N q u(x)Jx(x) · n(x) dSx. ∂Ω Using the boundary conditions (12.12) for the Ramo-Shockley function u(x) and the no-flux boundary condition for the x component of the flux at the insulating boundaries, give Z h h h h I = N q Jx(x) · n(x) dSx. (12.21) ∂Ωj If the electrode surface is a reflecting wall for trajectories, the total diffusive flux through ∂Ωj is zero, and by equation (12.21) the total electric current also vanishes. Indeed, in this case, the steady state of the system is equilibrium, which means that there is neither a diffusive nor an electric flux. If the electrodes also serve as a feedback mechanism that injects and absorbs ions, the system can be in steady state with a non vanishing flux Boundary Conditions and a Closure Relation for the Pair Correlation Function in 242 Non-Equilibrium Diffusion

[180]. In this case, the external circuitry will measure a steady state electrical current, that according to equations (12.21) and (12.15), is equal to the total diﬀusion current multiplied by the corresponding electrical charges, Z X h h h Itotal = q N J (x) · n(x)dSx. (12.22) h ∂Ωj This formula shows that the current measured by the external circuitry is exactly equal to the total ionic current ﬂowing between the two baths. An example for such a (hypothetical?) system is an Ag+Cl− solution with solid AgCl electrodes maintained at 1V (higher voltage can cause electrolysis of the water molecules).

12.4.2 Quasi Steady State We now consider the case of reﬂecting electrodes, before the system equilibrates. In this case, the solution for the time dependent probability density function, equation (9.11), can be expanded in eigenfunctions (similarly to the expansion (12.9))

−λ1t pN (x˜, v˜, t) ≈ a0p0(x˜, v˜) + a1p1(x˜, v˜)e . (12.23)

Treating the time-dependent problem as was done above for the steady state, we obtain that Z h h h ∂p I = N q ∇u(x1) · J x + u(x1) ∇xJx + ∇vJv + dx˜ dv˜ ∂t Z ∂p = N hqh u(x ) dx˜ dv˜, (12.24) 1 ∂t

where all integrals involving Jx and Jv vanish, due to the no flux boundary conditions on the domain boundaries. Inserting the approximation (12.23) into (12.24) integrating over all particles, except the first one of species h, we obtain that in the quasi steady state, the electrical current due to species h is given by Z h h h −λ1t h I = N q λ1e u(x)p1 (x) dx, (12.25) Ω h where p1 is the marginal density of p1(x˜, v˜) corresponding to a particle of species h. 12.4 The Connection between the Diffusion Current and the Electric Current 243

Note that p1 is the second eigenfunction, and being orthogonal to the ﬁrst eigenfunction p0 it is approximately a constant +1 in one bath and a constant −1 in the other, with additional boundary layers near the channel (up to a normalization factor). Therefore, the integral is a constant times the volume of the baths. However, since λ1 is proportional to a/|Ω| (see Chapter 5), the current is independent of the total volume of the baths, but depends on the small diameter of the channel. Chapter 13

Open Problems and Future Research

There are still many problems related to the topics of this dissertation that remain open. In this Chapter I list several open problems, and try to give a lead when possible. The main problem that arises from part I of this dissertation is how to simulate systems of interacting particles. That is, how to inject particles from a bath into the simulation, where there are interactions between the simulated particles and the bath particles, and between bath particles and themselves. In part I we have reasoned that all interactions can be replaced by a mean field approximation sufficiently far away from a channel. Deriving an estimate of the error such an assumption introduces, as well as the correction needed in the simulation algorithm, remain open problems. There are several unsolved problems related to Narrow Escape: (a) Does the error estimate O(ε log ε) remains valid for all three dimensional domains? A possible direction could be to use the maximum principle while bound- ing the given domain between two balls; (b) How do boundary singularities of three dimensional domains affect the MFPT? (c) The Narrow Escape problem is closely related to the homogenization problem of a domain with many small holes, that is, to the approximation of the complicated absorbing boundary conditions by a simpler killing measure [142, 93, 148]. This direction can possibly be even further developed. Part II raises many more questions. The problem of finding a “good” closure relation in equilibrium occupies many physical chemists even today, so there is no wonder that finding a closure relation for systems out of equi- 245 librium is a tough nut to crack. The pair correlation function of a few one dimensional systems can be determined analytically, without any closure relation [94, 95, 188, 70, 71] by using the Markov property of a related stochastic process. The resulting pair function has a WKB form for high densities. However, in higher dimensions the Markov property is no longer preserved: there is no meaning of “the next particle” as in one dimension (the closest particle interpretation fails too). A possible direction would be to look for a solution in a WKB form in high densities for the BBGKY equation, and to match it to the regular low density cluster expansions. Appendix A

