Diploma Thesis
On Brownian motion of asymmetric particles An application as molecular motor
MARTIN REICHELSDORFER
May 2010 Institut f¨urTheoretische Physik I Prof. Dr. Klaus Mecke Friedrich-Alexander Universit¨atErlangen-N¨urnberg Zusammenfassung
Asymmetrische Brownsche Teilchen k¨onnenunter Nichtgleichgewichtsbedingungen eine im Mittel ge- richtete Bewegung ausf¨uhren. Dies kann zur Konstruktion molekularer Motoren genutzt werden. Die vorliegende Arbeit befasst sich mit der Erweiterung und Analyse eines biologisch inspirierten Modells, welches von Susan Sporer et al. [42] eingef¨uhrt wurde. Aufgrund der oft sehr allgemeinen und abstrakten Darstellung sind die Betrachtungen jedoch auch im weiter gefassten Kontext der generellen Brownschen Bewegung asymmetrischer Objekte relevant. Das Modell besteht aus einem asymmetrischen zweidimensionalen K¨orper, der sich entlang einer Schiene in einem zweidimensionalen, unendlich ausgedehnten idealen Gas bewegt. Durch vor¨uber- gehendes Verlangsamen an Stoppstellen wird der Motor bzw. das zugeh¨origestatistische Ensemble aus dem Gleichgewicht gebracht. Anhand des Beispiels eines parabolischen Potentials mit exponentiell abfallender Federkonstante wird gezeigt, dass sich diese Bindungspl¨atzedurch zeitlich ver¨anderliche Potentialmulden darstellen lassen. Ahnlich¨ der K¨uhlung eines Gases durch Expansion, f¨uhrtdieser Ausschaltevorgang zu einer Verschm¨alerungder Geschwindigkeitsverteilung. Haupts¨achlich w¨ahrend der darauffolgenden Relaxation erh¨altdie zuf¨alligeBewegung eine Vorzugsrichtung bis sich erneut die Gleichgewichtsverteilung einstellt. Ein einmaliges L¨osendes Motors aus einer Stoppstelle erweist sich im betrachteten Beispiel als eher ineffizient, in dem Sinne dass die Breite der Ortsverteilung stets deutlich gr¨oßerals die Verschiebung ihres Mittelwertes ist. Durch periodische Anordnung der Bindungspl¨atzejedoch kann eine effektive Gleichrichtung der Bewegung erreicht werden, da eine Equilibrierung des Systems verhindert wird und sich stattdessen ein station¨arerNichtgleichgewichts- zustand einstellt. Da das Modell, bedingt durch seine Komplexit¨at,nur n¨aherungweise gel¨ostwerden kann, werden ferner eine Reihe reduzierter Modelle entwickelt. Diese weisen die gleichen charakteristischen Eigen- schaften wie das allgemeinere Modell auf, sind jedoch analytisch besser zug¨anglich. Hervorzuheben ist insbesondere ein System aus nur drei diskreten Geschwindigkeitszust¨anden,welches als Minimalmo- dell eines asymmetrischen Brownschen Teilchens in einer Dimension angesehen werden kann. Es zeigt sich, dass sich die Bewegung von asymmetrischen und symmetrischen Teilchen nicht nur im Nichtgle- ichgewicht (wie vielleicht aus den vorangegangen Beobachtungen zu vermuten gewesen w¨are)sondern auch im Gleichgewicht unterscheidet. Durch Erg¨anzungum diskrete Ortszust¨aendeerlaubt dieses Modell dar¨uber hinaus die bereits angesprochene Untersuchung sich wiederholender Stoppstellen. Hierbei lassen sich qualitative Ubereinstimmungen¨ zu experimentellen Daten eines monomerischen Kinesin KIF1A Motors feststellen. Zu ¨uberpr¨ufen, ob sich die Ergebnisse auch quantitativ repro- duzieren lassen, w¨areAufgabe weiterer Untersuchungen. Verh¨altnism¨aßiggroße thermische Fluktuationen sind die treibende Kraft des Motors, weshalb neue Methoden zu seiner thermodynamischen Beschreibung angewandt werden m¨ussen.Im Rahmen der sogenannten stochastischen Thermodynamik k¨onneneinzelnen Phasenraumtrajektorien Gr¨oßen wie Energie, Arbeit, W¨armeoder sogar Entropie zugeordnet werden, die nunmehr als stochastische Variablen mit ausgedehnten Wahrscheinlichkeitsverteilungen zu verstehen sind. Hinsichtlich dieser Verteilungen l¨asstsich eine Reihe von Relationen, genannt Fluktuationstheoreme, finden, die oft eine detailliertere Fassung bekannter thermodynamischer Aussagen, wie etwa des zweiten Hauptsatzes, darstellen und bereits bei der Analyse experimenteller Daten, z.B. bei der Messung von freien En- ergien, Anwendung gefunden haben. Die G¨ultigkeit dieser Theoreme f¨urdas vorliegende Modell kann durch direkte Rechnung best¨atigtwerden. Abstract
Asymmetric Brownian particles are able to conduct directed average motion under non-equilibrium conditions. This can be used for the construction of molecular motors. The present paper is concerned with the extension and analysis of a biologically inspired models, which was introduced by Susan Sporer et al. [42]. Due to the often quite abstract formulation, the considerations are also relevant in the broader context of general Brownian motion of asymmetric objects. The model consists of an asymmetric two-dimensional body, which moves along a rail through a two-dimensional, infinite ideal gas. By temporarily slowing it down at stopping sites, the motor or the corresponding statistical ensemble, respectively, is pushed away from equilibrium. The example of a parabolic potential with exponentially decreasing spring constant shows that these binding sites can be modelled by time-dependent potential wells. Similar to the cooling of a gas by expansion, this release process leads to a narrowing of the velocity distribution. Particularly during the subsequent relaxation the random motion is biased until the equilibrium distribution is established again. A singular release of the motor is found to be rather inefficient in the considered case, in the sense that the width of the distribution of positions is always profoundly larger than the drift of its mean value. By periodically distributing the binding sites, however, an effective rectification of the motion can be achieved, since the equilibration of the system is prevented and a stationary non-equilibrium sate establishes instead. As the model can only be solved approximately due to its complexity, a series of reduced models is developed. These systems exhibit the same characteristic features as the more general model, but are easier to access analytically. In particular, a system composed of only three discrete velocity levels can be regarded as a minimal model of an asymmetric Brownian particle in one dimension. It turns out that the motion of asymmetric and symmetric particles differ not only in non-equilibrium (as might have been anticipated from prior observations), but also in equilibrium. Moreover, augmenting the model with discrete position states allows for the already mentioned investigation of repeated stopping sites. Hereat, qualitative accordances to experimental data from a monomeric kinesin KIF1A motor are observed. To test, whether the results can also be reproduced quantitatively could be the subject of further studies. Comparatively large thermal fluctuations are the driving force of the motor, wherefore new methods have to be applied for its thermodynamic description. In the framework of so-called stochastic thermodynamics, quantities like energy, work, heat or even entropy may be attributed to individual phase space trajectories. Hence, they are now to be understood as stochastic variables with wide- stretching probability distributions. With respect to these distributions a multitude of relations, called fluctuation theorems, can be found, which often constitute a more detailed version of common thermodynamic statements like the second law, and which have already been applied in the analysis of experimental data (e.g. in the determination of free energies). The validity of these theorems for the present model can be confirmed by direct calculation. 4 Contents
1 Why Brownian motion is still interesting 7
2 Outline 9
3 Molecular motors in theory and reality – the context of this work 11 3.1 Two examples of theoretical models ...... 11 3.2 Kinesin ...... 11 3.3 A kinesin inspired model ...... 14
4 Brownian motion as stochastic process - the mathematical framework 17 4.1 Selected foundations of stochastic processes ...... 17 4.2 Description of the Model ...... 21 4.3 Modelling the stochastic process ...... 21 4.4 Time evolution of the phase space probability density ...... 25 4.5 Time evolution of moments of the phase space probability density ...... 29
5 Using potential wells as stopping sites 31 5.1 Relaxation without potential ...... 32 5.2 Potential-change without collisions: time-dependent undamped harmonic oscillator . . 37 5.3 Time evolution with potential and collisions ...... 41 5.4 A remark on re-opening and multiple stopping sites ...... 50
6 Minimal Models 53 6.1 Two-state model ...... 53 6.2 Three-state generic model of an asymmetric Brownian particle in one dimension . . . 57 6.3 Augmented three-state model ...... 64 6.4 A step towards bridging the gap back to reality ...... 70
7 Stochastic thermodynamics 73 7.1 Energy, work and heat ...... 73 7.2 Entropy ...... 75 7.3 Fluctuation theorems ...... 76
8 Conclusion and Outlook 81
A Derivations and Proofs 85 A.1 Coefficients of the differential Chapman-Kolmogorov equation ...... 85 A.2 Relation of dissipated heat and transition rates ...... 88 A.3 Expansion of the Kramers-Moyal coefficients ...... 89 A.4 Limit of the mean squared velocity of the time-dependent undamped harmonic oscillator 90 A.5 Moments of the phase space distribution of an harmonic oscillator in equilibrium . . . 91 A.6 Properties of the spectrum of the transition matrix ...... 91
5 Contents
6 1 Why Brownian motion is still interesting
For thou wilt mark here many a speck, impelled By viewless blows, to change its little course, And beaten backwards to return again, Hither and thither in all directions round.
Titus Lucretius Carus (ca. 60 BC) [5]
Actually, Brownian motion is a fairly old story. Observations date back as far as to ancient times, where, for instance, in the first century BC Titus Lucretius Carus [5] describes the jittering motion of small specks of dust in sunlight and – probably inspired by Epicurean atomistic physics – originates it in the underlying dynamics of the “primeval atoms”. In 1827, the Scottish botanist Robert Brown [4] finds erratically moving floating particles during his observations of pollen and succeeds in drawing the scientific community’s attention back to this phenomenon, bringing him the honour of coining its name. Some 80 years later, Albert Einstein [12] and Marian Smoluchowski [44] contribute greatly to clarify the physical background and present a quantitative analytical description, serving as well as a way of indirectly confirming the existence of atoms and molecules, being still disputed at that time. Their models successfully explain the characteristic observations of Brownian motion, namely a linearly growing mean squared displacement1 of the particle and and unchanged average position:
hXi = 0, (1.1) hX2i = 2dDt. (1.2)
The parameter d gives the number of dimensions and D is the diffusion constant. Moreover, Einstein and Smoluchowski are able to relate friction and diffusion to the same underlying mechanism, the collisions of fluid molecules with the Brownian particle. This fact is expressed by the so-called Einstein-Smoluchowski relation (cf. Reference [12] or [30] p. 120)
k T D = B , (1.3) γ which relates the diffusion constant D to the friction coefficient γ. As factor of proportionality serves the thermal energy, where kB is Boltzmann’s constant and T the absolute temperature. Now, again more then a hundred years later, still many physicists are working on this topic. One of the reason is an increasing interest in small (often also non-equilibrium) systems, where thermal fluctuations are comparably large and play a fundamental role. This is in particular true in the research field of biological physics, where biological processes are physically investigated down to molecular scales. The biological background of the present paper are so-called molecular motors,
1In practise, this linear growth law of hX2i does not hold for arbitrary small times, as Einstein already remarks in Reference [13]. Instead, one typically observes a transition from ballistic (hX2i ∝ t2) to diffusive (hX2i ∝ t) motion.
7 1 Why Brownian motion is still interesting which are known to carry out active transport by, for instance, dragging along loads or pumping ions through membranes. Even force generation in muscle cells is due to the co-operation of many individual motor molecules (for instance, see References [3, 30, 23, 22, 2]). Remarkably, many models for these system are based on the seemingly undirected Brownian motion, but at a closer look one finds that, indeed, under non-equilibrium conditions and in presence of a spatial asymmetry net motion may be obtained. This class of molecular motors is therefore called Brownian motors.
8 2 Outline
The focus of the present work lies on the analysis and the extension of a model of a Brownian motor, which was introduced by Susan Sporer et al. (cf. References [42, 41]), and which makes use of an intrinsic asymmetry of the Brownian particle itself. Non-equilibrium is established by stopping and releasing the motor at stopping sites. In course of the generalisation, the approach to the problem resembles in places the steps taken in Reference [41], and results therein obtained are retrieved as specialisations. As the treatment is often quite abstract, it has also relevance for the general issue of Brownian motion of asymmetric particles, especially in non-equilibrium states. The paper is organized as follows: Chapter 3 introduces the conceptional and the biological background of the present model. The major results of Reference [42] are outlined. In Chapter 4, the model is described in more detail and the mathematical framework is developed. The original description, which was based on a master equation for the time evolution of the velocity probability density, is extended to include positions and time-dependent potentials. Moreover, it is derived from a Poisson type stochastic process, which models individual collisions of particles of the medium with the motor. The probability of a specific realisation of this stochastic process is given, which may in principal serve as a starting point for path integral techniques an which is applied to directly verify so-called fluctuation theorems in Chapter 7. The inclusion of potentials allows to show in Chapter 5 how stopping sites may be realised by potential wells – a point completely left open in Reference [42]. Interestingly, not the trapping, but the release from the well by gradually turning it off stops the motor in the sense that the velocity distribution is narrowed. The efficiency of the process and different influences are studied. In general, a singular release from a potential well is found to have a rather weak rectifying effect. Since a complete solution of the “full” model is not feasible, a set of minimal models is developed in Chapter 6. These systems have the advantage of allowing for better analytical access, while still exhibiting the same characteristic behaviour. In particular, a system with three velocity states is devised as a minimal model for an asymmetric Brownian particle in one dimension. The effect of asymmetry, being perceivable even in equilibrium, and the role of detailed balance are examined. Additionally, the implementation of a discrete position space makes it possible to study the influence of (multiple) stopping sites that are able to bind a released particle again. This setup is capable of preventing the equilibration of the system successfully and the motion of the particle is effectively rectified. Finally, in Chapter 7, the model is viewed from the point of stochastic thermodynamics. As thermal fluctuations are the driving force of the motor, they have to be accounted for in thermodynamic considerations, leading to trajectory dependent formulations of quantities like work, heat or entropy, which – instead of single deterministic values – posses probability distributions, which obey the fluctuation theorems mentioned above. Using the path integral weights derived in Chapter 4, it is possible to verify these relations by direct calculation.
9 2 Outline
10 3 Molecular motors in theory and reality – the context of this work
In this chapter, two theoretical models are very briefly discussed. The first one is a very prominent example, showing very instructively how the combination of asymmetry and non-equilibrium can generate net motion. The second one has great similarities to the present system and served as an inspiration in fact. Additionally, also the biological background, a molecular motor called kinesin, is introduced. Eventually, the model from References [42, 41], which is the basis of the present work, is briefly sketched and relevant results are summarised.
3.1 Two examples of theoretical models
There are two basic ingredients, needed to rectify Brownian motion: asymmetry and non-equilibrium1 (cf., for instance, Reference [30] p. 422). In the last few decades, a multitude of models have been brought forward, all sporting different implementations of this concept. Reference [32] gives a very thorough overview of this topic. Some of these models have already successfully been used to explain experimental data of biological molecular motors. One of the first models that has been proposed features a periodic sawtooth potential that is turned off and on repeatedly (cf. Reference [33]). See Figure 3.1 for an illustration. The particles gather in the minima and start diffusing symmetrically when the potential is turned off. Since the distances to the neighbouring maxima on the left and on the right are different, more particles will have crossed the closer one when the potential is turned on again and slide into the next minimum. Thus, a net current is established. Another – not so intuitive – model is sketched in Figure 3.2. Two asymmetric objects, which are stiffly linked, are placed each in a separate heat bath, consisting of ideal gases. Different variants of this setup are studied in References [10, 29, 11, 46, 45, 9], including various object shapes, two- dimensional, three-dimensional, translational and rotational realisations, and even the application as heat pump or refrigerator, respectively. If the temperatures are different, the system is not in equilibrium and the motor performs a net motion while transferring heat (by collisions with gas particles) from the warmer to the cooler reservoir. In contrast to the first system, which uses an asymmetric external potential, this model possesses an intrinsic asymmetry. The same principle is used by the motor under consideration in the present paper, but another mechanism for perturbing thermal equilibrium is applied.
3.2 Kinesin
According to Reference [30] p. 58, the family of single-protein motors called kinesins are, for instance, involved in the transport of proteins to axon terminals and of the ingredients from which synaptic vesicles will be built. The following is based on Reference [30] p. 437 ff. A kinesin molecule usually possesses two heads or motor domains, with which according to common theory it “walks” down microtubules by binding and unbinding the heads at subsequent binding sites. During one cycle the energy from
1Throughout this paper, when speaking of the motor being in or out of equilibrium, or having a certain distribution, this, of course, actually refers to the whole corresponding statistical ensemble.
11 3 Molecular motors in theory and reality – the context of this work
(a) U
tential U
(b) concentration po
(c) concentration
Figure 3.1 Model of a Brownian motor using a sawtooth potential that is periodi- cally switched on and off (a). (b) In the “on”-state particles gather in the potential minima. (c) In the “off”-state particles diffuse freely. Due to the asymmetry of the po- tential, more particles cross the right-hand maximum (shaded area) than the left-hand maximum. (adapted from Reference [33])
Figure 3.2 Model of a thermal Brownian motor, making use of intrinsic asymmetry. Two (in this particular case identical) rigidly linked objects are placed in separated heat baths with different Temperatures T1 and T2. (adapted from Reference [29])
12 3.2 Kinesin
Figure 3.3 Details of the model for the stepping of two-headed kinesin. Each of the steps of this cyclic reaction is described in the text of Section 3.2. The steps form a loop, to be read clockwise from upper left. The grey symbols represent a microtubule, which is slightly polar, indicated by the “+” and the “-” sign. Strong physical bonds are denoted by multiple lines, weak ones by single lines. The symbols T, D and P denote ATP, ADP and inorganic phosphate, respectively. (from Reference [30]) the hydrolysis of one adenosine triphosphate molecule (ATP) is transduced and the motor crosses a distance of approximately 8 nm. In Figure 3.3 a proposed mechanism, some elements of which are still under debate, is graphically summarised. The cycle is not meant to be taken literally; it just shows some of the distinct steps in the enzymatic pathway and follows. Although taken from Reference [30], it originally follows an analysis in Reference [36]. Initially (top left panel of the figure), a kinesin dimer approaches the microtubule from solution and binds one head, releasing one of its adenosine diphosphates (ADP), meanwhile the other head cannot reach any binding site because its 0 tether is too short (E). Then, the bound head binds an ATP molecule from solution (ES1, ES1), which causes its so-called neck linker to dock on the head, biasing the other head’s random motion in the forward direction (see also Reference [26]). Being thrown forward by the bound head’s neck linker greatly increases the probability that the elastic tethers will momentarily stretch far enough for the free head to reach the next binding site. It may bind weakly, then detach, many times (ES2). Eventually, instead of detaching, the forward head releases its ADP and by that binds strongly to the microtubule (ES3). Meanwhile, the rear head splits its ATP and releases the resulting phosphate (EP). This reaction weakens its binding to the microtubule. The strain induced by the binding of the forward head then biases the rear head to unbind from the microtubule (rather than releasing its ADP). Obviously, kinesin has been cunningly designed to coordinate the action of its two heads, and one might wonder how such a complex motor could have evolved from something simpler. Or, putting the
13 3 Molecular motors in theory and reality – the context of this work
Figure 3.4 Model for single-headed kinesin motility. Bound ATP is denoted by T; ADP and P molecules are not shown. Other symbols are as in Figure 3.3. (a) Initially, the kinesin monomer is strongly bound to site n on the microtubule. (b) In the weakly bound state, the kinesin wanders freely along the microtubule. (c) When the kinesin reenters the strongly bound state, it is most likely to rebind to its original site, somewhat likely to rebind to the next site and least likely to bind to the previous site. Relative probabilities are represented by shading. (adapted from Reference [30]) matter differently, one could ask whether there is a simpler force-generating mechanism, perhaps not as efficient or as powerful as two-headed kinesin, which could have been its evolutionary precursor. In fact, a single-headed (monomeric) form of kinesin, called KIF1A, has been found to have single- molecule motor activity (cf. References [31, 30]). For this motor, experiments showed strongly stochastic trajectories with linearly increasing average displacement and variance. The authors of Reference [31] successfully modelled this system as a Brownian motor in an asymmetric ratchet potential (cf. Figure 3.1), which is assumed to originate from the polar structure of the microtubule. The chemical cycle of ATP binding, ATP hydrolysis and ADP release is comprised in two states (s and w), one corresponding to strong binding (s) (i.e, the motor feels the potential of the binding sites) and the other corresponding to weak binding (w) (i.e., the sawtooth potential is essentially turned off). After entering a state, the motor waits an average time ts or tw, respectively, before snapping back to the other state. This cycle is illustrated in Figure 3.4.
3.3 A kinesin inspired model
Inspired by the single headed kinesin Susan Sporer et al. [42, 41] brought forward a model of a Brownian motor, which picks up characteristic features of the biological system, but uses a different approach than Reference [31]. Instead of focussing on the slight asymmetry in the binding potentials, they based their model on the asymmetry of the Brownian particle itself. The microtubule is repre- sented by a rail with periodic stopping sites, where the motor is stopped and subsequently released, whenever it reaches one of them. This serves to sustain a non-equilibrium state. Figure 3.5 depicts a triangular two-dimensional motor in a Lennars-Jones fluid, which was used for simulations. As well by simulations as by analytical calculations it was possible to show that indeed an average motion could be generated (cf. Figures 3.6 and 3.7). Although it is unclear to what extent this mechanism plays a role in the motility of the kinesin KIF1A, the model constitutes an elegant concept to realise rectified Brownian motion in a completely symmetric environment and in a single heat bath.
14 3.3 A kinesin inspired model
Figure 3.5 An asymmetrically shaped motor (here a triangle) is built from fluid par- ticles and placed in a Lennard-Jones fluid. Its motion is restricted to a one-dimensional track with periodically spaced binding sites along the X axis. If the motors center of mass crosses a binding site, the velocity of the motor is set to zero. (adapted from Reference [42])
Figure 3.6 Average position (400 runs) of an asymmetrically oriented, isosceles triangle in a Lennard-Jones fluid for different distances δs between stopping sites (σ is the Lennard-Jones radius of a particle). Directed motion is possible if the triangle is oriented asymmetrically with respect to the direction of motion. Otherwise the motor performs symmetric fluctuations around its starting position. Inset: trajectories of individual motors. (from Reference [42])
15 3 Molecular motors in theory and reality – the context of this work
Figure 3.7 Relaxation of a motor to thermal equilibrium: the molecular dynamics simulations of the LJ-fluid (symbols; averages over 400000 runs) can be well described by the analytic results in the ideal gas limit (lines) if the data are normalized by the maximum velocity Vmax and the relaxation time. (from Reference [42])
16 4 Brownian motion as stochastic process - the mathematical framework
Due to the enormous amount of particles in the medium into which the Brownian particle is embedded, it is almost impossible to treat the system deterministically and a statistical approach is appropriate. This chapter is concerned with how the motion of the motor may be modelled as a stochastic process. While Section 4.1 provides some selected theoretical background, the remaining sections apply the latter to the specific model under consideration in a yet very general fashion, that is for still arbitrary potentials and motor shapes. Starting from the description as a Poisson type process in path integral representation, a time evolution equation of the phase space probability density function (phase space pdf) is derived. Additionally, techniques for obtaining approximate solutions of this equation are introduced, including a Kramers-Moyal expansion, an expansion in the mass ratio of motor and medium particles, and equations for the time evolution of the moments of the phase space pdf.
4.1 Selected foundations of stochastic processes
According to Reference [19] p. 42 ff., systems which evolve probabilistically in time or more precisely, systems in which a certain time-dependent random variable X(t) exists, can mathematically be described as stochastic processes. One can measure values x1, x2, x3, ..., etc., of X(t) at times t1, t2, t3, ... and it is assumed that a set of joint probability densities P (x1, t1; x2, t2; x3, t3; ...) exists which describe the system completely. In terms of these joint probability density functions, one can also define conditional probability densities
P (x1, t1; x2, t2; ...|y1, τ1; y2, τ2; ...) := P (x1, t1; x2, t2; ...; y1, τ1; y2, τ2; ...)/P (y1, τ1; y2, τ2; ...), (4.1) which may be used for the prediction of future values of X(t) (i.e., x1, x2, ... at times t1, t2, ...), given the knowledge of the past (values y1, y2, ..., at times τ1, τ2, ...).
4.1.1 Markov processes A stochastic process is called a Markov process if the conditional probability is determined entirely by the knowledge of the most recent condition, that is
P (x1, t1; x2, t2; ...|y1, τ1; y2, τ2; ...) = P (x1, t1; x2, t2; ...|y1, τ1), (4.2) given the times satisfy the ordering t1 ≥ t2 ≥ t3 ≥ ... ≥ τ1 ≥ τ2 ≥ ... . As a consequence, any arbitrary joint probability density may be decomposed into a product of simple conditional probabilities of the type P (xi, ti|yi, τi). For example, by definition of the conditional probability P (x1, t1; x2, t2|y1, τ1) = P (x1, t1|x2, t2; y1, τ1)P (x2, t2|y1, τ1) and using the Markov assumption (4.2), one finds P (x1, t1; x2, t2|y1, τ1) = P (x1, t1|x2, t2)P (x2, t2|y1, τ1). (4.3) Integrating over all events of one kind in a joint probability density eliminates that variable, that is Z P (x1, t1|x3, t3) = dx2 P (x1, t1; x2, t2|x3, t3) Z = dx2 P (x1, t1|x2, t2; x3, t3)P (x2, t2|x3, t3). (4.4)
17 4 Brownian motion as stochastic process - the mathematical framework
Application of the Markov assumption (4.2) then yields Z P (x1, t1|x3, t3) = dx2 P (x1, t1|x2, t2)P (x2, t2|x3, t3), (4.5) which is the Chapman-Kolmogorov equation.
4.1.2 The differential Chapman-Kolmogorov equation of a Markov process In this paper, the formalism developed here is going to be applied to a Brownian particle with one degree of freedom, X(t), and corresponding velocity V (t) (see Section 4.2 for the detailed definition of the system). The phase space position of the motor, Γ(t) := (X(t),V (t)), thus takes over the role of the random variable. The sample paths of X(t) are expected to be continuous, whereas V (t) will posses discontinuities due to collisions with instantaneous hard body interaction. According to Reference [19] p. 46, it can be shown that with probability one the sample paths of a Markov process are continuous functions of t, if for any > 0 1 Z lim dx P (x, t + ∆t|z, t) = 0 (4.6) ∆t→0 ∆t |x−z|> uniformly in z and t. On page 47 it is stated that the Chapman-Kolmogorov equation can be reduced to a differential equation under appropriate assumptions, which are closely connected with the continuity properties of the process under consideration. Because of the form of the continuity condition (4.6), one is led to consider a method of dividing the differentiability conditions into parts, one corresponding to continuous motion of a representative point and the other to discontinuous motion. The following conditions are required for all > 0:
lim P (x, t + ∆t|z, t)/∆t = W (x|z, t) (4.7) ∆t→0 uniformly in x, z and t for |x − z| ≥ ; 1 Z lim dx (xi − zi)P (x, t + ∆t|z, t) = Ai(z, t) + O(); (4.8) ∆t→0 ∆t |x−z|< 1 Z lim dx (xi − zi)(xj − zj)P (x, t + ∆t|z, t) = Bij(z, t) + O(); (4.9) ∆t→0 ∆t |x−z|< the last two being uniform in z, and t. It can be shown that all higher-order coefficients of the above form must vanish. According to the condition for continuity (4.6), the process can only have continuous paths if W (x|z, t) vanishes for all x 6= z. Thus, this function must in some way describe discontinuous motion, while the quantities Ai and Bij must be connected with continuous motion. The derivation of the differential Chapman-Kolmogorov equation, which is to be found in detail in Reference [19] p. 48 ff., is only sketched here. It starts with considering the time evolution of the expectation value of a function f(z) which is twice continuously differentiable. Thus, Z Z 0 1 0 0 ∂t dx f(x)P (x, t|y, t ) = lim dx f(x)[P (x, t + ∆t|y, t ) − P (x, t|y, t )] ∆t→0 ∆t 1 Z Z = lim dx dz f(x)P (x, t + ∆t|z, t)P (z, t|y, t0)− ∆t→0 ∆t Z − dz f(z)P (z, t|y, t0) , (4.10)
18 4.1 Selected foundations of stochastic processes where the Chapman-Kolmogorov equation has been used in the second line. The next steps are to divide the integral over x into two regions |x − z| ≥ and |x − z| < , to expand f(x) in the terms where |x − z| < to second order around z with a remainder that vanishes as |x − z| → 0, to introduce a one in the form of R dx P (x, t + ∆t|z, t) into the last term of Equation (4.10), to identify the quantities W , Ai and Bij, and finally to take the limit → 0. Equation (4.10) now reads Z Z X ∂f(z) 1 X ∂2f(z) ∂ dz f(z)P (z, t|y, t0) = dz A (z, t) + B (z) P (z, t|y, t0)+ t i ∂z 2 ij ∂z ∂z i i i,j i j Z Z + dz f(z) dx [W (x|z, t)P (x, t|y, t0) − W (x|z, t)P (z, t|y, t0)] . (4.11)
Notice that the integral over x actually is a principal value integral lim R dx. The last thing →0 |x−z|> to do, is to integrate by parts, where an apt limitation of the stochastic process and f to some finite regions lets surface terms vanish, and which yields the differential form of the Chapman-Kolmogorov equation,
X ∂ ∂ P (z, t|y, t0) = − [A (z, t)P (z, t|y, t0)] t ∂z i i i X 1 ∂2 + [B (z, t)P (z, t|y, t0)] 2 ∂z ∂z ij i,j i j Z + dx [W (z|x, t)P (x, t|y, t0) − W (x|z, t)P (z, t|y, t0)], (4.12) onto which large parts of the analysis in this paper are based.
4.1.3 Detailed balance According to Reference [19] p. 148 f., a Markov process satisfies detailed balance if, roughly speak- ing, in the stationary situation each possible transition balances with the reversed transition. The mathematical formulation of this statement reads
0 0 0 P (x, t|x , 0)Ps(x ) = P (x , t|x, 0)Ps(x), (4.13) where x := (1x1, 2x2,...) represents the behaviour of x under time reversal and Ps(x) signifies the stationary distribution. The variable xi transforms to the reversed variable according to the rule
xi → ixi (4.14)
i = ±1 (4.15) depending on whether the variable is odd or even under time reversal. Positions X, for instance, are even, whereas velocities V are odd. This concept is motivated by physical considerations. For exam- ple, take a one-dimensional system in classical mechanics. A transition (X, V, t) → (X0,V 0, t + ∆t) is not simply reversed by interchanging primed and unprimed quantities. Rather, it is (X0, −V 0, t) → (X, −V, t + ∆t), which corresponds to the time reversed transition and requires the velocities to be reversed because the motion from X0 to X is in the opposite direction from that from X to X0. It is noteworthy that for t = 0 Equation (4.13) implies
Ps(x) = Ps(x), (4.16) since the conditional probabilities reduce to delta functions.
19 4 Brownian motion as stochastic process - the mathematical framework
On page 151 ff. of Reference [19] it is shown that for the differential Chapman-Kolmogorov equation necessary and sufficient conditions of detailed balance are given by 0 0 0 W (x|x )Ps(x ) =W (x |x)Ps(x), (4.17) X ∂ A (x)P (x) = − A (x)P (x) + [B (x)P (x)], (4.18) i i s i s ∂x ij s j j
ijBij(x) =Bij(x). (4.19) These conditions are in fact fulfilled by the present model as demonstrated in Section 4.4.2.
4.1.4 The Poisson process A key assumption of the model will be that collisions of particles of the medium with the motor occur independently of each other. The rates of these events will depend strongly on the current motor velocity, the velocity of the incoming particle and the location of the point of impact on the motor surface. For the derivation of basic properties, however, it is convenient to study the elementary process with only one sort of event and one corresponding rate. The generalisation and adaptation to the model is left to Section 4.3. The following is largely based on Reference [19] p. 12 f. Consider a stochastic process that consists of some kind of events occurring independently of each other and at constant rate λ. If N is the number of events counted up to a time t, then the probability that N increases by one in an infinitesimal interval ∆t is given by P (N + 1, t + ∆t|N, t) = λ∆t, (4.20) and one has for the total probability of having N counts at t + ∆t P (N, t + ∆t) =(1 − P (N + 1, t + ∆t|N, t))P (N, t) + P (N, t + ∆t|N − 1, t)P (N − 1, t) =(1 − λ∆t)P (N, t) + λ∆tP (N − 1, t), (4.21) for N > 0. The validity can be extended to N ≥ 0 by setting P (−1, t) = 0. Taking the limit ∆t → 0 yields ∂ P (N, t) = λ[P (N − 1, t) − P (N, t)], (4.22) ∂t which is solved by the Poisson distribution (λt)N P (N, t) = e−λt. (4.23) N! It is very instructive to reconsider this last formula in a slightly different way, as it allows for the assignment of illustrative interpretations to its individual components and for setting up detailed expressions for individual series of events. For instance, one finds that the probability that nothing happens during an interval t is given by
−λt 1 Pvoid(t) := P (0, t) = e . (4.24) Conversely, the probability of an event to occur in the infinitesimal interval dt is λdt. Having these expressions at hand, one can now construct the joint probability of a series of events taking place at times 0 < t1 < t2 < . . . < tN < t,
−λt1 −λ(t2−t1) −λ(tN −tN−1) −λ(t−tN ) dP ({ti}) = e λdt1 e λdt2 ... e λdtN e = N −λt = λ e dt1 ... dtN . (4.25)
By integrating out t1 to tN one eventually recovers Equation (4.23). 1An alternative way to arrive at this result is to divide t into m parts of length t/m. Then P (0, t) = (1 − λt/m)m, which goes to exp(−λt) as m → ∞.
20 4.2 Description of the Model
U(X,t) m ê M ê θ X motor V SF(θ)dθ X T,�ρ
Figure 4.1 Sketch of the system: The two-dimensional Figure 4.2 A line element of convex motor particle is immersed in an ideal gas and length SF (θ)dθ on the motor sur- free to move along the X axis with velocity V . M is face, parametrised by the angle θ the mass of the motor, whereas the gas particles have between the tangent vector eˆ|| and masses of m. Their density is ρ and the temperature of the X axis. the gas is T . In addition to collisions with gas particles the motor experiences forces due to a time-dependent potential U(X, t).
4.2 Description of the Model
The system under consideration resembles largely the one studied in Reference [42] and is sketched in Figure 4.1. A two-dimensional convex body, called the motor, of mass M, having one translational degree of freedom X, is placed inside an infinite two-dimensional ideal gas at constant temperature T with density ρ and particle masses m. The gas is assumed to be in thermal equilibrium at any time. Concerning interactions, motor and gas particles behave as fully elastic hard objects. The limitation to convex bodies has two reasons. First, Minkowski functionals may be applied to describe the shape of the motor, which was carried out in detail in Reference [41], and second, multiple collisions of gas particles with the body are mostly omitted. In combination with the ideal behaviour of the medium, impacts of particles on the motor may thus be treated as independent random events, which is be the basis of the stochastic process describing its motion. The model is extended by allowing the body to additionally experience forces due to a time-dependent potential U(X, t).