Algebra Solution of the Albedo Problem

A.1 The Stationary Albedo Problem

The stationary albedo problem was solved by K losek [103] using the half- range expansion technique developed by Hagan, Doering and Levermore [64, 65]. The albedo problem stands for finding the exit probabilities and the first passage time of a Langevin trajectory that is initiated with a positive velocity exactly at the location of the absorbing boundary. Our interest in the albedo problem arises from the need to connect a discrete simulation to a continuum bath in simulating ion permeation through protein channels [47, 180], though there are many other physical applications [96, 198]. The non-characteristic case means that the Langevin particle moves in a non-vanishing force field. The major part of the solution presented by K losek consists of analyzing the following boundary layer problem

f(˜x/γ) Q + −v + Q − vQ = 0 (A.1) vv γ v x˜ Q(˜x = 0, v > 0) = 0 where f(z) is the force field near the boundary, and γ is a large parameter that stands for the friction. The analysis [103] is accurate but quite involved, so one wishes to find a simpler solution. Here we present an alternative solution to the boundary layer problem (A.1), which is perhaps simpler. We construct the non-characteristic case A.2 The Algebra Solution 247 solution from the characteristic case solution of Hagan et al [64] by using the 2 algebra of the differential operator L = ∂v − v∂v.

A.2 The Algebra Solution

The operator L satisﬁes the commute relations

[L, ∂v] = ∂v, (A.2)

[L, v] = 2∂v − v, (A.3) where [A, B] = AB − BA is the commutator. For example, the identity

L∂vQ = LQv = Qvvv − vQvv = ∂v(Qvv − vQv) + Qv

= ∂vLQ + ∂vQ, (A.4) proves relation (A.2). Substituting the Taylor expansion of f(z)

∞ X f (n)(0) x˜n f(˜x/γ) = . (A.5) n! γn n=0 into (A.1) results in

∞ ! X f (n)(0) x˜n Q + −v + Q − vQ = 0. (A.6) vv n! γn+1 v x˜ n=0

We look for a solution of the form 1 1 Q = Q0 + Q1 + Q2 + ... (A.7) γ γ2

Substituting (A.7) in (A.6) we ﬁnd that Q0 should satisfy

0 0 0 Qvv − vQv − vQx˜ = 0, (A.8) and the recursion formula

n X f (n−m)(0) Qn+1 − vQn+1 − vQn+1 = − x˜n−mQm. (A.9) vv v x˜ (n − m)! v m=0 248 Algebra Solution of the Albedo Problem

Equations (A.8) and (A.9) can be rewritten as

0 (L − v∂x˜)Q = 0, (A.10) and n X f (n−m)(0) (L − v∂ )Qn+1 = − x˜n−mQm, (A.11) x˜ (n − m)! v m=0 The solution to problem (A.10) was found by Hagan et al [64]

" ∞ √ # 1 X N( n) √ Q0(˜x, v) = E −ζ − (v − x˜) − (−1)n √ V −(v)e− nx˜ . (A.12) 2 2n! n n n=1 We now proceed to ﬁnd Q1. Setting n = 0 in equation (A.11) follows

1 0 (L − v∂x˜)Q = −f(0)Qv. (A.13) Using the algebra of L (eqs. (A.2)-(A.3)) we ﬁnd that Q1 is given by f(0) Q1 = − (v − x˜) Q0 + AQ0, (A.14) 2 where A is an arbitrary constant that should be determined from the match- ing condition [103]. Indeed, f(0) (L − v∂ )Q1 = − (L − v∂ )(v − x˜) Q0 + A(L − v∂ )Q0 x˜ 2 x˜ x˜ f(0) = − {(v − x˜)(L − v∂ ) + [L − v∂ , v − x˜]} Q0 2 x˜ x˜ f(0) = − {[L, v] + v [∂ , x˜]} Q0 2 x˜ f(0) = − [2∂ − v + v] Q0 2 v 0 = −f(0)Qv. (A.15) Clearly, Q1 also satisﬁes the boundary condition