4.3 Modelling the stochastic process
As impacts of gas particles on the body are regarded independent and random, the chronology of these events may be modelled as a Poisson-type stochastic process with the phase space position Γ(t) := (X(t),V (t)) being the random variable. However, the rates of velocity-changes will be highly dependent on the current motor velocity. Generally, the faster the motor is travelling into one direction, the likelier a collision which gives it a push in the opposite direction. This is the microscopic realisation of friction and ensures a stable equilibrium distribution. Moreover, the shape and especially its asymmetry translate into these rates, which is in fact the crucial detail enabling asymmetric motion in non-equilibrium. Section 4.3.1 shows how the transitions rates explicitly derive from the shape of the motor. After- wards they are used to construct expressions for the probabilities of individual trajectories and the phase space density in Section 4.3.2.
21 4 Brownian motion as stochastic process - the mathematical framework
4.3.1 Collision rates An infinitesimal line element of the motor’s surface is characterised by an angle θ and a length factor F (θ) (cf. Figure 4.2). θ is defined as the angle between the tangent unit vector eˆ|| := (cos θ, sin θ) and the X axis. Hence, the corresponding normal vector is given by eˆ⊥ = (sin θ, − cos θ). The length of the line element SF (θ)dθ, where S is the total perimeter, implicitly defines the function F (θ) representing the fraction of the surface with tangent angle θ. Let v˜ be the velocity vector of a gas particle with respect to the motor’s rest frame. The current density of particles having a velocity between v˜ and v˜ + dv˜ heading towards the body’s surface is then given by
j(˜v) d˜v = ˜vρΦ(˜v + V)Θ(−˜veˆ⊥) d˜v, (4.26) where V := V eˆx, Φ(v) is the velocity probability density of the gas and the Heaviside Θ ensures that no particles are coming from inside the body. Accordingly, the flux or impact rate on a line element with tangent angle θ is
λi(θ, ˜v,V ) d˜vdθ = −SF (θ)eˆ⊥j(˜v) d˜vdθ. (4.27)
If one demands conservation of total energy, total momentum in X direction and conservation of the particle’s momentum tangential to the body’s surface, the velocity-change caused by a collision takes the form
∆V = αv˜⊥, (4.28)
m 2 M sin θ withv ˜⊥ := ˜veˆ⊥ and α := m 2 (cf. References [41, 11]). Usage of the two dimensional Maxwell- 1+ M sin θ Boltzmann velocity distribution
m mv2 Φ(v) = exp − , (4.29) 2πkBT 2kBT eventually leads to an expression for the rate of events inducing a velocity-change between ∆V and ∆V + d∆V at motor velocity V
2π ∞ Z Z d∆V λ(∆V,V ) d∆V := dθ d˜v λ (θ, (˜v , v˜ = ∆V/α),V ) || i || ⊥ |α| 0 −∞ 2π Z ∆V ∆V = dθ a Θ − exp −b(∆V c + V )2 d∆V (4.30) α|α| α 0 q with a := −ρSF (θ) m , b := m sin2 θ and c := 1 . An exemplary plot of λ(∆V,V ) is 2πkBT 2kBT α sin θ presented in Figure 4.3.
4.3.2 The phase space probability density
∗ Consider an interval t0 ≤ t ≤ t during which the motor experiences N collisions at times {ti|i = ∗ 1, ..., N; t0 ≤ t1... ≤ t } causing velocity-changes {∆Vi|i = 1, ..., N}. As a shorthand notation, this trajectory of velocity-jumps will be denoted by the symbol ∆V (t). In this section, expressions for three probabilities will be developed: first, the conditional proba- bility density for the system to take a specific path ∆V (t), P [∆V (t)|Γ0, t0], given the starting point Γ0 at t0, second, the conditional probability density of finding it a the phase space position Γ at time ∗ ∗ ∗ t , P (Γ , t |Γ0, t0), given the same starting point, and third, the phase space probability density ∗ ∗ function (pdf), P (Γ , t ), given a certain initial distribution P0(Γ) at t0.
22 4.3 Modelling the stochastic process
q 1 λ(∆V,V )/ ρS 2π /100 30 1.2
20 1
10 0.8 T B M k 0 0.6 q V/ -10 0.4
-20 0.2
-30 0 -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 q kBT ∆V/ M
Figure 4.3 The rate λ(∆V,V ) of velocity-changes ∆V at current motor velocity V (cf. Equation (4.30)). The plotted data correspond to the test motor of Chapter 5 (cf. Figure 5.1), an equilateral triangle with one side perpendicular to the X axis and the opposite corner pointing in positive X direction. Moreover, δ := pm/M = 0.1.
23 4 Brownian motion as stochastic process - the mathematical framework
Similar to Equation (4.24), the void probability of no collision to occur between ti and tj is defined as tj ∞ Z Z Pvoid(tj, ti) := exp − dt d∆V λ(∆V,V (t)) . (4.31)
ti −∞ R ∞ The quantity −∞ d∆V λ(∆V,V ) is the combined rate of all possible collisions at motor velocity V . The integration over t becomes necessary as V (t) may vary in between two collisions due to the potential U(X, t) and can be motivated by considering the continuum limit of the product of many void probabilities for subsequent small time intervals with constant V or λ, respectively. With the definition of Vi, the velocity just before collision i,
Vi := lim V (t), (4.32) t%ti the probability density of the velocity-change ∆Vi to happen at ti becomes λ(∆Vi,Vi). It is now straightforward to write down the first sought conditional probability density (cf. Equation (4.25)):
N Y ∗ P [∆V (t)|Γ0, t0] := [Pvoid(ti, ti−1)λ(∆Vi,Vi)] Pvoid(t , tN ) i=1 ∗ t ∞ N Z Z Y = exp − dt d∆V λ(∆V,V (t)) λ(∆Vi,Vi). (4.33) i=1 t0 −∞
∗ For integrations over all possible collision histories between t0 and t , it is convenient to define another symbol:
∗ ∗ ∗ ∞ t t t ∞ ∞ ∞ Z X Z Z Z Z Z Z D[∆V (t)] := dt1 dt2... dtN d∆V1 d∆V2... d∆VN . (4.34) N=0 t0 t1 tN−1 −∞ −∞ −∞
By this means, the normalisation condition of P [∆V (t)|Γ0, t0] reads Z D[∆V (t)] P [∆V (t)|Γ0, t0] = 1, (4.35) but turns out to be difficult to prove. In Section 6.1.2 it is shown for a simpler, but similarly constructed process, which shall serve as a motivation for the assumption of the validity of the above equation. The second desired quantity is easily obtained by integrating over all paths ∆V (t) weighted by their probabilities and keeping only those which end at Γ∗: Z ∗ ∗ ∗ ∗ P (Γ , t |Γ0, t0) := D[∆V (t)] P [∆V (t)|Γ0, t0]δ(Γ − Γ(t )). (4.36)
Finally, integrating out Γ0 yields the phase space pdf: Z ∗ ∗ dΓ0 ∗ ∗ P (Γ , t ) := P (Γ , t |Γ0, t0)P0(Γ0) 2π~/M Z Z dΓ0 ∗ ∗ = D[∆V (t)] P [∆V (t)|Γ0, t0]P0(Γ0)δ(Γ − Γ(t )), (4.37) 2π~/M where the factor M/(2π~) is needed to render the expression dimensionless and assure consistency with quantum mechanics.
24 4.4 Time evolution of the phase space probability density
4.4 Time evolution of the phase space probability density 4.4.1 The time evolution equation The collision probability densities or rates λ(∆V,V ) only depend on the current phase space position Γ. Obviously, the same is true for the deterministic motion between collisions. Thus, the condi- tional probability density P (Γ, t|Γ0, t0) is generated by a Markov process and consequently obeys the Chapman-Kolmogorov equation Z P (Γ, t|Γ0, t0) = dΓ1P (Γ, t|Γ1, t1)P (Γ1, t1|Γ0, t0). (4.38)
Following Section 4.1.2, the latter can be transformed into a differential time evolution equation. 0 If the uniform convergence of the limits is assumed, the quantities W (Γ|Γ , t), Ai(Γ, t) and Bij(Γ, t) are found to be W (Γ|Γ0, t) = W (V |V 0)δ(X − X0); W (V |V 0) := λ(V − V 0,V 0), (4.39)
A1(Γ, t) = V, (4.40) 1 ∂U(X, t) A (Γ, t) = − and (4.41) 2 M ∂X Bij(Γ, t) = 0, (4.42) as demonstrated in Appendix A.1. In fact, the vector A := (A1,A2) simply corresponds to Γ˙ if dis- continuous changes due to collisions are excluded. With these coefficients, the differential Chapman- Kolmogorov equation (4.12) takes the form ∂P (Γ, t|Γ , t ) ∂ 1 ∂U(X, t) ∂ 0 0 = −V + P (Γ, t|Γ , t )+ ∂t ∂X M ∂X ∂V 0 0 ∞ Z 0 0 0 0 + dV [W (V |V )P ((X,V ), t|Γ0, t0) − W (V |V )P (Γ, t|Γ0, t0)] . (4.43) −∞
R dΓ0 By means of the definition P (Γ, t) := P (Γ, t|Γ0, t0)P0(Γ0), the evolution equation of the 2π~/M phase space pdf is obtained: ∞ ∂P (Γ, t) ∂ 1 ∂U(X, t) ∂ Z = −V + P (Γ, t)+ dV 0 [W (V |V 0)P ((X,V 0), t) − W (V 0|V )P (Γ, t)] . ∂t ∂X M ∂X ∂V −∞ (4.44) The first term of the right hand side corresponds to the Liouville equation of a classical particle in the potential U(X, t) and accounts for the deterministic motion between collisions. The second term has the form of a so-called master equation and represents the stochastic part of the evolution of P (Γ, t) due to the random collisions. By setting U(X, t) = const and integrating out X, the master equation for velocities, which was used in References [42, 41, 10], can be recovered.
4.4.2 Stationary solution and detailed balance For a time invariant potential U(X) the canonical distribution 1 1 1 2 Peq(Γ) := exp − MV + U(X) (4.45) Z kBT 2 with the partition function Z dΓ 1 1 Z := exp − MV 2 + U(X) (4.46) 2π~/M kBT 2
25 4 Brownian motion as stochastic process - the mathematical framework is a stationary solution of the time evolution equation (4.44). One convinces oneself easily that
∂ 1 ∂U(X) ∂ −V + P (Γ) = 0. (4.47) ∂X M ∂X ∂V eq
To show that the integral part of Equation (4.44) vanishes as well, it is written as
∞ Z d∆V [W (V |V + ∆V )Peq((X,V + ∆V )) − W (V + ∆V |V )Peq((X,V ))]. (4.48) −∞
By noticing that
∆q Peq((X,V + ∆V )) = Peq((X,V ) exp , where (4.49) kBT 1 ∆q := − M (V + ∆V )2 − V 2 , (4.50) 2 which can be associated with the heat dissipated during a collision (cf. Section 7.1), and
∆q W (V |V + ∆V ) exp = W (−(V + ∆V )| − V ), (4.51) kBT proved in Appendix A.2, one finds (using W (a|b) = λ(a − b|b))
∞ Z (4.48) = d∆V [λ(−∆V | − V ) − λ(∆V |V )]Peq((X,V )) = −∞ 2π Z = −ρV Peq((X,V )) dθ SF (θ) sin(θ) = 0, (4.52) 0 where Equation (4.30) was used for the second equality. The integral over θ vanishes because dθSF (θ) sin θ is the projection perpendicular to the X axis of the line element dθSF (θ) weighted with a sign that reflects the notion of an inside and an outside of the body, or in other words, that indicates whether the surface element is oriented to the right or to the left. As the cross section of the body is the same no matter whether it is looked at from positive or negative X direction, the sum of the projections – labelled with appropriate signs – along the complete outline of the body adds up to zero. It is now easy to show that this stationary state exhibits detailed balance (cf. Section 4.1.3). With 1 = 1, 2 = −1, A = (V, −∂X U(X)/M) (cf. Equations (4.40) and (4.41)) and Bij(Γ) = 0 (cf. Equation (4.42)), one readily finds that conditions (4.19) and (4.18) hold, that is
ijBij(Γ) =Bij(Γ) and (4.53)
iAi(Γ) = − Ai(Γ). (4.54)
Condition (4.17) reads
0 0 0 W (V |V )Peq((X,V )) = W (−V | − V )Peq((X,V )). (4.55)
By substituting V 0 = V + ∆V and applying identity (4.49), one finds that the above equation transforms into a relation similar to Equation (4.51), also proved in Appendix A.2.
26 4.4 Time evolution of the phase space probability density
4.4.3 Kramers-Moyal expansion of the time evolution equation
A common way of tackling master equations is to perform a so-called Kramers-Moyal expansion. For instance, expanding a master equation generated by a simple undirected random walk produces the diffusion equation and almost all expansion coefficients vanish. In case of the master equation like integral part of the time evolution equation, none of the coefficients disappear in general. Still, this method proves to be useful since for example only certain coefficients play a role if moments of the phase space pdf are considered (cf. Section 4.5). By means of a smart substitution and the help of an auxiliary function, f, the second part of Equation (4.44) can be expanded in powers of the step size V 0 − V :
∞ Z dV 0 [W (V |V 0)P ((X,V 0), t) − W (V 0|V )P (Γ, t)] =
−∞ ∞ ∞ Z Z = dy W (V |V − y)P ((X,V − y), t) − dy W (V + y|V )P (Γ, t) = | {z } | {z } −∞ f(y,V −y) −∞ =:f(y,V ) ∞ Z = dy [f(y, V − y)P ((X,V − y), t) − f(y, V )P (Γ, t)] =
−∞ ∞ ∞ Z X (−y)n ∂n = dy [f(y, V )P (Γ, t)] (4.56) n! ∂V n −∞ n=1
Re-substitution of V 0 − V for y yields the Kramers-Moyal expansion of the time evolution equation:
∞ ∂P (Γ, t) ∂ 1 ∂U(X, t) ∂ X (−1)n ∂n = −V + P (Γ, t) + [α (V )P (Γ, t)] (4.57) ∂t ∂X M ∂X ∂V n! ∂V n n n=1 with the n’th Kramers-Moyal coefficient being defined by
∞ Z 0 0 n 0 αn(V ) := dV (V − V ) W (V |V ). (4.58) −∞
q If one inserts Equation (4.30), Equation (4.39), a = −ρSF (θ) m , b = m sin2 θ, c = 1 2πkBT 2kBT α sin θ m 2 M sin θ q m 0 and α = m 2 into Equation (4.58) and substitutes R := (V − V )/α, one readily finds 1+ M sin θ 2kBT
r n+1 3n+1 kBT ρS n−1 α (V ) = − 2 2 √ δ × n M π 2π 0 !2 Z sin θ n Z r M δ × dθ F (θ) dRRn+1 exp − R + √ V sin θ , (4.59) 2 2 1 + δ sin θ kBT 2 0 −∞
p m where δ := M .
27 4 Brownian motion as stochastic process - the mathematical framework
4 6 exact 2 5 O(δ1) 3 ) 0 2
T O(δ )
T 4 B
M 3 B M k O(δ )
k -2
( 3 q /
-4 / 1 2 g 2 -6 exact g -8 O(δ1) 1 2 -10 O(δ ) 0 O(δ3) -12 -1 -20 -15 -10 -5 0 5 10 15 20 -20 -15 -10 -5 0 5 10 15 20 q q kBT kBT V/ M V/ M
Figure 4.4 Different approximations of the integrands g1(V, θ = π/2) and g2(V, θ = π/2) (cf. Equation (4.61)) of the first two Kramers-Moyal coeffiencts for δ = 0.1. For θ = π/2 deviations are maximal. Guided by thermal equilibrium, typical values of V q kBT are expected to be of a few M .
4.4.4 Series expansion of the Kramers-Moyal coefficients
The integral over R in Equation (4.59) can be solved analytically (cf. Reference [41] p. 87), but results in a rather complicated V dependence involving the Gauss error function. Luckily, a series expansion in the square root of the mass ratio, δ, which usually will be a small quantity, turns αn(V ) into a polynomial in V . In detail one has
2π r n+1 Z n 3n−1 kBT ρS n−1 n α (V ) = (−1) 2 2 √ δ dθ F (θ) sin θ× n M π 0 ∞ " l k l−k−m k−2m n r 2k X X X (−1) 2 (n − 1 + l − k)!Γ( + 1 + k − m) M × δ2l sin2l θ 2 V 2k+ (l − k)!(n − 1)!(2k − 2m)!m! kBT l=0 k=0 m=0 l k l−k−m k−2m+ 1 n+3 r 2k+1 # X X (−1) 2 2 (n − 1 + l − k)!Γ( + k − m) M +δ2l+1 sin2l+1 θ 2 V 2k+1 . (l − k)!(n − 1)!(2k − 2m + 1)!m! kBT k=0 m=0 (4.60)
The derivation of this formula is sketched in Appendix A.3. Truncating the series then yields approximate expressions for the Kramers-Moyal coefficients. To give an impression of its performance, the quantity gn(V, θ), implicitly defined via
Z 2π αn(V ) = ρS dθ F (θ)gn(V, θ), (4.61) 0 is considered. As the order of sin θ increases with the order of δ, the approximation to gn(V, θ) will be worst if sin θ = 1. In Figure 4.4.4, expansions up to different orders of δ of g1(V, π/2) and g2(V, π/2) are compared with the exact solutions. The focus is set on the first two coefficients as they will reappear in the analysis in Section 5.3. It is noteworthy that it is possible to express the integrals over θ in terms of tensorial Minkowski
28 4.5 Time evolution of moments of the phase space probability density functionals, namely (cf. References [42, 41]) Z 1 n (0,n) dθSF (θ) sin θ = W1 (K) . (4.62) 2 ∂K X...X | {z } n times K signifies the body the functional is applied to, being the motor in this case. The n-fold subscript (0,n) X denotes the X...X entry of W1 (K), that is the element of the rank n tensor solely associated with the X direction. In general, the first tensorial Minkowski functional of a convex body K is defined as n times Z m times (m,n) 1 z }| { z }| { W1 (K) := dO r ⊗ ... ⊗ r ⊗ eˆ⊥ ⊗ ... ⊗ eˆ⊥, (4.63) d ∂K where d is the dimension, dO denotes a surface integral, r is the position and eˆ⊥ the normal vector on the surface (see References [42, 28, 27]).
4.5 Time evolution of moments of the phase space probability density
As already hinted in Section 4.4.3, one of the virtues of the Kramers-Moyal expansion is that the time evolution equations of the moments of the phase space pdf, Z dΓ hXiV ji(t) := XiV jP (Γ, t), (4.64) 2π~/M may be expressed in terms of the expansion coefficients αn(V ). By taking the time derivative of Equation (4.64) and applying Equation (4.44), one finds Z i j dΓ i j ∂thX V i(t) := X V ∂tP (Γ, t) 2π~/M Z Z dΓ i j+1 1 dΓ i j =− X V ∂X P (Γ, t) + X V ∂X U(X, t)∂V P (Γ, t)+ 2π~/M M 2π~/M | {z } | {z } (i) (ii) Z dΓ Z Z dΓ Z + dV 0 XiV jW (V |V 0)P ((X,V 0), t) − dV 0 XiV jW (V 0|V )P (Γ, t). 2π~/M 2π~/M | {z } (iii) (4.65) Term (i) and (ii) are easily integrated by parts, where surface terms are assumed to vanish. If V and V 0 are swapped in the first integral of part (iii), the latter can be written as j Z Z X j hXi dV 0 (V 0j − V j)W (V 0|V )i = hXi dV 0 V j−n(V 0 − V )nW (V 0|V )i. (4.66) n n=1 The last equality is readily appreciated by writing V 0j = ((V 0 − V ) + V )j and applying the binomial identity. Now it is possible to identify the expression for αn(V ) in (iii), which yields in combination with (i) and (ii): j j X j ∂ hXiV ji = ihXi−1V j+1i − hXiV j−1∂ U(X, t)i + hXiV j−nα (V )i. (4.67) t M X n n n=1
With the expansion of αn(V ) in δ or V , respectively, (cf. Equation (4.60)) approximate equations for the moments may be obtained (see Section 5.3 for a concrete example).
29 4 Brownian motion as stochastic process - the mathematical framework
30 5 Using potential wells as stopping sites
In the original model (cf. References [42, 41]) the velocity of the motor is set to zero, whenever it reaches a stopping site. In terms of the statistical ensemble this means turning the velocity distribution into a delta peak, but, striving to make a different point, the author did not dwell on how this may actually be achieved. In fact, this turns out to be very hard if not impossible. Thinking of binding sites, one usually pictures some sort of potential well, where strong attractive forces keep the bound particle in place. Leaving aside quantum effects, one can by that means in principle confine the motor to a more or less arbitrary small region. However, the equilibrium velocity distribution of such a trapped particle is independent of the potential. It is simply given by the Gaussian shaped Maxwell-Boltzmann distribution, the width of which is determined by temperature alone. Nevertheless, potential wells can still be used to narrow the velocity distribution – namely by turning them off. If the motor is gradually released and given enough time to perform work against the “walls” of the potential well, the effect is quite similar to the cooling of a gas by expansion. With respect to real systems such a change in binding energy may probably be induced by ATP hydrolysis or some other biochemical reaction. To get deeper insight into the release process, a case study with a parabolic potential 1 U(X, t) := k(t)X2 (5.1) 2 is carried out in this chapter. The release from the well is realised by an exponentially decreasing spring constant −t/τ k(t) := k0e (5.2) with typical decay time τ. Moreover, the analysis is mostly restricted to moments of the phase space pdf, making use of the approximations introduced in Section 4.4.4. Any quantitative data presented in this chapter is obtained by using the test motor particle depicted in Figure 5.1. It is an equilateral triangle of circumference S with one side perpendicular to the X axis and the opposite corner pointing in positive X direction. For this setup, the elements of the Minkowski tensors (see Equation (4.62) for a definition) adopt values of ! S 1n−1 W (0,n)(K) = + (−1)n . (5.3) 1 X...X 6 2 | {z } n times
The shape dependence of the motor’s motion was studied in Reference [41] and is therefore not subject of the analysis. All numerical solutions to differential equations were obtained using the NDSolve algorithm of Wolfram Mathematica 6.0 with default options.
S/3
30° X
Figure 5.1 The test motor particle, used for any quantitative calculations.
31 5 Using potential wells as stopping sites
The following considerations are divided into three parts. Initially, the process is studied either without potential or without collisions, allowing for better analytical access, before both are combined. In Section 5.1, the free relaxation towards equilibrium out of a given initial distribution at zero potential is investigated, yielding that the difference of the starting value of hV 2i from the equilibrium value plays a crucial role in the efficiency of the process. It is therefore desirable to bring the velocity distribution as close to the delta peak as possible. This is important information for the design of the release process studied under negligence of collisions in Section 5.2. There, the importance of the correct choice of time scales is pointed out. Finally, Section 5.3 comprises both aspects. A method of deriving approximate sets of equations for the moments of the phase space pdf is proposed, and different approximation levels are numerically solved and compared. It turns out that many features of the combined system (i.e., collisions and potential) can already be understood from the reduced systems of the first two parts. Moreover, if the time scales of potential-change and relaxation in the medium are well separated, an analytical approximation may be obtained by treating both processes as being independent of each other and occurring subsequently, restoring to some extend the assumptions used in References [42, 41]. Additionally, a short outlook comments on the binding process, where in contrast to releasing some extra dissipation mechanism might be needed, and a way to incorporate multiple or re-opening stopping sites is outlined.
5.1 Relaxation without potential 5.1.1 Evolution of moments of the phase space probability density To investigate the collision driven evolution of the system in the case of zero potential, the following set of equations is studied:
∂thXi = hV i, (5.4) k T ∂ hV i = AδhV i − Bδ2 B − hV 2i , (5.5) t M 2 ∂thX i = 2hXV i, (5.6) k T ∂ hV 2i = −2Aδ B − hV 2i , (5.7) t M 2 ∂thXV i = hV i + AδhXV i, (5.8) with r r 2 kBT (0,2) A := − 4 ρ W1 (K) , (5.9) π M XX (0,3) B := − 2ρ W1 (K) . (5.10) XXX It is obtained by use of the δ expansion of the Kramers-Moyal coefficients (4.60) and constitutes a good approximation for small values of the mass ratio (δ . 0.1). It shall not be deduced here, since, in fact, it is the ω(t) = 0 case of Equations (5.86) to (5.90) in Section 5.3. For their derivation, however, it is necessary to include a potential, as it is reasonable to demand of any approximate equations to posses stationary solutions that match the corresponding equilibrium distribution for a non-zero time-invariant potential. To zeroth order in δ the above equations are those of a free particle. With δ1 merely linear damping terms proportional to Aδ are introduced and one can identify
γ := −AδM (5.11)
32 5.1 Relaxation without potential as the first order approximation of the linear friction coefficient. Due to this negative feedback, one expects exponential decays of initial values with a typical relaxation time of M −1 τ := = . (5.12) r γ Aδ Note that A is insensitive to asymmetry due to the sin2 θ terms. Asymmetry comes into play in the form of B when going to the second order in δ. In this case, an additional force term arises in the equation of hV i, which, however, vanishes in case of a symmetric object (B = 0) or if hV i has reached kBT its equilibrium value of M . Solutions to Equations (5.4) to (5.8) are found using the method of variation of constants:
2 hV i(t0) B kBT hXi(t) = hXi(t ) − 1 − eAδ(t−t0) − − hV 2i(t ) 1 − eAδ(t−t0) , (5.13) 0 Aδ 2A2 M 0 | {z } translation due to asymmetry and non-equilibrium Bδ k T Aδ(t−t0) B 2 Aδ(t−t0) Aδ(t−t0) hV i(t) = hV i(t0)e + − hV i(t0) 1 − e e , (5.14) | {z } A M exp. dissipation of initial mean vel. | {z } velocity boost
kBT 2 2 2hXV i(t0) − hV i(t0) hX2i(t) = hX2i(t ) − 1 − eAδ(t−t0) − M 1 − eAδ(t−t0) − 0 Aδ A2δ2 2 kBT kBT − M 1 − eAδ(t−t0) + 2 M (t − t ), (5.15) A2δ2 −Aδ 0 | {z } =:D kBT kBT hV 2i(t) = − − hV 2i(t ) e2Aδ(t−t0), (5.16) M M 0 | {z } | {z } equil. value exp. relaxation back to equilibrium
kBT 2 kBT − hV i(t0) hXV i(t) = hXV i(t )eAδ(t−t0) + M 1 − eAδ(t−t0) eAδ(t−t0) − M 1 − eAδ(t−t0) . 0 Aδ Aδ | {z } lim hXV i t→∞ (5.17) Exemplary plots of these functions are shown in Figure 5.2. There are several aspects that are worth a remark.
• First of all, net motion is possible even from a symmetric initial distribution (hV i(t0) = 0) if B 6= 0. This is true for most asymmetric objects, due to the sin3 θ term in the integral. However, this effect dies away as the velocity distribution approaches equilibrium as already anticipated from the differential equations.
• The typical time scale of the process is indeed set by τr. For instance, this quantity appears in the exponential dissipation of the initial mean velocity as well as the exponential relaxation of hV 2i back to its equilibrium value. • For large times (i.e., when the equilibrium distribution of velocities has been re-established) the mean square displacement goes linear with t, which corresponds to normal diffusion. The quantity k T D := B (5.18) −AδM may thus be identified with the first order approximation of the diffusion constant, and one recovers the Einstein-Smoluchowski relation (1.3) k T k T D = B τ = B (5.19) M r γ
33 5 Using potential wells as stopping sites
0 0 -2 -1 -4 -2 -6 1000 100 -8 × -3 × T
-10 B -4 M k ρS
i -12 -5 q X -14 / h i -6
-16 V -18 h -7 -20 -8 0 1 2 3 4 5 6 7 8 0 1 2 3 4 5 6 7 8
t/τr t/τr
6 1 5 4 0.8 3 T B ρS 2 M 0.6 k i slope 2 1 3 / X i h
0 2 0.4
-1 2 V lg -2 h 0.2 -3 1 -4 0 -2 -1 0 1 2 3 0 0.5 1 1.5 2 2.5 3 3.5 4
lg(t/τr) t/τr
14 12 T B M 10 k
q 8
ρS/ 6 i 4 XV h 2 0 0 1 2 3 4 5 6 7 8
t/τr
Figure 5.2 Evolution of the first five moments of the phase space pdf in absence of a potential U(X, t), as given by Equations (5.13) to (5.17). Parameter values q q 2 kBT 1 are A = − π ρS M and B = 4 ρS, corresponding to the triangular test motor (cf. Figure 5.1). Moreover, δ = 0.1 and t0 = 0. Initially, all moments are set to zero, which constitutes an ideal “stopped” state and matches the case discussed in References [42, 41].
34 5.1 Relaxation without potential
1.1 2.5 η × 100 t /τ 1 max r approximation 2 f(σ, t = tmax) 0.9 0.8 1.5 0.7 0.6 1 0.5 0.5 0.4 0.3 0 0 1 2 3 4 5 6 7 8 -2 -1.5 -1 -0.5 0 0.5 1
t/τr σ
Figure 5.3 Left: The efficiency as defined in Equation (5.20) for the same set of parameters as in Figure 5.2. The approximation (5.24) is in perfect agreement with the original curve. Right: Dependence of the temporal position tmax (given by Equa- tion (5.26)) and hight (∝ f(σ, tmax); cf. Equation (5.25)) of the efficiency maximum on the initial deviation σ of hV 2i from the equilibrium value.
within the limits of the given approximation.
• The intensity of the velocity boost and consequently the final value of hXi, too, are – among 2 kBT others – controlled by the initial difference of hV i from its equilibrium value M . Therefore, it is desirable to drive the system away from equilibrium as far as possible to obtain maximum net motion. Moreover, the magnitude of the boost and its duration vary with δ reciprocally to each other, which is why the contribution of the integrated boost to the long time limit of hXi does not depend on δ itself.
• Last, one might notice that hXV i does not approach its equilibrium value of zero. This is explained by the fact that for an unbounded system there indeed exists no equilibrium distri- bution for the positions, or at least none that can be reached in finite time. However, this does not hinder the equilibration of the velocities.
5.1.2 Efficiency
As a measure of randomness or “directedness” one may regard the quantity
|hXi(t)| η(t) := , (5.20) phX2i(t) − hXi(t)2 which will be further referred to as efficiency. An exemplary plot is presented in Figure 5.3 (a), which shows that after having reached a maximal value after a comparatively short time, η declines towards zero, as random diffusive motion begins to dominate. In terms of optimizing the process it is desirable to estimate the position and height of η’s maximum. The analysis is be restricted to cases where hXi(t0) = 0 and hV i(t0) = 0, to focus solely on the effect of the asymmetric relaxation process. Moreover, t0 = 0. With the definition of M 2 σ := 1 − hV i(t0) , (5.21) kBT
35 5 Using potential wells as stopping sites
Equations (5.13) and (5.13) take the form:
2 B kBT hXi(t) = − σ 1 − e−t/τr , (5.22) 2A2 M neglect
z }| { 2 2 2 2 −t/τr kBT −t/τr −t/τr A δ hX i(t0) − 2AδhXV i(t0) 1 − e − M σ 1 − e + 2 1 − e − t/τr hX2i(t) = . A2δ2 (5.23)
Compared to hX2i, hXi2 can be neglected. If higher orders of δ in the expression for hX2i are left out, too, one finds that η separates nicely into a product of terms with different dependences:
r δ B kBT η(t) ≈ f(σ, t), (5.24) 2 A M where
2 |σ| 1 − e−t/τr f(σ, t) := q . (5.25) −t/τ 2 −t/τ −σ 1 − e r − 2 1 − e r − t/τr
This approximation is plotted in Figure 5.3 (a) as well and matches the exact curve excellently for the given set of parameters. Setting its first derivative with respect to t equal to zero, shows that f(σ, t) posses a minimum at t = 0 and is maximal when
3e−t/τr − et/τr + 4t/τ − 2 σ = r . (5.26) 2 1 − e−t/τr
Being a transcendental equation, this cannot be solved for t analytically to find the location of the maximum tmax. However, one may simply plot σ versus tmax and swap the axes, which is how Figure 5.3 (b) was obtained. Similarly, the plot of f(σ, t = tmax) (also contained in Figure 5.3 (b)) was generated. Notice that σ cannot be greater than one, as hV 2i ≥ 0, which is why the corresponding axis terminates at that value. If the initial distribution is narrower than the equilibrium pdf, then 0 ≤ σ ≤ 1, if it is broader, then σ ≤ 0. In the latter case f(σ, t = tmax) grows unbounded with decreasing σ which allows the conclusion that in this sense pushing the motor randomly (to increase its average kinetic energy) has the potential of being more efficient than trying to slow it down or stop it. Unfortunately tmax will increase as well. In summary on finds that, within this approximation, the efficiency may be optimised by choosing a motor design that maximises |B/A| and by developing a mechanism that initially pushes hV 2i away from equilibrium as far as possible. Concerning the mass ratio δ, larger values seem preferable, but one has to be aware that there is a trade off with σ as a higher damping makes it generally more difficult to obtain highly non-equilibrium states (cf. Section 5.3.2). Last, the temperature T , density p ρ and motor size S are found to have no direct influence (since A ∝ ρS kBT/M and B ∝ ρS), but may enter through σ.
36 5.2 Potential-change without collisions: time-dependent undamped harmonic oscillator
5.2 Potential-change without collisions: time-dependent undamped harmonic oscillator
The trajectory of the motor is composed of parts of deterministic motion disrupted by discrete jumps caused by collisions. To describe the deterministic segments, it is necessary to study the time- dependent undamped harmonic oscillator, which is the subject of this section. Moreover, this will give an impression of how the system is pushed away from its equilibrium state during the release process.