Q1(˜x = 0, v > 0) = 0, (A.16)

because f(0) Q1(˜x = 0, v > 0) = A − v Q0(˜x = 0, v > 0) = 0. (A.17) 2 A.2 The Algebra Solution 249

We see that ﬁnding Q1 which is the O (γ−1) term is an immediate consequence of the algebra that L generates and the vanishing force fundamental solution Q0 that was found by Hagan et al [64]. The use of the algebra saved us the complicated complex variable analysis of the half range expansion. Appendix B

Appendix of Chapter 5

B.1 Estimate of kKk2

B.1.1 Estimate of the kernel A rough estimate of the kernel, for 0 ≤ u, v ≤ ε, is obtained from equation (5.58) as

2 2 5 1 1 K (u, v) ≤ cos (v + u) log 2 sin (v + u) 4π2 2 2 2 5 1 1 + cos (v − u) log 2 sin (v − u) . 4π2 2 2

Furthermore,

Z ε 2 Z 2 sin 1 (v+ε) 1 1 2 2 cos (v + u) log 2 sin (v + u) du = (log x) dx 2 2 1 0 2 sin 2 v 2 2 1 1 1 1 1 ≤ 2 sin (v + ε) − sin v log 2 sin v ≤ ε cos v log 2 sin v 2 2 2 2 2 and

ε 2 2 sin 1 ε Z 1 1 Z 2 ε cos v log 2 sin v dv = ε (log x)2 dx ≤ 2ε2 log2 ε. 0 2 2 0 B.2 Elliptic Hole 251

Similarly,

Z ε 2 1 1 cos (v − u) log 2 sin (v − u) dv = 0 2 2

1 1 Z 2 sin 2 u Z 2 sin 2 (ε−u) (log x)2 dx + (log x)2 dx ≤ 0 0

2u log2 u + 2(ε − u) log2(ε − u).

It follows that Z ε 2u log2 u + 2(ε − u) log2(ε − u) du ≤ 4ε2 log2 ε, 0 because u log u is an increasing function in the interval 0 ≤ u ≤ e−2. Alto- gether, we obtain √ 30 1 kKk ≤ ε log for ε e−2, (B.1) 2 2π ε which is (5.68).

B.2 Elliptic Hole

We present here, for completeness, Lure’s [116] solution to the integral equation (5.16) in the elliptic hole case. We deﬁne for y = (x, y)

x2 y2 L(y) = 1 − − (b ≤ a) a2 b2 and introduce polar coordinates in the ellipse ∂Ωa

x = y + (ρ cos θ, ρ sin θ), with origin at the point y. The integral in eq.(5.16) takes the form

Z Z 2π Z ρ0(θ) g0(x) g˜0 dρ dSx = dθ p , (B.2) ∂Ωa |x − y| 0 0 L(x) 252 Appendix of Chapter 5

where ρ0(θ) denotes the distance between y and the boundary of the ellipse in the direction θ. Expanding L(x) in powers of ρ, we ﬁnd that (x + ρ cos θ)2 (y + ρ sin θ)2 L(x) = 1 − − = L(y) − 2φ ρ − φ ρ2, (B.3) a2 b2 1 2 x cos θ y sin θ cos2 θ sin2 θ where φ = + and φ = + . Solving the quadratic 1 a2 b2 2 a2 b2 equation (B.3) for ρ, taking the positive root, we obtain

1 n 2 1/2o ρ(x) = −φ1 + φ1 + φ2 (L(y) − L(x)) , (B.4) φ2 therefore, for ﬁxed y and θ, 1 dL(x) dρ(x) = − , (B.5) 2 2 1/2 [φ1 + φ2 (L(y) − L(x))] and the integral takes the form

Z Z 2π Z L(y) g0(x) 1 dL(x) g˜0 dSx = dθ |x − y| 2 2 1/2 p ∂Ωa 0 0 [φ1 + φ2 (L(y) − L(x))] L(x) Z 2π Z L(y) 1 g˜ dz = dθ 0 . 2 p 2 p 0 0 φ1 + φ2z L(y) − z z φ2 Substituting s = and setting ψ = 1 , we ﬁnd that L(y) φ2L(y) Z Z 2π Z 1 g0(x) g˜0 ds dSx = dθ √ √ √ = ∂Ωa |x − y| 0 2 φ2 0 ψ + s 1 − s Z 2π r 1 g˜0 ψ + s dθ √ 2 arctan = 0 2 φ2 1 − s 0 Z 2π g˜ √0 π − 2 arctan pψ dθ = 0 2 φ2  x cos θ y sin θ  2π + Z g˜ dθ 2 2 0 π − 2 arctan a b  . r 2 2  r 2 2  0 cos θ sin θ  cos θ sin θ  2 + + L(y) a2 b2 a2 b2 B.3 A Pathological Example 253