5.2.1 Linear dynamics
1 2 The motion of an undamped oscillator in the potential U(X, t) := 2 k(t)X is governed by the equation
Γ˙ = A(t)Γ, (5.27) with 0 1 A(t) := −k(t) . (5.28) M 0
Due to the linearity of Equation (5.27), its solution depends linearly on the initial value Γ(t0) and can be written as
Γ(t) = M(t, t0)Γ(t0), (5.29)
R t 0 0 where M(t, t0) is a 2 × 2 matrix. It might by worth noticing that M(t, t0) 6= exp dt A(t ) t0 in general, as R t dt0 A(t0) and A(t) not necessarily commute. Nonetheless, it is always possible to t0 write the solution in the form (5.29), which is readily verified by direct insertion into Equation (5.27), yealding also a conditional equation for M(t, t0):
∂tM(t, t0) = A(t)M(t, t0). (5.30)
The matrices M(t0, t) have the following properties:
M(t, t) = I2, (5.31) (M(t0, t))−1 = M(t, t0), (5.32) M(t00, t0)M(t0, t) = M(t00, t), (5.33) det (M(t0, t)) = 1, (5.34) where I2 denotes the 2 × 2 identity matrix. Equation (5.34) is no other than conservation of phase ∂ 0 space and can be obtained by noticing that ∂t0 det (M(t , t)) = 0, using Equation (5.30), and that det (M(t, t)) = 1, obviously. It also leads to the useful relations
0 0 M11(t , t) =M22(t, t ), (5.35) 0 0 M12(t , t) = − M12(t, t ), (5.36) 0 0 M21(t , t) = − M21(t, t ), (5.37)
0 0 where Mij(t , t) denotes the ij’th matrix element of M(t , t).
37 5 Using potential wells as stopping sites
5.2.2 Exponentially decreasing spring constant Let −t/τ k(t) = k0e . (5.38) With k(t) ω(t)2 := (5.39) M one easily obtains from Equation (5.27)
X¨ + ω(t)2X = 0. (5.40)
Substituting y := 2τω(t), Equation (5.40) is found to be equal to the Bessel differential equation (cf. Reference [1] p. 102) d2 d y2 + y + y2 − n2 X = 0 (5.41) dy2 dy with n = 0, which is solved by X(y) = C1J0(y) + C2Y0(y). (5.42)
J0 and Y0 are Bessel functions of the first and second kind, respectively, and C1 and C2 are integration constants. Re-substitution for y and specialisation to X˙ (t0) = V (t0) eventually lead to
t0 t t t0 M11(t, t0) = ω(t0)πτ J1 Y0 − J0Y1 , (5.43) t t0 t0 t M12(t, t0) = πτ J0Y0 − J0 Y0 , (5.44) t0 t t t0 M21(t, t0) = ω(t0)ω(t)πτ J1 Y1 − J1Y1 , (5.45) M22(t, t0) = M11(t0, t), (5.46) with
t Ji := Ji (2τω(t)) , (5.47) t Yi := Yi (2τω(t)) . (5.48)
5.2.3 Evolution of the phase space probability density Consider now an ensemble of harmonic oscillators as described above with an initial phase space probability density function P (Γ, t0) = P0(Γ). The pdf at any given (later) time t can then be obtained by tracing back the trajectory of each point in phase space Γ to its original position at t0.
∂M(t0, t)Γ P (Γ, t) = P (M(t0, t)Γ, t0) , (5.49) ∂Γ where the Jacobian of the coordinate transformation Γ 7→ M(t , t)Γ, ∂M(t0,t)Γ = det (M(t , t)) = 1, 0 ∂Γ 0 due to the conservation of phase space volume. If one chooses ~ω(t0) M 2 2 2 P0(Γ) = Peq(Γ, ω(t0)) := exp − V + ω(t0) X , (5.50) kBT 2kBT the canonical distribution of thermal equilibrium, one finds that P (X, t) := R dV P (Γ, t) and 2π~/M P (V, t) := R dXP (Γ, t) keep their Gaussian shapes, whereas mixed terms of X and V appear in P (Γ, t). For an example see Figure 5.4. An analytical description is postponed until Section 5.2.4, where moments of the distributions are considered, which completely characterise P (X, t) and P (V, t).
38 5.2 Potential-change without collisions: time-dependent undamped harmonic oscillator
P ((X,V ), t) × kBT ~ω0 4 1 3 t = 0 t = 2τ 2 1 0 0.8 -1 -2 0.6
T -3 B M -4 k
q 4
3 t = 4τ t = 6τ 0.4 2 V/ 1 0 -1 0.2 -2 -3 -4 0 -1.5 -1 -0.5 0 0.5 1 1.5 -1 -0.5 0 0.5 1 1.5 q kBT X/ τ M
Figure 5.4 Deterministic evolution of the phase space probability density for a po- M 2 2 2π t tential U(X, t) = 2 ω(t) X according to Equation (5.49), with ω(t) = τ exp − 2τ and t0 = 0. The initial distribution corresponds to thermal equilibrium and collisions with gas particles are neglected.
Qualitatively speaking, P (X, t) spreads infinitely with time, as the confining potential U(X, t) van- ishes. P (V, t), however, approaches a narrower distribution, the width of which is determined by how rapidly U(X, t) is switched off. The faster ω(t) is decreasing, the broader P (V, t) will remain, as the oscillator needs some time to interact with the changing potential, performing work and interchanging kinetic and potential energy. This is readily appreciated by considering the limiting case of an in- stantaneous disappearence of U(X, t), where there is no reason for P (V, t) to change at all. Certainly, this behaviour is a generic feature of almost any bound system, the potential of which is relaxed, and not confined to the special case treated here. The whole process resembles the cooling of a gas by expansion with the difference that the individual members of the ensemble cannot interact (e.g. by collisions as in a gas), with the effect that no energy can be exchanged and no new equilibrium with lesser temperature can be established.
5.2.4 Evolution of moments of the phase space probability density
As collisions come into play the analysis is restricted to moments of the phase space pdf. Hence, it is appropriate to examine their behaviour in this simpler case as well, which will also be helpful for the approximation in Section 5.3.3. From the definition hXiV ji(t) := R dΓ XiV jP (Γ, t) it is fairly straightforward to see that 2π~/M
i j i j hX V i(t) = h(M11(t, t0)X + M12(t, t0)V ) (M21(t, t0)X + M22(t, t0)V ) i(t0), (5.51)
0 by using P (Γ, t) = P0(M(t0, t)Γ) (Equation (5.49)) and substituting Γ := M(t0, t)Γ in the integral.
39 5 Using potential wells as stopping sites
2 0 2
1/(πτω(t )) 1.8 hX2i/ kBT τ 0 M T B 1.6 M -0.5 k 2 kBT 1.4 hV i/ M /
1.2 ) -1 t
kBT τ ( 1 hXV i/ M i 2
0.8 V -1.5 0.6 h →∞ 0.4 lim -2 t
0.2
0 lg -2.5 0 2 4 6 8 10 0.1 1 10 100
t/τ τω(t0)
Figure 5.5 Evolution of the second order Figure 5.6 The narrowing of the velocity dis- p moments for t0 = 0 and ω0 := k0/M = tribution depends on the ratio of decay time 2π/τ (cf. Equations (5.54) to (5.56)). The τ and initial cycle time 2π/ω(t0) (cf. Equa- first order moments hXi and hV i remain zero, tion (5.57)). For large values of τω(t0) the long-time limit of hV 2i goes like kBT . as the corresponding distributions are sym- Mπτω(t0) metric at any time.
Explicitly, one has for the first few moments:
P0=Peq hXi(t) =M11(t, t0)hXi(t0) + M12(t, t0)hV i(t0) = 0, (5.52)
P0=Peq hV i(t) =M21(t, t0)hXi(t0) + M22(t, t0)hV i(t0) = 0, (5.53)
2 2 2 2 2 P0=Peq hX i(t) =M11(t, t0) hX i(t0) + M12(t, t0) hV i(t0) + 2M11(t, t0)M12(t, t0)hXV i(t0) = (5.54) kBT 2 1 2 = M11(t, t0) 2 + M12(t, t0) , M ω(t0) 2 2 2 2 2 P0=Peq hV i(t) =M21(t, t0) hX i(t0) + M22(t, t0) hV i(t0) + 2M21(t, t0)M22(t, t0)hXV i(t0) = (5.55) kBT 2 1 2 = M21(t, t0) 2 + M22(t, t0) and M ω(t0) 2 2 hXV i(t) =M11(t, t0)M21(t, t0)hX i(t0) + M12(t, t0)M22(t, t0)hV i(t0)+
P0=Peq + (M11(t, t0)M22(t, t0) + M12(t, t0)M21(t, t0))hXV i(t0) = (5.56) kBT 1 = M11(t, t0)M21(t, t0) 2 + M12(t, t0)M22(t, t0) , M ω(t0) where Peq denotes the equilibrium distribution for ω = ω(t0). Exemplary curves are presented in Figure 5.5. The narrowing of the velocity distribution is characterised by hV 2i(t). In the long-time limit one finds
2 kBT 2 2 τω(t0)→∞ kBT lim hV i(t) = J0(2τω(t0)) + J1(2τω(t0)) −→ , (5.57) t→∞ M Mπτω(t0)
(cf. Appendix A.4) which is the smaller, the larger the value of τω(t0), that is the slower the decrease of ω(t) or, the larger the initial frequency ω(t0) (cf. Figure 5.6). The speed of the change of U(X, t) thus has to be measured with respect to the time scale induced by the oscillator’s initial cycle time.
40 5.3 Time evolution with potential and collisions
i + j N0 N1 N2 N3 N4 N5 1 δ0 δ1 δ2 δ3 δ4 δ5 2 - - δ1 δ1 δ2 δ3 3 - - - δ0 δ1 δ2 4 - - - - - δ1 5 - - - - - δ0
Table 5.1 Orders of δ up to which the equations of motion of the moments hXiV ji are expanded in the approximations N0 to N5.
5.3 Time evolution with potential and collisions 5.3.1 Approximate time evolution equations Time evolution equations of the moments hXiV ji of the phase space probability density that account 1 2 2 for both, deterministic motion due to inertia and the parabolic potential U(X, t) = 2 Mω(t) X , and random discontinuous increments or decrements of the velocity V due to collisions with gas particles, are obtained by plugging the Kramers-Moyal coefficients αn(V ) into Equation (4.67). If one uses the expanded form of αn(V ) (cf. Equation (4.60)), an infinite set of coupled, inhomogeneous, ordinary first order differential equations with non-constant coefficients is generated. As an exact analytical solution is out of question, an appropriate approximation has to be found. To this end, the expansion in the square root of the mass ratio δ := pm/M, usually a small quantity, comes in handy. Additionally, the following condition is imposed: In case of a temporarily constant, non-vanishing potential, any approximated moment should approach the value corresponding to the equilibrium distribution, which is given by Z i j dΓ i j hX V ieq := X V Peq(Γ, ω) 2π~/M i+j 1 (i+j) r 2 2 i + 1 j + 1 k T = Γ Γ 1 + (−1)i 1 + (−1)j ω−i B , (5.58) 4π 2 2 M where Peq(Γ, ω) is the equilibrium distribution defined in Equation (5.50). The derivation of this formula is given in Appendix A.5. A common ansatz would now be to write the moments as power series in δ, too, hXiV ji(t) = P∞ (ij) n n=0 cn (t)δ , and solve each equation for each order of δ separately. However, this method does not seem to fulfil the above condition (5.58) in general if one wishes to truncate the series at any given finite n. The reason for its failure has to be sought in the fact that δ directly enters the typical time scale of the relaxation process towards thermal equilibrium, which takes place in an exponential kind of way, consequently requiring all orders of δ.1 Luckily another approach proves to be more fruitful. Instead of expanding the solutions, the time evolution equations themselves are approximated in a way that a natural hierarchy emerges and closed sets of equations can be constructed. The idea is as follows: First, a power of δ, n, up to which the equations of the lowest order2 moments hXi and hV i are to be expanded, is chosen. In these equations, higher order moments appear under certain orders of δ. If hχi is one of these moments and m the corresponding power of δ, the time evolution equation of hχi is then expanded only to order n − m, proceeding in a similar way. If n − m < 0, it is simply left out. Moreover, additional powers of δ are added when needed for the equations to have stationary solutions agreeing with
1It should be noted that for constant ω(t), however, a power series ansatz appears to be promising if one substitutes t0 := δt. Problems arise if ω(t) is time-dependent, as additional δs will appear in the expression for ω(t) when going from t to t0. 2The order of hXiV j i is defined as i + j.
41 5 Using potential wells as stopping sites condition (5.58). Different levels of approximation, N0 to N5, which are constructed by this means and which are compiled in Table 5.1, are compared. The following set of equations corresponds to N5, the other approximations may be obtained by skipping the appropriate terms.
∂thXi = hV i, (5.59) k T k T k T ∂ hV i = −ω(t)2hXi + A − 6 B C + B Eδ2 δ2 δhV i + B + Dδ2 δ2 hV 2i − B + t M M M k T E + C + B Eδ2 δ3hV 3i + δ5hV 5i, (5.60) M 20
2 ∂thX i = 2hXV i, (5.61) k T k T k T ∂ hV 2i = −2 B A − 12 B Cδ2 δ − 2ω(t)2hXV i − 8 B Bδ2hV i+ t M M M k T + 2 A − 15 B Cδ2 δhV 2i + 2Bδ2hV 3i + 2Cδ3hV 4i, (5.62) M k T k T ∂ hXV i = hV 2i − ω(t)2hX2i − B Bδ2hXi + A − 6 B Cδ2 δhXV i + Bδ2hXV 2i + Cδ3hXV 3i, t M M (5.63)
3 2 ∂thX i = 3hX V i, (5.64) k T 2 k T k T ∂ hV 3i = 12 B Bδ2 − 3ω(t)2hXV 2i − 6 B AδhV i − 21 B Bδ2hV 2i + 3AδhV 3i + 3Bδ2hV 4i, t M M M (5.65) k T ∂ hX2V i = 2hXV 2i − ω(t)2hX3i − B Bδ2hX2i + AδhX2V i + Bδ2hX2V 2i, (5.66) t M k T k T ∂ hXV 2i = hV 3i − 2ω(t)2hX2V i − 8 B Bδ2hXV i + 2AδhXV 2i + 2Bδ2hXV 3i − 2 B AδhXi, t M M (5.67)
4 3 ∂thX i = 4hX V i, (5.68) k T ∂ hV 4i = −4ω(t)2hXV 3i + 4AδhV 4i − 12 B AδhV 2i, (5.69) t M 3 2 2 2 4 3 ∂thX V i = 3hX V i − ω(t) hX i + AδhX V i, (5.70) k T ∂ hXV 3i = hV 4i − 3ω(t)2hX2V 2i + 3AδhXV 3i − 6 B AδhXV i, (5.71) t M k T ∂ hX2V 2i = 2hXV 3i − 2ω(t)2hX3V i + 2AδhX2V 2i − 2 B AδhX2i, (5.72) t M
5 4 ∂thX i = 5hX V i, (5.73) 5 2 4 ∂thV i = −5ω(t) hXV i, (5.74) 4 3 2 2 5 ∂thX V i = 4hX V i − ω(t) hX i, (5.75) 4 5 2 2 3 ∂thXV i = hV i − 4ω(t) hX V i, (5.76) 3 2 2 3 2 4 ∂thX V i = 3hX V i − 2ω(t) hX V i, (5.77) 2 3 4 2 3 2 ∂thX V i = 2hXV i − 3ω(t) hX V i. (5.78)
42 5.3 Time evolution with potential and collisions
The coefficients are defined by
r r 2 kBT (0,2) A := − 4 ρ W1 (K) , (5.79) π M XX (0,3) B := − 2ρ W1 (K) , (5.80) XXX r r 2 2 M (0,4) C := − ρ W1 (K) , (5.81) 3 π kBT XXXX (0,5) D :=2ρ W1 (K) , (5.82) XXXXX r r 3 2 2 M (0,6) E := ρ W1 (K) , (5.83) 3 π kBT XXXXXX where Z (0,n) 1 n W1 (K) = dθSF (θ) sin θ (5.84) X...X 2 ∂K | {z } n times is an element of the first Minkowski tensor of the motor (cf. Section 4.4.4). Note that
(0,1) W1 (K) = 0 (5.85) X as discussed in Section 4.4.2, wherefore the corresponding coefficient is missing above. In N0, no collisions are taken into account and it is simply composed of the equations for hXi and hV i for the time-dependent oscillator, already studied in Section 5.2. In N1 only a damping term AδhV i is added. The first non-trivial set of equations which includes interactions with higher moments is N2:
∂thXi = hV i, (5.86) k T ∂ hV i = −ω(t)2hXi + AδhV i + Bδ2 hV 2i − B , (5.87) t M 2 ∂thX i = 2hXV i, (5.88) k T ∂ hV 2i = −2ω(t)2hXV i + 2Aδ hV 2i − B , (5.89) t M 2 2 2 ∂thXV i = hV i − ω(t) hX i + AδhXV i. (5.90)
In Figure 5.7, numerically obtained exemplary solutions of N2 to N5 are compared for a moderate value of δ = 0.1. The initial state corresponds to thermal equilibrium. Results suggest that no additional features emerge by including higher orders of δ, regarding the first and second order moments, and even quantitatively N2 performs very well. Note that there is an exception if ω(t) is periodically modulated. Then, for specific values of the modulation frequency, parametric resonances occur, in case of which the different approximation levels begin to deviate. Most curves qualitatively resemble very much the ones for the case of zero potential and an “ideally stopped” initial state, which were presented in Figure 5.2. The main difference lies in the behaviour of hV 2i, which is firstly pushed away from its equilibrium value to which it subsequently relaxes again when the potential has more or less been turned off. As a consequence of this retarded relaxation, all other moments evolve in a delayed fashion as well and change only little at small times.