The arctan term changes sign when θ is replaced by θ + π, therefore its integral vanishes, and we remain with Z Z 2π g0(x) πg˜0 dθ dSx = r 2 2 ∂Ωa |x − y| 2 0 cos θ sin θ + a2 b2 π Z 2 dθ = 2πbg˜0 r 2 2 0 a − b 1 − sin2 θ b2

= 2πbg˜0K(e), (B.6) where K(·) is the complete elliptic integral of the ﬁrst kind, and e is the eccentricity of the ellipse r b2 e = 1 − , (a > b). (B.7) a2 We note that the integral (B.6) is independent of y, so we conclude that (5.17) is the solution of the integral equation (5.16).

B.3 A Pathological Example

We have derived an integral equation for the leading order terms of the flux and the MFPT in the case where the MFPT increases indefinitely as the relative area of the hole decreases to zero. However, the MFPT does not necessarily increase to infinity as the relative area of the hole decreases to zero. This is illustrated by the following example. Consider a cylinder of length L and radius a. The boundary of the cylinder is reflecting, except for one of its bases (at z = 0, say), which is absorbing. The MFPT problem becomes one dimensional and its solution is z2 v(z) = Lz − . (B.8) 2 Here there is neither a boundary layer nor a constant outer solution; the MFPT grows gradually with z. The MFPT, averaged against a uniform ini- L2 tial distribution in the cylinder, is Eτ = and is independent of a, that is, 3 the assumption that the MFPT becomes infinite is violated. Appendix C

Appendix of Chapter 6

C.1 Maximal Exit Time for the Circular Disk

Using equation (6.15) we ﬁnd

∞ a0 X v = u(1, 0) = + a (C.1) max 2 n n=1 Z π−ε ∞ a0 1 X = + √ h (t) [P (cos t) + P (cos t)] dt. 2 1 n n−1 2 0 n=1 Recall the generating function of the Legendre polynomials [1]

∞ 1 X n √ = Pn(x)t , (C.2) 2 1 − 2tx + t n=0 from which it follows that

∞ X 1 1 Pn(cos t) = √ = . (C.3) 1 − 2 cos t + 1 t n=0 2 sin 2 Together with equation (6.16), this gives   a 1 Z π−ε 1 1 Z π−ε h (t) dt v = 0 + √ h (t)  − 1 dt = √ 1 . (C.4) max 1  t  t 2 2 0 sin 2 0 sin 2 2 C.1 Maximal Exit Time for the Circular Disk 255

Combining with equation (6.21) and integrating by parts, we get

u 1 Z π−ε 1 1 d Z t u sin du v = √ √ 2 dt (C.5) max t 2 0 π sin dt 0 cos u − cos t 2 u 1 Z t u sin du π−ε = √ 2 √ t 2π sin 0 cos u − cos t 0 2 t u 1 Z π−ε cos Z t u sin du + √ 2 dt √ 2 . t 2 2π 0 sin2 0 cos u − cos t 2

Equations (6.22) and (6.24) show that

√ u 2 Z t u sin du t t √ 2 = −2 log cos + 2 log 1 + sin + k(t), (C.6) π 0 cos u − cos t 2 2 where t  q  Z sin arcsin s2 + cos2 t 4 2  2  k(t) = −   ds. (C.7) π  r t  0 s2 + cos2 2

Therefore,

t t √ u −2 log cos + 2 log 1 + sin + k(t) 2 Z t u sin du 2 2 lim √ 2 = lim t t t→0 π sin 0 cos u − cos t t→0 sin 2 2 4 = 2 − arcsin(1) = 0. (C.8) π 256 Appendix of Chapter 6

Hence

vmax =

  r   ε 2 2 ε Z cos arcsin s + sin 1  ε ε 4 2  2   2 log 1 + cos − 2 log sin −   ds ε  2 2 π  r ε   2 cos 0 s2 + sin2 2 2 t u 1 Z π−ε cos Z t u sin du + √ 2 dt √ 2 . t 2 2π 0 sin2 0 cos u − cos t 2 For ε 1 t u ε 1 Z π cos Z t u sin du v = − log + √ 2 dt √ 2 + O(ε). (C.9) max t 2 2 2π 0 sin2 0 cos u − cos t 2 Changing the order of integration, we get t ε 1 Z π u Z π cos dt v = − log + √ u sin du 2 + O(ε). max t √ 2 2 2π 0 2 u sin2 cos u − cos t 2 (C.10) Substituting rcos u − cos t s = (C.11) 2 in the inner integral results in t u Z π cos dt √ cos 2 = 2 2 . t √ u u sin2 cos u − cos t sin2 2 2 Therefore, ε 1 Z π u ε 2 Z π/2 vmax = − log + u du = − log − log sin v dv 2 2π 0 tan 2 π 0 2 ε = − log + log 2. (C.12) 2 C.2 Exit Times along the Ray 257

C.2 Exit Times along the Ray

Along the ray θ = π the MFPT is given by

2 ∞ 1 − r a0 X v (r) ≡ v(r, θ = π) = + + a (−r)n ray 4 2 n n=1 2 Z π−ε ∞ 1 − r a0 1 X = + + √ h (t) [P (cos t) + P (cos t)](−r)n dt. 4 2 1 n n−1 2 0 n=1

Using the generating function (C.2) of the Legendre polynomials to sum the inﬁnite series, we obtain

2 Z π−ε 1 − r 1 − r h1(t) dt vray(r) = + √ √ . (C.13) 2 4 2 0 1 + 2r cos t + r

Combining with equation (6.21), integrating by parts, and hanging the order of integration gives

1 − r2 1 − r vray(r) = + √ a0 4 2 1 − 2r cos ε + r2 r(1 − r) Z π−ε u Z π−ε sin t dt − √ u sin du √ . 2 3/2 2π 0 2 u (1 + 2r cos t + r ) cos u − cos t √ √ The substitutions s = cos u − cos t and x = 2r s lead to √ Z π−ε sin t dt 2 cos u + cos ε √ = √ , 2 3/2 2 2 u (1 + 2r cos t + r ) cos u − cos t (1 + 2r cos u + r ) 1 − 2r cos ε + r which implies that

1 − r2 1 − r vray(r) = + √ a0 (C.14) 4 2 1 − 2r cos ε + r2 √ √ 2 r(1 − r) Z π−ε u cos u + cos ε − √ u sin du. 2 2 π 1 − 2r cos ε + r 0 2 1 + 2r cos u + r 258 Appendix of Chapter 6

The substitution (6.23) gives √ Z π−ε u cos u + cos ε u sin 2 du = 0 2 1 + 2r cos u + r r ε ε arccos s2 + sin2 s2 ds √ Z cos 2 4 2 2 , r ε 0 (1 − 2r cos ε + r2 + 4rs2) sin2 + s2 2

and we obtain the exact form of vray(r) as 1 − r2 1 − r vray(r) = + √ a0 (C.15) 4 2 1 − 2r cos ε + r2 r ε ε arccos s2 + sin2 s2 ds 8r(1 − r) Z cos 2 − √ 2 . π 1 − 2r cos ε + r2 r ε 0 (1 − 2r cos ε + r2 + 4rs2) sin2 + s2 2 √ For ε 1 and 1 − r ε equation (C.15) becomes

1 − r2 ε 8r Z 1 s arccos s ds vray(r) = − log − 2 2 + O(ε). (C.16) 4 2 π 0 (1 − r) + 4rs To evaluate the integral in (C.16), we write π arccos s = − arcsin s, (C.17) 2 and obtain Z 1 8r π s ds 2 2 2 = −2 log(1 − r) + log(1 + r ). π 2 0 (1 − r) + 4rs The function q(r), deﬁned by

8r Z 1 arcsin s s ds q(r) = 2 2 (C.18) π 0 (1 − r) + 4rs in the interval 0 ≤ r ≤ 1, has the endpoint values

q(0) = 0, q(1) = log 2. (C.19) C.3 Flux Proﬁle 259

Therefore,

ε 1 − r2 v (r) = − log + 2 log(1 − r) + − log(1 + r2) + q(r) + O(ε), (C.20) ray 2 4 √ is the MFPT for ε 1 and 1 − r ε. In particular,

ε 1 v = v (0) = − log + + O(ε), center ray 2 4 as asserted in (6.2).