43 5 Using potential wells as stopping sites
0 0 N2 -2 -1 100 N3
× -4 N4 -2 1000
N5 -6 × -3 T B M T k
-8 B -4 M k q -10 -5 τ
q N2 / -12 i -6 N3 / i V N4 -14 h -7 X
h N5 -16 -8 0 20 40 60 80 100 0 20 40 60 80 100 t/τ t/τ
16 1 N2 14 N3
100 0.8
/ 12 N4 N5 T 2 B 10 M 0.6 k T τ M 8 B / k i
2 0.4 6 N2 / V i h N3
2 4 0.2 N4 X
h 2 N5 0 0 0 20 40 60 80 100 0 20 40 60 80 100 t/τ t/τ
10
8 T τ M B 6 k / i 4 N2
XV N3 h 2 N4 N5 0 0 20 40 60 80 100 t/τ
Figure 5.7 Comparison of the different approximation levels N2 to N5 (cf. Table 5.1, Equations (5.59) to (5.78) and (5.86) to (5.90)). Parameter values are ω0 = 10 · 2π/τ, q δ = 0.1 and ρS = p π / M /τ . The shape of the motor is that of the triangle 2 kBT depicted in Figure 5.1 and the corresponding elements of the Minkowski tensors are given by Equation (5.3). Initially, at t = t0 := 0, the system is in thermal equilibrium (cf. Equation (5.58)).
44 5.3 Time evolution with potential and collisions
5.3.2 Variation of parameters In the following, the influence of different parameters on the evolution of the system is studied with the help of numerical solutions of N2. The shape of the motor particle is again that of the triangular test motor (cf. Figure 5.1 and Equation (5.3)). By defining the dimensionless quantity r r 2 k T r := ρSτ B , (5.91) π M which is proportional to ρS, the inverse of the typical length scale of the problem, the coefficients A and B become r A = − , (5.92) τ 1rπ r M r B = . (5.93) 4 2 kBT τ
Moreover, t0 is always chosen to be zero and the system starts from the thermal equilibrium state 1 2 2 corresponding to the potential U(X, t = 0) = 2 Mω0X . Thus, the initial values of the moments of the phase space pdf are given by Equation (5.58). If the geometry is to be left constant and times are measured in units of τ, energies in units of kBT and masses in units of M, the initial oscillator frequency of the potential well f0 := ω0/(2π), the square root of the mass ratio δ and the typical length scale, represented by r, are identified as the remaining parameters that can be varied independently. As the following plots show, many features and dependences are already well explained by the observations made in the study of either the undamped oscillator (cf. Section 5.2) or the collision driven evolution without potential (cf. Section 5.1). As the typical time scales τ and τr = −1/(Aδ) approach each other, interactions between both processes grow. On the other hand, if the time scales are well separated, the processes may be regarded more or less independently, which is discussed in Section 5.3.3.
Variation of the initial oscillator frequency f0 Figure 5.8 shows the time evolution of the first and second order moments, and of the efficiency η for a wide range of values of f0.
• One clearly notices that the effect of rectification dies down rapidly if f0 drops below a value of approximately 1/τ, whereas it changes only weakly for larger f0. This relates to the de- 2 1 pendence of how far hV i is pushed away from its equilibrium value on the ratio of f0/ τ (cf. Equation (5.57)).
• Likewise, the height of the efficiency maximum varies very little for sufficiently high f0, which 2 characterises the range where the neglect of hX i(t0) in the approximation of η (5.24) is valid. • A feature that is presumably based on the interplay of potential change and collisions is the observation that the individual curves appear to be more or less temporally shifted in a rather regular manner. A somewhat crude attempt to understand this behaviour is to define a critical time tc := 2τ ln(τrf0), (5.94) which signifies the time when the characteristic time scale of the oscillator potential, 2π/ω(t), and that of the gas - motor interaction, τ, are identical. It might be interpreted as loosely describing the transition point from a potential dominated to a collision dominated regime, where the relaxation process back to equilibrium sets in. The logarithm in the expression for tc coincides with the uniform spacing of the curves as their corresponding values of f0 vary exponentially.
45 5 Using potential wells as stopping sites
0 0 -2 -1 100
× -4 -2 1000
f τ = 0.1 -6 0 × -3 T B M f0τ = 1 T k
-8 B -4 M
f0τ = 10 k q -10 -5 τ f τ = 100 0 q f0τ = 0.1 / -12 i -6 f0τ = 1 / i V f τ = 10 -14 h -7 0 X
h f0τ = 100 -16 -8 0 20 40 60 80 100 0 20 40 60 80 100 t/τ t/τ
1 4 0.9
slope 2 0.8
2 T τ T M 0.7 B B M k 0 k 0.6 8 / / 0.5 i i 2 2 -2 0.4 f0τ = 0.1 V
X 4 h h f0τ = 1
0.3 -4 f0τ = 10
lg 0.2 f0τ = 100 -6 0.1 0 1 2 3 4 0 5 10 15 20 25 30 35 40 lg(t/τ) t/τ
10 12 9 f0τ = 0.1 8 10 f0τ = 1 7 f0τ = 10
T τ 8
M f0τ = 100 B 6 k 5 1000 6 / i 4 ×
f τ = 0.1 η 3 0 4 XV f0τ = 1 h 2 f0τ = 10 2 1 f0τ = 100 0 0 0 20 40 60 80 100 -1 0 1 2 3 4 5 6 t/τ lg(t/τ)
Figure 5.8 Variation of the initial oscillator frequency f0 for the setup described at the beginning of Section 5.3.2. The values of the remaining parameters are δ = 0.1 and r = 1.
46 5.3 Time evolution with potential and collisions
0 -2 -2 -3 100 -4 T × -4 B M k -6 -5 T q B
M -6 / k -8
i| -7 q
-10 V τ
|h -8
-12 / -9 i -14 lg
X -10 h -16 -11 1 10 100 1000 10000 0 1 2 3 4 5 t/τ lg(t/τ)
10 1 δ = 0.001 8
δ = 0.01 0.8 2 6 δ = 0.1 T τ T M slope B B
4 M
k 0.6 k 2 6 / / i i
2 0.4 2 0 4 V X h h -2
2 0.2
lg -4 -6 0 0 1 2 3 4 5 6 1 10 100 1000 10000 lg(t/τ) t/τ
3 0 2 -2 1
T τ -4 M B
k 0 )
-6 η
/ -1 i
lg( -8 -2 XV h -3 δ = 0.001 -10 δ = 0.001 δ = 0.01 δ = 0.01 lg -4 -12 δ = 0.1 δ = 0.1 -5 -14 1 10 100 1000 10000 -2 -1 0 1 2 3 4 5 6 t/τ lg(t/τ)
Figure 5.9 Variation of the mass ratio δ for the setup described at the beginning of Section 5.3.2. The values of the remaining parameters are f0 = 10/τ and r = 1.
47 5 Using potential wells as stopping sites
Variation of the mass ratio δ := pm/M Figure 5.9 features the same plots as in the previous paragraph, now for different values of δ. As the reliability of the used equations N2 becomes doubtful for larger δs, the value 0.1 is used as upper boundary.
• Almost all properties of the graphs can be explained based on the system without potential (cf. Section 5.1), which is due to the fact that for δ = 0.1 the two major time scales are already pretty well separated and become even more so, as δ decreases. These dependences include the roughly linear change of the height of the velocity boost, the efficiency maximum, the final value of hXV i and the long-time behaviour of hX2i (D ∝ δ). Moreover, the saturation level of hXi hardly depends on δ at all.
• In the plots of hV 2i one can observe how increased damping (larger values of δ) hinders leaving the equilibrium state.
• The temporal spacing between the individual curves now is not so much due to the change of tc as to the alteration of the relaxation time τr := −1/(Aδ).
• An interesting detail is the bump that evolves in the slope of the double logarithmic plot of hX2i for small δs. The interpretation might be that in this case the damping effect of the medium is so weak that the initially fast oscillations of the slope, which one can clearly see at small times, survive very long.
Variation of the typical length scale, associated with 1/r Eventually, the changes induced by a variation of r are shown in Figure 5.10.
−1 τ • Since τr = Aδ = rδ , the relaxation time changes in just the same manner as if δ was modified. Moreover, in the equations of the second order moments in N2, r appears only in combination with δ as rδ. Hence, the plots of hX2i, hV 2i and hXV i are virtually the same as in Figure 5.9. The difference is that r is not bounded in principal and thus also comparably small relaxation times can be explored.
• As τr approaches τ or even falls below, it becomes more and more difficult to push away the system from equilibrium, since relaxation is to fast. This can be nicely seen in the plots of hV 2i, for instance. The limit case would be that of a quasi static process along equilibrium states, as in classical thermodynamics.
• It is an interesting observation that the linear dependence of the saturation value of hXi also prevails for large r, that is in the regime of similar relaxation and potential-decay time scales, where one should actually be careful about transferring results from the isolated processes.
• Considering the intensity of the velocity boost and the efficiency maximum, one might get the impression that for separated time scales there is no very pronounced dependence on r, as predicted by the formulas for the isolated processes (cf. Equations (5.14) and (5.24)), whereas there are strong changes when time scales become comparable or reversed. However, the plots might as well be misleading, since r itself is altered in a highly non-linear way from curve to curve.
48 5.3 Time evolution with potential and collisions
2 0 -1 0 T B
M -2 1000 k -2
× -3 q τ T
-4 B -4 M
r = 0.01 k / -5 i| -6 r = 0.1 q / X r = 1 i -6 |h -8 r = 10 V h -7 r = 100 lg -10 -8 -1 0 1 2 3 4 5 1 10 100 1000 10000 lg(t/τ) t/τ
10 1 8 0.9
0.8
2 6 0.7 T τ T M 4 B B M
k 0.6 2 slope k 0.5 / / 0 i i 6
2 0.4 2 -2 V X 4 0.3 h h
-4 2 0.2
lg -6 0.1 -8 0 0 1 2 3 4 5 6 -2 -1 0 1 2 3 4 5 lg(t/τ) lg(t/τ)
4 12 3
10 2 T τ M
B 1 8 k
0 1000 / 6 i -1 ×
-2 η 4 XV h
-3 2 lg -4 -5 0 0 2 4 6 8 10 12 14 -2 -1 0 1 2 3 4 5 6 lg(t/τ) lg(t/τ)
Figure 5.10 Variation of r, representing the typical length scale, for the setup de- scribed at the beginning of Section 5.3.2. The values of the remaining parameters are δ = 0.1 and f0 = 10/τ.
49 5 Using potential wells as stopping sites
5.3.3 Separation of time scales
If τ τr, hardly any collision will occur during the first crucial phase of the decrease of the potential strength. Likewise, the potential will more or less be turned off and change only imperceptibly3 by the time the interactions between motor and gas begin to play a role. This motivates dividing the process into two parts to obtain an analytical approximation: first, a very fast change of the potential to almost U(X) = 0, where collisions are neglected and where the analysis of Section 5.2 applies, and second, the slow relaxation towards equilibrium (at least concerning the velocities), where the potential is treated as being constantly zero and where the findings of Section 5.1 may be employed. The results of the first phase are handed on as initial values to the second phase. In Figure 5.11, solutions of the isolated processes are compared with numerical solutions of N2. τ The parameter values are r = 0.1, δ = 0.1 and f0 = 10/τ. Hence, τr = rδ = 100 · τ. The curves which neglect collisions start at t0 = 0 from the corresponding equilibrium values. The curves of the free relaxation without potential were manually shifted to the right by an amount of 10 τ, to achieve a good match with the numerical data. A consistent way of choosing the initial values of the second phase would then be to use the values the solutions of the first part have at the very moment that the second stage starts, t = 10 τ that is. However, using tc = 2 ln(1000)τ ≈ 13.8τ instead, leads to a better agreement with the numerical solutions of N2. The reason might simply be that the division into two isolated processes is of course still but an approximation.4 Likewise, one should certainly not attach to much importance to these specific values or the fact that tc works nicely here, as these choices are still rather arbitrary. Nonetheless, tc might be a good guide for where to set the cut approximately. A detail that should be stressed is that the value of the crucial quantity hV 2i luckily is not very sensitive to the exact time it is taken a,t at the end of phase one, as it quickly approaches its long-time limit. In fact, the latter could indeed be used as the desired initial value as well. With regard to the application as Brownian motor one usually is more interested in the second phase, where the system tries to equilibrate again and most of the net motion originates from. Similarly, the focus lies on time scales of the order of τr. If τ τr, it is then possible to replace the first part simply by an instantaneous drop of hV 2i and appropriate changes of the other moments. The amount of these alterations can be based on the specific model of the stopping site, as demonstrated above. This situation comes very close to the mode of stopping used in References [42, 41], where the motor velocity is set to zero whenever a binding site is reached.
5.4 A remark on re-opening and multiple stopping sites
So far only a singular release process has been studied and it turned out that its capability of rectifying the Brownian motion of an asymmetric particle is rather weak. Typical values of η were clearly below one – meaning that a significant proportion of motors headed towards the “wrong” direction – and − 1 decreased like t 2 for larger times as normal diffusion set in. As will be demonstrated in Section 6.3 with a minimal model, this problem can be overcome if re-opening stopping sites are used, that is stopping sites that are able to bind a once released particle again, and which, moreover, are preferably distributed over the whole system. In this section an idea is briefly sketched how this scheme could – in principle – be included into the present model. First of all, it should be noted that simply opening up a potential well might not suffice to trap a particle successfully. Disturbing equilibrium effectively when the potential is relaxed again, requires a typical oscillation period in the well much shorter than the velocity relaxation time in the medium (cf. Section 5.3.2). Hence, in most cases the particle to be trapped would simply traverse the stopping site without losing energy in a collision. Consequently, an additional dissipation mechanism is needed. A
3 Of course the potential will never be exactly zero, and stays a parabola for all times. However, it dilates so quickly√ that after a few τ it is virtually flat within the region explored by the motor (roughly proportional to t) in comparison to the thermal energy. 4A first step towards carefully blending both regions while retaining rather simple equations could be to include a linear damping term into the equations of the harmonic oscillator.
50 5.4 A remark on re-opening and multiple stopping sites
0 4 numerical data -0.2 2 no collisions -0.4
T no potential 1000 B M 0 k
-0.6 × T q
-0.8 B -2 τ M k -1 / q -4 i /
-1.2 numerical data i X h no collisions V -6 -1.4 h no potential -1.6 -8 0.1 1 10 100 1000 1 10 100 1000 t/τ t/τ
8 1.2 1
6
2 0.8 T τ T M 4 B B
M 0.6 k k 2 0.4 / / i i 2 2 0 0.2 V X h h numerical data 0 -2 no collisions
lg -0.2 no potential -4 -0.4 0.1 1 10 100 1000 10000 0.1 1 10 100 1000 t/τ t/τ
3 2 1 T τ M
B 0 k
-1 / i -2 -3 XV
h numerical data -4 no collisions lg -5 no potential -6 0.1 1 10 100 1000 t/τ
Figure 5.11 Approximation of the numerical solutions of N2 (cf. Equations (5.86) to (5.90)) by the analytical solutions of the isolated processes either without collisions (cf. Equations (5.52) to (5.56)) or without potential (cf. Equations (5.13) to (5.17)) in case of separated time scales. Here, τr = 100 · τ, r = 0.1, δ = 0.1 and f0 = 10/τ.
51 5 Using potential wells as stopping sites suggestion is that during the binding process and according to Newton’s third law, the motor of course also exerts forces on the stopping site, wherefore oscillatory modes of the filament or microtubule the binding site is part of may be excited. Assuming to have successfully coped with this problem, the easiest way of trapping and releasing the motor again an again is to close and open up the binding sites periodically after given times tw and ts, corresponding to transitions between a weak binding state (w) and a strong binding state (s). This proceeding is similar to the ratchet mechanism, which was applied to single headed kinesin in Section 3.2. To make this model more realistic, one could switch from deterministic transitions between s and w to stochastic ones, governed by a rate equation. Transitions rates could be made dependent on the current position (e.g. to distinguish binding sites from the rest), and, moreover, one could also work in concentrations of the involved chemical agents. As the transitions have to be smooth to allow the motor to perform work and lose kinetic energy, one needs another variable to control these processes. Say, for instance, the motor has just switched from w to s, then this variable is set to an initial value corresponding to a still closed potential well, but has to evolve deterministically to another value implying an open well.
52 6 Minimal Models
The analysis of the model presented in Section 4.2 proved to be rather cumbersome, and only ap- proximate results were obtained (cf. Chapter 5). In this chapter, a series of simple generic systems is introduced, which provide better analytical accessibility while already exhibiting crucial features of the more complex scenario. A model with only two states or two velocities, respectively, is presented in Section 6.1, whereat the main purpose lies in motivating the normalisation of the path-dependent probability density defined in Equation (4.33) on page 24. In Section 6.2, a three-state system is studied, which constitutes a minimal and analytically fully solvable model for the motion of an asymmetric Brownian particle in one dimension. Great qualitative similarities to the “full” model are found already at this very abstract stage. The influence of asymmetry, which is noticeable even in equilibrium, is discussed, and in a small excursus differences to systems with even state variables (i.e., variables which do not change sign upon time reversal) are briefly examined. Moreover, in Section 6.3, the three-velocity model is augmented by a discrete position variable, allowing for the investigation of the effect of stopping sites. Large differences are found with respect to the singular release process of Chapter 5 if the stopping sites are able to trap a released particle again and are periodically positioned. This is in accordance with previous results from molecular dynamics simulations (cf. References [42, 20]). Additionally, ideas to further extend the model are proposed. Finally, some thoughts on connections between the minimal models and the full system, and moreover to experimental data of the monomeric kinesin (cf. Section 3.2) are expressed in Section 6.4. Please note that for convenience only dimensionless quantities are used in this chapter. Moreover, as there are only three symmetrically distributed discrete velocities at maximum, they always can be normalised to −1, 0 and 1.