C.3 Flux Proﬁle

In this appendix we calculate the ﬂux proﬁle given by equation (6.36). Sub- stituting equation (6.21) for h1 in equation (6.36) gives

 u  1 d Z t u sin du  √ 2  1 d  θ Z π−ε   π dt 0 cos u − cos t  f(θ) = − + cos √ dt . 2 dθ  2 0 cos t − cos θ   

Integration by parts and changing the order of integration, we ﬁnd that

θ u " cos Z π−ε u sin du 1 1 d 2 2 f(θ) = − + p √ 2 π dθ cos(π − ε) − cos θ 0 cos u + cos ε # 1 θ Z π−ε u Z π−ε sin t dt − cos u sin du 3/2 1/2 . 2 2 0 2 u (cos t − cos θ) (cos u − cos t) √ We evaluate the inner integral by making the substitution x = cos u − cos t, √ Z π−ε sin t dt 2 cos u + cos ε √ 3/2 1/2 = . u (cos t − cos θ) (cos u − cos t) (cos u − cos θ) − cos θ − cos ε 260 Appendix of Chapter 6

Therefore

" θ u 1 1 d cos Z π−ε u sin du f(θ) = − + √ 2 √ 2 2 π dθ − cos ε − cos θ 0 cos u + cos ε θ u√ cos Z π−ε u sin cos u + cos ε # −√ 2 2 du − cos θ − cos ε 0 cos u − cos θ  θ cos 1 1 d 2 π = − + √ √ a0− 2 π dθ  − cos ε − cos θ 2

θ u√  cos Z π−ε u sin cos u + cos ε √ 2 2  du . − cos θ − cos ε 0 cos u − cos θ

The substitution (6.23) gives u√ Z π−ε u sin cos u + cos ε du 2 = 0 cos u − cos θ r ε ε arccos s2 + sin2 s2 ds √ Z cos 2 2 2 2 . ε θ r ε 0 s2 + sin2 − cos2 s2 + sin2 2 2 2

Therefore, the ﬂux takes the form

 θ 1 1 d cos f(θ) = − + √ √ 2 × 2 2π dθ  − cos θ − cos ε

 r  2 2 ε 2 Z cos(ε/2) 4 arccos s + sin s ds  2  πa0 −  ,  ε θ r ε  0 s2 + sin2 − cos2 sin2 + s2 2 2 2 C.3 Flux Proﬁle 261 which is rewritten as

 θ √ √  cos Z 1−a2 2 2 2 ! 1 1 d 2 arccos s + a s ds f(θ) = − +  πa0 − 4 √  ,  2 2 2 2  2 2π dθ b 0 a + s (s + b ) (C.21) ε where a = sin and 2b2 = − cos θ − cos ε. Writing 2 √ 2 2 ∞ arccos s + a X 2n φ(a, s) = √ = φ2n(a) s , 2 2 s + a n=0 we find the Taylor coefficients arccos a arccos a 1 φ0(a) = , φ2(a) = − + √ , a 2a3 2a2 1 − a2 and so on. For all n ≥ 0 we find the asymptotic behavior c 1 φ (a) ∼ 2n + O as a → 0. 2n a2n+1 a2n To see this, consider the Taylor expansions

r !2n+1 ∞ s2 X s2m 1 + = cm a n a2m m=0 r ! ∞ r !2n+1 s2 π X s2 arccos a 1 + = + α a2n+1 1 + a 2 n a n=0 ∞ ∞ π X X s2m = + α a2n+1 cm 2 n n a2m n=0 m=0 ∞ ∞ π X s2m X = + cmα a2n+1, 2 a2m n n m=0 n=0

m where αn and cn are (known) constants, and ∞ 1 1 X (2n)! s2n = (−1)n . (C.22) r s2 a (2nn!)2 a2n a 1 + n=0 a 262 Appendix of Chapter 6