6.1 Two-state model
6.1.1 Description of the model
In Section 4.3.2, an expression for the conditional probability density of a specific sequence of N velocity changes {∆Vi} at times {ti}, abbreviated by ∆V (t), was proposed:
N Y ∗ P [∆V (t)|Γ0, t0] := [Pvoid(ti, ti−1)λ(∆Vi,Vi)] Pvoid(t , tN ) i=1 ∗ t ∞ N Z Z Y = exp − dt d∆V λ(∆V,V (t)) λ(∆Vi,Vi). (6.1) i=1 t0 −∞
The main objective of this section is to motivate the assumption that P [∆V (t)|Γ0, t0] is a well defined probability density and in particular properly normalised. For this purpose a reduced model, which consists of just two discrete (velocity) states and features no potential, is studied. Transitions from state 1 to 2 and vice versa occur at constant rates λ1 and λ2. Between the initial time t0 and the final time t∗ the system runs through a sequence of N + 1 states, being an alternation of state 1 and 2 in fact, characterisable by the corresponding rates {λ(i)}, i = 1 ...N + 1. Transitions take places
53 6 Minimal Models
2
λ1 λ2
1
Figure 6.1 The two-state model: Transitions between the levels occur randomly, but at constant rates λ1 and λ2, respectively.
at times {ti}, i = 1 ...N. Equation (6.1) may thus be written as
N h (i) i (N+1) ∗ (i) Y −λ (ti−ti−1) (i) −λ (t −tN ) P [∆V (t)|Γ0, t0] = P (N, {λ }, {ti}) := e λ e (6.2) i=1 for the model under consideration.
6.1.2 Normalisation The normalisation condition reads
t∗ t∗ t∗ ∞ Z Z Z X (i) dt1 dt2 ... dtN P (N, {λ }, {ti}) = 1. (6.3) N=0 0 t1 tN−1
(i) N If λ1 = λ2 = λ, the probability density P (N, {λ }, {ti}) is reduced to λ exp(−λ). Since this expression is independent of the times {ti}, the integrals merely contribute a factor of 1/N! and one (i) easily convinces oneself that P (N, {λ }, {ti}) in fact is properly normalised in this particular case. For unequal rates things become more complicated. If one chooses to name the level the process (1) starts at “state 1”, that is λ = λ1, any trajectory is uniquely identified solely by the set of transition times {ti}. Moreover, t0 and t1 can be gauged to 0 and 1, respectively, without loss of generality. As (i) (i) λ is λ1 if i is an odd number, and λ2 if i is even, P (N, {λ }, {ti}) may then be rewritten in the form
λnλne−(λ1s+λ2(1−s)) if N = 2n, n ∈ P (N, {λ(i)}, {t }) = P (N, s) := 1 2 N , (6.4) i n+1 n −(λ1s+λ2(1−s)) λ1 λ2 e if N = 2n + 1, n ∈ N
P2n+1 i ∗ with s := − i=0 (−1) ti, where tN+1 := t = 1. The new variable s is the total time the system spends in state 1, (1 − s) the total time it spends in state 2. Since P (N, s) does obviously not depend on the specific times {ti}, but only on s, the nested time integrals may be transformed:
1 if N = 0 1 1 1 1 n n−1 Z Z Z s (1−s) R n! (n−1)! ds if N = 2n, n ∈ N/{0} dt1 dt2 ... dtN → 0 . (6.5) 1 n n 0 t1 tN−1 s (1−s) R n! n! ds if N = 2n + 1, n ∈ N 0 This regrouping is best illustrated in an example as in Figure 6.2. One can now rewrite the normali- sation condition as
1 " ∞ # Z X ds PP(λ1, 0, 1) + (λ1PP(λ1, n, s) + λ2PP(λ1, n + 1, s)) PP(λ2, n, 1 − s) = 1, (6.6) 0 n=0
54 6.1 Two-state model
state 2
1 t 0 1 =
+ +
s 1-s (i) (ii) (iii)
Figure 6.2 Any specific trajectory of the two-state system may be regrouped and decomposed into two simple Poisson processes (i) and (ii) with events (i.e., jumps) occurring at rates λ1 and λ2, respectively. To connect both parts, an additional transition (iii) is required. with (λt)k P (λ, k, t) := e−λt (6.7) P k! being the Poisson distribution. One notices that as PP(λ, k, t) is normalised with respect to k, the above expression is always smaller than or equal to exp(−λ1) + λ1 + λ2. Consequently, for any finite choice of λ1 and λ2 the probability distribution is normalisable. Actually the normalisation constant should already equal unity independently of λ1 and λ2, of course. If one plugs in the expressions for the Poisson distributions and carries out the sums, one finds that the normalisation condition becomes (cf. Reference [1] p. 108)
1 p p p Z λ1 λ2s I1 2 λ1λ2s(1 − s) + λ1λ2s(1 − s)I0 2 λ1λ2s(1 − s) ds e−λ1 + = 1, (6.8) p λ s+λ (1−s) λ1λ2s(1 − s) e 1 2 0 where I0 and I1 are modified Bessel functions of the first kind. This integral was numerically evaluated i (using the default routine of Maplesoft’s Maple 9.5) for λ1/2 = 10 , where i = −3, −2, ..., 2, 3 and −9 −9 λ1 6= λ2. In all cases the result was either 1.0000000000 ± 0.5 · 10 or 0.9999999999 ± 0.5 · 10 . The 3 2 −9 only exception was the combination λ1 = 10 and λ2 = 10 , which yielded 0.9998955312 ± 0.5 · 10 , but it might well be that the desired numerical accuracy could not be achieved anymore due to the large numbers. In summary, the results indicate that the normalisation condition holds indeed and suggest that this conclusion may also be extended to the more general yet similarly constructed expression (6.1).
6.1.3 Time evolution Although the two-state system is in fact too primitive to represent an asymmetric particle, it is still instructive to solve this simplest possible model and see how it relates to the more realistic case. Let Pj(t) be the probability of finding the system in state j at a given time t. Following Section 4.1.2 and 4.4.1, the time evolution equation adopts a particularly simple form:
P˙1 = − λ1P1 + λ2P2, (6.9)
P˙2 =λ1P1 − λ2P2. (6.10)
55 6 Minimal Models
Note that the parts attributed to continuous motion in Equation (4.44) have vanished and the integrals have transformed into sums, as states are discrete. Unlike the case of the complex model, these equations are easily solved analytically:
λ 2 −(λ1+λ2)t −(λ1+λ2)t P1(t) = 1 − e + P1(0) e , (6.11) λ1 + λ2 λ 1 −(λ1+λ2)t −(λ1+λ2)t P2(t) = 1 − e + P2(0) e , (6.12) λ1 + λ2
where t0 = 0 was chosen for simplicity. In the limit t → ∞ the system approaches the stationary distribution
λ2 P1,stat = , (6.13) λ1 + λ2 λ1 P2,stat = . (6.14) λ1 + λ2
Bearing in mind the physical background, one is led to assign velocities V1 and V2 to the states 1 and 2. With respect to the canonical distribution it is reasonable to demand a symmetric stationary state, wherefore V1 = −V2 =: −1 and λ1 = λ2 =: λ. With these definitions, it is possible to compare this minimal model to the “full” system (cf. Section 4.2) by calculating moments of the phase space pdf. However, it is useful to consider the velocity auto-correlation function (VACF) CVV (t, s) first, which is needed in the subsequent calculations.
2 2 X X CVV (t, s) := hV (t)V (t + s)i = ViVjPji(t + s, t)Pi(t). (6.15) i=1 j=1
Here, Pji(t + s, t) denotes the conditional probability of finding the system in state j at time t + s given it was in state i at time s. Obviously, Pji(t + s, t) is equal to Pj(s) if Pj(0) = δij is used as boundary value. From Equations (6.11) and (6.12) thus follows
1 P (t + s, t) = 1 − e−2λs + δ e−2λs. (6.16) ji 2 ij
A short calculation implying the normalisation condition P 1 + P 2 = 1 then yields
−2λs CVV (t, s) = e . (6.17)
Another quantity needed is the correlation function of positions and velocities
Z 2 2 X X −2λs hX(t)V (t + s)i = dXX Pi(X, t) VjPji(t + s, t) = hXV i(t)e , (6.18) i=1 j=1
where Pi(X, t) denotes the joint probability of position X and state i at time t. Upon calculating
56 6.2 Three-state generic model of an asymmetric Brownian particle in one dimension the first few moments of the phase space probability distribution one eventually finds
−2λt hV i(t) = P2 − P1 = hV i(0) e , (6.19) t Z 1 − e−2λt hXi(t) = hXi(0) + dt0 hV i(t0) = hXi(0) + hV i(0) , (6.20) 2λ 0 2 hV i(t) = P1 + P2 = 1, (6.21) t t t Z Z Z hX2i(t) = hX2i(0) + 2 dt0 hX(0)V (t0)i + dt0 dt00 hV (t0)V (t00)i =
0 0 0 t t t−t0 Z Z Z 2 0 0 0 0 = hX i(0) + 2 dt hX(0)V (t )i + 2 dt ds CVV (t , s) = 0 0 0 1 1 − e−2λt = hX2i(0) + 2hXV i(0)e−2λt + 2 t − , (6.22) 2λ 2λ2 |{z} =:D t t Z Z 0 0 0 0 0 hXV i(t) = hX(0)V (t)i + dt hV (t )V (t)i = hX(0)V (t)i + dt CVV (t , t − t ) = 0 0 1 − e−2λt = hXV i(0)e−2λt + . (6.23) 2λ These equations correspond precisely to the solutions of the approximation N2 (cf. Equations (5.13) to (5.17)) in case of a symmetric motor, that is B = 0, and zero potential. The role of Aδ is taken over by −2λ. One easily convinces oneself by expanding the exponential that, at least for uncorrelated initial positions and velocities, hX2i goes like t2 for small times, which corresponds to ballistic motion, whereas for large times diffusion predominates, where hX2i(t) ≈ 2Dt. This behaviour, as well as the exponential dissipation of velocities as shown by hV i and CVV (t, s), is a typical feature of undirected (not overdamped) random motion. Rectification cannot appear, as the requirement of a symmetric equilibrium distribution interdicts asymmetric rates in case of a two-state system. Introducing a third state, however, resolves this problem, as presented in the following section.
6.2 Three-state generic model of an asymmetric Brownian particle in one dimension 6.2.1 Description of the model and general solution A three-state model as sketched in Figure 6.3 is the smallest system which can be designed to feature a symmetric equilibrium distribution with detailed balance while exhibiting a profound asymmetry concerning the transition rates. It therefore constitutes a minimal model of an asymmetric Brownian particle. The asymmetry of rates is the crucial prerequisite for the system to relax from even a symmetric non-equilibrium distribution into an again symmetric equilibrium through a series of asymmetric intermediate states. If the levels are again interpreted as velocities, these intermediate states are precisely what causes a net motion. If Pi(t) denotes again the probability of finding the system in state i at time t and Wij is the transition rate from state j to i, the master equation of this three-state system reads
P˙ = WP, (6.24)
57 6 Minimal Models
3
W23 W32 W31
2
W13 W12 W21
1
Figure 6.3 The three-state model: Transitions from level i to j occur randomly at a rate of Wji. where P1 −(W21 + W31) W12 W13 P := P2 , W := W21 −(W12 + W32) W23 . (6.25) P3 W31 W32 −(W13 + W23)
Note that the sum of each column of the transition matrix W is zero, which implies conservation of probability. To quantify the asymmetry of the rates (or the motor, respectively) the asymmetry ratio
W a := 23 (6.26) W21 is introduced and the transition matrix may be decomposed as
W = W0 + (a − 1)W˜ (a), (6.27) where W0 is independent of a. The general solution of Equation (6.24) is
2 Wt X µit P(t) = e P(0) = Cie , (6.28) i=0 where Ci and µi are eigenvectors and eigenvalues of W and again t0 = 0 for convenience. The second equality holds because W is diagonalisable. If one requires all rates to be positive, one can show that µ0 = 0 and all other eigenvalues have negative real parts, using the Perron-Frobenius theorem (cf. 1 Appendix A.6.1). Thus, the system decays to a stationary distribution given by C0 independent of P(0). The norms of the remaining eigenvectors can be adjusted to fit boundary values. To adapt this still very general formulation to the problem of the Brownian motor, the stationary state is required to be symmetric and to exhibit detailed balance, which imposes further relations on the transition rates. For the three-state system the detailed balance condition (4.17) on page 20 can be written in the form WijPj,stat = W(j)(i)Pi,stat, (6.29) where the function (i) maps a sate i to its corresponding time-reversed state. If one again assigns velocities −1, 0, 1 to states 1, 2, 3, or in fact uses any odd quantity which lies symmetric around state 2, the time reverse function is given by (i) = 4−i. To shed light on the role of this very special property, differences to even detailed balance with (i) = i will be briefly addressed. Moreover, there probably are asymmetric three-state systems (e.g. chemical reaction networks) where in fact the even version is more appropriate.
1If rates are allowed to be zero, similar statements can be found, but the eigenvalue zero may be degenerate.
58 6.2 Three-state generic model of an asymmetric Brownian particle in one dimension
6.2.2 Odd detailed balance Odd detailed balance implies by Equation (6.29)
W21P1,stat = W32P2,stat, (6.30)
W23P3,stat = W12P2,stat, (6.31)
W31P1,stat = W31P3,stat, (6.32)
W13P3,stat = W13P1,stat. (6.33)
Thus, the probability current from state 1 to 2 is the same as from state 2 to 3. This is also true for the opposite direction (i.e., from 3 to 2 to 1), but unless a = 1 the current is not equal to the first case. Therefore, a net flux through the sequence of states emerges, which is balanced by a current from the last to the first state such that the total probability is conserved, leading to a circular net current in the stationary state in the asymmetric case a 6= 1. Considering the rates, one obtains from the above equations the relations W W 23 = 12 , (6.34) W21 W32 W13 − W31 = W21 − W23, (6.35) where for the second one the conservation of total probability is taken into account, too. One then has −(W13 + W21) W12 W13 W0 := W21 −2W12 W21 (6.36) W13 W12 −(W13 + W21) and −W21 0 0 ˜ W := 0 W12/a W21 . (6.37) W21 −W12/a −W21 The eigenvalues of W are √ 2W a + 2a2W + W (a + 1) ∓ K µ = − 13 21 12 (6.38) 1/2 2a with
2 2 2 2 2 K := 4W13a − 4W13W12a(a + 1) + W12(a + 1) + 4(W13 − W12)W21a (a − 1). (6.39) For a = 1 they reduce to
µ1 = −(W21 + 2 min(W12,W13)), (6.40)
µ2 = −(W21 + 2 max(W12,W13)), (6.41) or simpler
µ1,sym := −(W21 + 2W12), (6.42)
µ2,sym := −(W21 + 2W13). (6.43)
For a 6= 1 the eigenvalues may be complex and oscillatory solutions can emerge. The eigenvectors are rather complicated in general, wherefore they shall not be given here, except for
1 C0 ∝ aW21/W12 , (6.44) 1
59 6 Minimal Models
which is directly deduced from Equations (6.30) to (6.33). As C0 is a constant part of any solution, the sum of the elements of each other eigenvector has to vanish, since the probability distribution P(t) has to be normalised. In case of a = 1 the eigenvectors adapt a particularly simple form:
1 1 1 C0,sym ∝ W21/W12 , C1,sym ∝ −2 , C2,sym ∝ 0 . (6.45) 1 1 −1
To consider the moments of the probability distribution, velocities −1, 0, 1 are assigned to the levels 1, 2, 3. A useful observation is that in terms of conditional probabilities Pji(t + s, t), one may write P1i(t + s, t) δ1i (i) Wt P (s) := P2i(t + s, t) = e δ2i . (6.46) P3i(t + s, t) δ3i (i) T Since the vectors P are the transform of the standard orthonormal basis vectors eˆi := (δ1i, δ2i, δ3i) , the solution of Equation (6.24) is the linear combination
3 X (i) P(t) = Pi(0)P (t). (6.47) i=1 Mean values can be decomposed analogously,
3 X (i) h·i = Pi(0)h·i , (6.48) i=1 where h·i(i) means that P (i) is used for averaging. The expressions for h·i(i) are in general more compact than h·i, but quite often still rather unhandy, wherefore only the average velocity and the equilibrium VACF are studied here. However, there are no principle objections to performing a more complete analysis, as done for the two-state model in Section 6.1.3. Concerning the average velocity, one finds W a(1 − a) (1) (3) 21 µ2t µ1t hV i (t) = P31(t, 0) − P11(t, 0) = −hV i (t) + √ e − e , (6.49) K W (a − 1) (2) 12 µ2t µ1t hV i (t) = P32(t, 0) − P12(t, 0) = √ e − e , (6.50) K 2W a − W (a + 1) 1 (3) 13 12 µ2t µ1t µ2t µ1t hV i (t) = P33(t, 0) − P13(t, 0) = √ e − e + e + e . (6.51) 2 K 2 These equations express the very intuitive fact that the asymmetry of the motor leads to a direction dependent dissipation of velocities. If the initial distribution is already symmetric, the average velocity adapts the simple form a − 1 µ2t µ1t hV i(t) = √ (W12P2(0) − W23P3(0)) e − e . (6.52) K The first bracket vanishes if P(0) already is the equilibrium distribution due to detailed balance (cf. Equation (6.31)), which means there is no rectification in this case. From the above equations one nicely sees how the ability for net motion ceases with the motor approaching perfect symmetry, as for a = 1 one has (1) (3) µ2,symt (2) hV isym(t) = −hV isym(t) = e , hV isym(t) = 0 (6.53) and therefore µ2,symt hV isym(t) = hV i(0) e . (6.54)
60 6.2 Three-state generic model of an asymmetric Brownian particle in one dimension
If the rates are symmetric, the time evolution thus merely consists in dissipating any initial velocity with a typical time constant −1/µ2,sym. Interestingly a second time scale −1/<(µ1,sym) appears as a leaves unity. The same effect is visible in the VACF. Analogous to Equation (6.15) one obtains
CVV (t, s) = (P11(t + s, t) − P31(t + s, t))P1(t) − (P13(t + s, t) − P33(t + s, t))P3(t) = (3) (1) = hV i (s)P3(t) − hV i (s)P1(t). (6.55)
In equilibrium P1(t) = P3(t) = P1/3,stat, wherefore the VACF only depends on s, leading to
W (a + 1) − 2W a + W a(1 − a) 12 13 21 µ1s µ2s µ1s µ2s CVV,stat(s) = P1/3,stat √ (e − e ) + (e + e ) . (6.56) K For a = 1 again one eigenvalue vanishes and
µ2,syms CVV,stat,sym(s) = 2P1/3,state . (6.57)
This result is particularly remarkable, as it shows that, even in equilibrium, there are differences between symmetric and asymmetric bodies. This is not suggested by the findings in Chapter 5, where only moments of the phase space pdf are considered and where one might in fact be led to assume that the effect of asymmetry dies out as soon as detailed balance is established. Moreover, the stationary VACF may oscillate if the eigenvalues are complex and even become negative. As an illustrating example consider the transition matrix
−5 1 3 W = 4 −3 2 , (6.58) 1 2 −5
1 which has an asymmetry ratio of a = 2 . Besides the trivial eigenvalue and eigenvector, one finds √ −13 ± i 7 µ = (6.59) 1/2 2 and √ √ 17−3i√ 7 17+3i√ 7 1+5i 7 1−5i 7 √ √ 5+3i 7 5−3i 7 C1 ∝ √ , C2 ∝ √ . (6.60) 2(2−i 7) 2(2+i 7) 1 1
For the initial distribution Pi(0) = δi2, being a perfect stopped state, so to speak, this results in √ ! √ !! 1 1 − 13 t 5 7 7 P1(t) = − e 2 √ sin t + cos t , (6.61) 4 4 7 2 2 √ ! √ !! 1 1 − 13 t 1 7 7 P2(t) = + e 2 √ sin t + cos t , (6.62) 2 2 7 2 2 √ ! √ !! 1 1 − 13 t −3 7 7 P3(t) = − e 2 √ sin t + cos t . (6.63) 4 4 7 2 2
In Figure 6.4, the time evolution of the probabilities P1(t), P2(t) and P3(t), and of hV i(t) = P3(t) − R t 2 P1(t), hXi(t) = 0 ds hV i(t) and hV i(t) = P3(t) + P1(t) are plotted. One observes a striking resemblance to the corresponding curves of the “full” model (cf. Figure 5.2 in Section 5.1.1). Due to the complex eigenvalues there are in fact oscillations, which, however, are exponentially damped in a way that they are not visible in the plots.