Therefore ∞ ∞ ∞ ! 1 X (2n)! s2n π X s2m X φ(a, s) = (−1)n + cmα a2n+1 , (C.23) a (2nn!)2 a2n 2 a2m n n n=0 m=0 n=0 from which it follows that ∞ X π (2n)! s2n φ(a, s) = (−1)n + O(a) . (C.24) 2 (2nn!)2 a2n+1 n=0 This shows that π (2n)! φ (a) ∼ (−1)n + O a−2n , (C.25) 2n 2 (2nn!)2a2n+1

as asserted. The asymptotic behavior (C.25) of the coeﬃcients φ2n(a) can be used to estimate the integral in equation (C.21), √ √ 2 ∞ 2 Z 1−a φ(a, s) s2 ds X Z 1−a s2n+2 ds = φ (a) . (C.26) s2 + b2 2n s2 + b2 0 n=0 0 To extract the asymptotic behavior of the integral as b → 0, we use the long division n s2n+2 X (−1)n+1b2n+2 = (−1)jb2js2n−2j + (C.27) s2 + b2 s2 + b2 j=0 and integrate it to yield √ Z 1−a2 s2n+2 2 2 ds = (C.28) 0 s + b √ √ n " 2 # 2 X Z 1−a Z 1−a ds (−1)jb2j s2n−2j ds + (−1)n+1b2n+2 s2 + b2 j=0 0 0 n √ √ X ( 1 − a2)2n−2j+1 1 − a2 = (−1)jb2j + (−1)n+1b2n+1 arctan . 2n − 2j + 1 b j=0 The Taylor expansion

√ ∞ 1 − a2 π X (−1)m+1 b2m+1 arctan = + √ (C.29) b 2 2m + 1 2 2m+1 m=0 ( 1 − a ) C.3 Flux Proﬁle 263 gives the Taylor expansion of the integral (C.28) in powers of b as √ 2 n √ Z 1−a s2n+2 X ( 1 − a2)2n−2j+1 ds = (−1)jb2j s2 + b2 2n − 2j + 1 0 j=0

∞ ! π X (−1)m+1 b2m+1 +(−1)n+1b2n+1 + √ 2 2m + 1 2 2m+1 m=0 ( 1 − a ) n √ X ( 1 − a2)2n−2j+1 = (−1)j b2j 2n − 2j + 1 j=0 ∞ π X (−1)n+m +(−1)n+1 b2n+1 + √ b2m+2n+2. 2 2 2m+1 m=0 (2m + 1)( 1 − a ) Therefore, the Taylor expansion of the integral (C.26) is √ 2 ∞ n √ Z 1−a φ(a, s) s2 ds X X ( 1 − a2)2n−2j+1 = φ (a) (−1)j b2j s2 + b2 2n 2n − 2j + 1 0 n=0 j=0 ∞ π X (−1)n+m +(−1)n+1 b2n+1 + √ b2m+2n+2 . 2 2 2m+1 m=0 (2m + 1)( 1 − a ) Rearranging in powers of b, we find that √ 2 ∞ Z 1−a φ(a, s) s2 ds X = β (a) bn, (C.30) s2 + b2 n 0 n=0 where the first three coefficients are √ ∞ √ 2 X ( 1 − a2)2n+1 Z 1−a π β (a) = φ (a) = φ(a, s) ds = a , 0 2n 2n + 1 4 0 n=0 0 πφ (a) π arccos a β (a) = − 0 = − , 1 2 2a √ ∞ 2 2n−1 X ( 1 − a ) φ0(a) β2(a) = − φ2n(a) + √ 2n − 1 2 n=1 1 − a √ Z 1−a2 φ(a, s) − φ (a) φ (a) = − 0 ds + √ 0 2 2 0 s 1 − a 264 Appendix of Chapter 6

and all other coeﬃcients βn are recovered in a similar fashion,

π π2 (2j)! β = (−1)j+1 φ (a) = − + O(a−2j), 2j+1 2 2j 4 (2jj!)2a2j+1

∞ √ X ( 1 − a2)2n−2j+1 β = (−1)j φ (a) 2j 2n 2n − 2j + 1 n=j

j−1 ! X 1 − φ2n(a) √ 2 2j−2n−1 n=0 (2j − 2n − 1)( 1 − a ) √ 2 ∞ Z 1−a 1 X = (−1)j φ (a)s2n ds s2j 2n 0 n=j

j−1 ! X 1 − φ2n(a) √ 2 2j−2n−1 n=0 (2j − 2n − 1)( 1 − a ) √ Z 1−a2 Pj−1 2n j φ(a, s) − n=0 φ2n(a)s = (−1) 2j ds 0 s

j−1 ! X 1 − φ2n(a) √ . 2 2j−2n−1 n=0 (2j − 2n − 1)( 1 − a )

We see that extra effort should be put in finding the even coefficients β2n. Expanding

∞ π 1 X (2n)! (s2 + a2)n φ(a, s) = √ − , (C.31) 2 2 2 (2nn!)2 2n + 1 s + a n=0

and noting that the following inﬁnite sum has a regular contribution

√ 2 ∞ Z 1−a 1 X (2n)! (s2 + a2)n lim ds = Cj, (C.32) a→0 s2j (2nn!)2 2n + 1 0 n=j C.3 Flux Proﬁle 265

where Cj are constants (also can be written in term of hypergeometric functions), we ﬁnd an alternative representation for the even coeﬃcients,

√ Z 1−a2 Pj−1 2n j φ(a, s) − n=0 φ2n(a)s β2j = (−1) 2j ds 0 s

j−1 ! X φ2n(a) − √ 2 2j−2n−1 n=0 (2j − 2n − 1)( 1 − a ) j = (−1) −Cj + O(a) +

j−1 2 2 n j−1 π 1 X (2n)! (s + a ) X 2n √ √ − − φ2n(a)s Z 1−a2 2 2 2 (2nn!)2 2n + 1 s + a n=0 n=0 2j ds 0 s

j−1 X 1 − φ2n(a) √ 2 2j−2n−1 n=0 (2j − 2n − 1)( 1 − a )

j−1 2 2 n j−1 π 1 X (2n)! (s + a ) X 2n √ √ − − φ2n(a)s Z 1−a2 2 s2 + a2 (2nn!)2 2n + 1 j n=0 n=0 = (−1) 2j ds 0 s π (2j − 2)! 1 1 −(−1)j−1 + O . 2 (2j−1(j − 1)!)2 a2j−1 a2j−2

The integrals are given in [145],

j−1 n−j+1/2 Z ds 1 X (−1)n j − 1 s2 √ = . (C.33) 2j 2 2 a2j 2n − 2j + 1 n s2 + a2 s s + a n=0

The binomial expansion gives

n Z (s2 + a2)n ds X 1 n = s2k−2j+1a2n−2k. (C.34) s2j 2k − 2j + 1 k k=0 266 Appendix of Chapter 6

Altogether, we ﬁnd that the integral term in equation (C.33) is

j−1 2 2 n j−1 π 1 X (2n)! (s + a ) X 2n √ √ − − φ2n(a)s Z 1−a2 2 2 2 (2nn!)2 2n + 1 s + a n=0 n=0 2j ds = 0 s

j−1 π 1 X (−1)n j − 1 1 = + O . 2 a2j 2n − 2j + 1 n a2j−1 n=0 This sum has the closed form [145]

k X (−1)i k (2kk!)2 = , (C.35) 2i + 1 i (2k + 1)! i=0 and we have obtained the asymptotic form of the even coeﬃcients

π 1 (2j−1(j − 1)!)2 1 β = + O . (C.36) 2j 2 a2j (2j − 1)! a2j−1

We are now able to find the asymptotic expansion of the flux profile (C.21),

 θ √ √  cos Z 1−a2 2 2 2 ! 1 1 d 2 arccos s + a s ds f(θ) = − +  πa0 − 4 √   2 2 2 2  2 2π dθ b 0 a + s (s + b )

" ∞ # 1 2 d θ X = − − cos β bn . 2 π dθ 2 n+1 n=0 Setting εα = π −θ, we obtain after some manipulations that to leading order in small ε the ﬂux is given in the interval −1 < α < 1 by

∞ " 2 # α2 1 X (2n+1(n + 1)!) (2nn!)2 f(α) = − √ − α2 − (1 − α2)n+1/2 2 ε (2n + 2)! (2n + 1)! ε 1 − α n=0 ∞ π X (2n)! (2n + 2)!(2n + 2) − − α2 (1 − α2)n + O(1). (C.37) 2ε (2nn!)2 (2n+1(n + 1)!)2 n=0 Appendix D

Appendix of Chapter 7

D.1 Laplace Beltrami Operator on 2-Sphere

The Laplace Beltrami operator on a manifold is given by

1 X ∂ √ ∂f ∆ f = √ gij det G , (D.1) M ∂ξ ∂ξ det G i,j i j where ∂r ij −1 ti = , gij = hti, tji,G = (gij), g = gij . (D.2) ∂ξi In spherical coordinates we have

2 2 2 gθθ = R , gφφ = R sin θ, gθφ = gφθ = 0. (D.3)

Therefore, for a function w = w(θ, φ)

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