61 6 Minimal Models
1 1 P1 0.8 P2 0.8 P3 hV i/hV i 0.6 0.6 max 2 2 hV i/hV imax 0.4 0.4 hXi/hXimax 0.2 0.2
0 0 0 0.5 1 1.5 2 0 0.5 1 1.5 2 t t
Figure 6.4 Exemplary solution of the three-state model for the transition ma- trix (6.58), corresponding to odd detailed balance and given by Equations (6.61) to (6.63). The system starts from the peaked distribution Pi(0) = δi2 and evolves through asymmetric intermediate states with P1 6= P3 into the again symmetric equi- librium. The asymmetric distribution leads to a momentary net velocity causing an average displacement.
6.2.3 Even detailed balance As the focus of this paper lies on odd detailed balance, merely a brief review of differences and similarities between even and odd detailed balance is given in this section. The most prominent similarity probably is that, in fact, both cases are identical for symmetric rates, that is if a = 1. This is readily appreciated from the relations derived from the even detailed balance condition and the request of a symmetric stationary distribution, W W 23 = 32 , (6.64) W21 W12 W13 − W31 = 0, (6.65) or the transition matrix W = W0 + (a − 1)W˜ , where W0 is the same as in the odd case (cf. Equa- tion (6.36)) and 0 0 0 ˜ W := 0 −W12 W21 . (6.66) 0 W12 −W21 Differences emerge as symmetry is broken. For instance, the currents between each two states are balanced in the stationary regime, wherefore there never is a net current. With odd detailed balance a circular probability current begins to flow as a = 1 is left. The non-zero eigenvalues of W are given by √ −(W + W )(a + 1) − W ± K µ = 12 21 13 (6.67) 1/2 2 with 2 2 2 2 2 K := (W21 + 2W21W12)(a − 1) + 4W13 − 4W13W12(a + 1) + W12(a + 1) . (6.68) In contrast to the odd case, they are always real as can be proven by direct calculation carried out in Appendix A.6.2. Oscillatory solutions are hence not possible. The eigenvector of the stationary solution is found to be 1 C0 ∝ W21/W12 (6.69) 1
62 6.3 Augmented three-state model
1 1 P1 0.8 P2 0.8 P3 hV i/hV i 0.6 0.6 max 2 2 hV i/hV imax 0.4 0.4 hXi/hXimax 0.2 0.2
0 0 0 0.5 1 1.5 2 0 0.5 1 1.5 2 t t
Figure 6.5 Exemplary solution of the three-state model for the transition ma- trix (6.70) corresponding to even detailed balance and given by Equations (6.74) to (6.76). To be able to compare the results to the case of odd detailed balance (cf. Figure 6.4) the states are again interpreted as velocities although in a physical system velocity distributions obeying even detailed balance are rather unlikely. and does not depend on the asymmetry ratio a as is the case for odd detailed balance. As a concrete example to compare with the odd case, the transition matrix
−5 2 1 W = 4 −3 2 (6.70) 1 1 −3
1 is chosen, having the same asymmetry ratio of a = 2 and the same stationary distribution as used in the test scenario with odd detailed balance. Upon solving the eigenvalue problem, one finds
µ1 = −4, (6.71)
µ2 = −7 (6.72) and −1 −1 C1 ∝ −2 , C2 ∝ 1 , (6.73) 3 0 leading to
1 1 1 P (t) = + e−4t − e−7t, (6.74) 1 4 12 3 1 1 1 P (t) = + e−4t + e−7t, (6.75) 2 2 6 3 1 1 P (t) = − e−4t, (6.76) 3 4 4 if again Pi(0) = δi2 is used as initial distribution. Repeating the plot of Figure 6.4, as done in Figure 6.5, results largely in a good qualitative agreement with the odd case. At a closer look one notices that the curves of P1 and P3 appear to have more or less swapped places, which causes the motor to head for the opposite direction.
63 6 Minimal Models
V=1 P(1|1) P(0|1) P(1|0)
P(1|-1) V=0 P(-1|1) P(0|0) + P(-1|0) P(0|-1) stopping�sites�{X�i} P(-1|-1) V=-1
Figure 6.6 The augmented three-state model: In contrast to the ordinary three- state system time is discretised, wherefore the rates Wji are replaced by conditional probabilities P (j|i) for transitions from level i to j at one time step. Additionally, a discrete position variable X is introduced and the motor can maximally jump one site to the left or to the right during one time step depending on its velocity. A subset {Xi} of all sites is defined as stopping sites, where the motors velocity is instantly set to zero.
6.3 Augmented three-state model
The virtue of the three-state model of Section (6.2) is that it reproduces the crucial features of the complex system in a reduced and exactly solvable scheme. Yet one may even go one step further by appropriately expanding this minimal model to explore issues which are much more difficult to access else. The talk is of course of the effect of stopping sites, in particular of stopping sites that open up again after having released a trapped motor or that are arranged periodically.
6.3.1 Introducing a position space and stopping sites To include stopping sites, the three-state model is supplemented by adding a second discrete state variable, X. The three original states are identified with the velocities V = −1, V = 0 and V = 1. Moreover, time is discretised as well. Each time step of length ∆t is composed of a combined collision and streaming step, and a stopping step. To find the description of a pure collision step without streaming, one may start from the continuous master equation
1 X P˙ (X, V, t) = [W (V |V 0)P (X,V 0, t) − W (V 0|V )P (X, V, t)]. (6.77) V 0=−1 V 06=V Crossing over to discretised times by replacing the time derivative by the difference quotient leads to
1 X P (X, V, t + ∆t) = P (V |V 0)P (X,V 0, t), (6.78) V 0=−1 where W (V |V 0)∆t =: P (V |V 0) is identified as the transition probability per step and
1 X P (V |V ) := 1 − P (V 0|V ). (6.79) V 0=−1 V 06=V On the other hand, a pure streaming step without collisions is represented by
P (X, V, t + ∆t) = P (X − ∆tV, V, t), (6.80)
64 6.3 Augmented three-state model
(a) tree diagram (b) transitions to regular (c) transitions to stopping site site
Figure 6.7 (a) The tree diagram of the augmented three-state model in the vicinity of a stopping site Xi. States are depicted by nodes, which are linked by the allowed transitions (straight lines). Around Xi some connections are of a lighter grey to indicate that they are not or rarely going to be used. (b) Detail of the tree outlining possible transitions to a regular site with corresponding transition probabilities. (c) Possible transitions to a stopping site. which simply means that particles with velocity V travel - with probability one - each time step a distance of ∆tV . The discrete X values should of course be chosen to comply with this step size. One could now simulate the time evolution by alternating collision and streaming steps, however it is more convenient to use a combined step of the form
1 X P (X, V, t + ∆t) = P (V |V 0)P (X − ∆tV, V 0, t). (6.81) V 0=−1
Last, the stopping sites have to be accounted for. In the simplest case they consist of a set of sites {Xi} where the velocity distribution P (Xi, V, t) is set to δV 0 at each time step. Of course, more complex behaviour may be incorporated (cf. Section 6.3.3), but Section 5.3.3 suggests that the above realisation might already be a good approximation and may serve well for finding fundamental effects. By adding this remaining part, the temporal recursion relation of the probability distribution becomes
" 1 X X 0 0 P (X, V, t + ∆t) = (1 − δXXi ) P (V |V )P (X − ∆tV, V , t) + i V 0=−1 1 1 # X X 00 0 00 0 +δXXi δV 0 P (V |V )P (X − ∆tV ,V , t) , (6.82) V 00=−1 V 0=−1 which is displayed as a tree diagram in Figure 6.7. Each path through this graph corresponds to a specific realisation of the underlying stochastic process.
65 6 Minimal Models
V 0 P (V 0| − 1) P (V 0|0) P (V 0|1) −1 1 − 5c c 3c 0 4c 1 − 3c 2c 1 c 2c 1 − 5c
Table 6.1 Transition probabilties P (V 0|V ) used in the test scenario (c = 0.1).
As in Section 6.2 the transition probabilities are required to conform to the conditions
P (−1|1) − P (1| − 1) = P (0| − 1) − P (0|1), (6.83) P (0|1) P (−1|0) = , (6.84) P (0| − 1) P (1|0) so as to velocities are distributed symmetrically in the equilibrium state and detailed balance is fulfilled.
6.3.2 Results In a test scenario, the following choice of parameters was used. The length of a time step was set to ∆t = 1, yielding a spatial distance of 1 between two neighbouring discrete X values. The transition probabilities are summarised in Table 6.1, where a new parameter c = 0.1 is introduced that in some sense sets the time resolution. As initial probability distribution again the “ideal stopped” state P (X, V, t = 0) = δX0δV 0 was chosen. Based on this setup, three different configurations of stopping sites were investigated in terms of numerically, iteratively evaluating Equation (6.82):
• “vanishing stop”: {Xi} = {}. As the system starts from a stopped state, having no stopping site actually corresponds to having one site where the motor is trapped initially and which vanishes as the observation starts. This case is equivalent to the situations studied in Chapter 5, but also in the test case of Section 6.2.
• “persistent stop”: {Xi} = {0}.
• “periodic persistent stop”: {Xi} = {X|X = 10i; i ∈ Z}. Besides the typical plots (cf. Figures 5.2 and 5.3 on page 34, for instance) of the first moments and the efficiency, shown in Figure 6.8, it is now possible to access the phase space distribution directly as presented in Figure 6.9. One can clearly observe that the ability of a stopping site to bind a once released particle again, as soon as it comes into its vicinity, has great effect on the rectification properties of the system. While in accordance with previous results (for example see Section 5.1.1), the net motion ceases shortly after the particle has been set free from the vanishing stop, there is a long running increasing of hXi in the other cases, since an equilibration of the system is prohibited. Note that there is no difference between the results of the single and the periodic persistent stopping sites for the first ten time steps, as this is the minimum time the motor needs to reach the next stopping site in the latter case. In comparison to the multiple persistent stopping sites the single persistent stopping site is not able to sustain non-equilibrium effectively, as the probability to find the particle near X = 0 decreases with the spreading of the distribution. This feature is best observed in the plots of hV i, hV 2i and P (V ) where the curves slowly approach the ones corresponding to the vanishing stop configuration. In −1/2 detail, hV√i seems to decay roughly like t , which can be deduced from the fact that hXi more or less goes like t, which itself can be concluded from noticing that the efficiency η := |hXi|/phX2i − hXi2 stays approximately constant and phX2i − hXi2 grows linearly.
66 6.3 Augmented three-state model
1.8 1.2 1.6 1 1.4 0.8 1.2 10
i 1 0.6 X i × h 0.8 0.4 V
0.6 h 0.2 0.4 0.2 0 0 -0.2 0 10 20 30 40 50 0 10 20 30 40 50 t t
6 3.5 5 3
4 2 2.5 i X ) 3 2 i 2 2 1.5 X
slope i − h h
1 2 2 1 lg( 0 X 0.5 h -1 1.5 0 -2 lg -0.5 1 -3 -1 1 10 100 1000 10000 1 10 100 1000 t t
0.55 1 0.5 0.5 0 0.45 -0.5 ) i η 2 0.4 -1 V h lg ( 0.35 -1.5 vanishing stop -2 0.3 persistent stop -2.5 periodic persistent stop 0.25 -3 1 10 100 1000 1 10 100 1000 10000 t t
Figure 6.8 Solution of the augmented three-sate model obtained by iteratively eval- uating Equation (6.82) for the transition probabilities given in Table 6.1. For reasons of clarity the discrete data points, which are located at t = i, i ∈ Z, are connected by straight lines and not explicitly plotted themselves. If {Xi} denotes the set of stopping sites, the three configurations correspond to {Xi} = {} (“vanishing stop”), {Xi} = {0} (“persistent stop”) and {Xi} = {X|X = 10i; i ∈ Z} (“periodic persistent stop”). In all cases the initial distribution is P (X,V, 0) = δX0δV 0. The√ additional line in the plot of the efficiency η is a guide to the eye proportional to t.
67 6 Minimal Models
1 6 P (V = 0) 5 t = 200 4 100
P (V = 1) × 3 ) X ( 2 P 1 P (V = −1) 0.1 0 1 10 100 -60 -40 -20 0 20 40 60 t X
(a) Evolution of the velocity distribution. (b) Snapshot of the distribution of positions.
1.2 t = 0 1 t = 5000 0.8
100 t = 10000
× 0.6 t = 15000 ) X ( 0.4 P 0.2 0 -100 0 100 200 300 400 500 600 700 X
(c) Evolution of the distribution of positions for “periodic persistent stop”.
Figure 6.9 Continuation of Figure 6.8: The labels of the curves in (a) and (b) are the same as in Figure 6.8. In (c) only the result of the configuration with periodic persistent stopping sites is displayed. The peaks in (b) and (c) are caused by stopping sites.
68 6.3 Augmented three-state model
Figure 6.10 Tree diagram of a delay line which realises stopping sites that release a trapped particle after three time steps have passed. The upper part of the sketch corresponds to a section of the main graph of the process (cf. Figure 6.7 (a)) in the vicinity of a stopping site Xi. If a path hits the stopping site, it is passed on to the delay line in the lower part of the figure and is fed back not until a certain number of time steps have elapsed. Note that only the links between nodes being crucial to explain the concept are depicted.
The situation changes as stopping sites are placed periodically all over the system and the particle is prevented from diffusing a long way without being trapped. Now it is easy to intervene in equi- libration, and, in fact, regarding the plot of P (V ), an asymmetric non-equilibrium steady state is found to establish. Consequently, hV i approaches a constant value and hXi increases linearly after some time, which is in accordance with MD simulations performed with a Lennard-Jones fluid (cf. References√ [42, 20]). Since the mean√ displacement grows faster than the standard deviation, which goes like t, the efficiency rises with t as well. The plots of P (X) in Figure 6.9 very nicely show how the entire distribution is shifted to the right, while spreading. At the stopping sites the probability to find the motor is always higher and peaks are formed in P (X). As a whole the evolution has some resemblance to a Gaussian wave packet with dispersion. Interestingly, while there are huge differences in the behaviour of the mean displacement between the individual setups, the standard deviation evolves very similar throughout all cases in fashion of a typical diffusion process.
6.3.3 Possible extensions of the model The results presented in the previous section were obtained at low computational costs and the used model is much more efficient than MD simulations, as it is able to treat the whole probability distri- bution rather than sample individual trajectories only. It is therefore very well suited to investigate principle mechanisms. Whether it is possible to use it for quantitative predictions, too, remains to be tested, but it is conceivable that a set of effective states and rates may be found which are a good representation of the continuous, more realistic system. This subject is addressed in Section 6.4. In this section, three extensions of the model for further studies are suggested, but more could be found, of course. More precisely, ways are outlined to have the stopping sites create some other velocity distribution than a delta peak, to include stopping sites that release the particle not until a certain delay time has passed, and to simulate stopping sites that open up only after a dead time.
69 6 Minimal Models
The first task can be accomplished fairly easily by replacing the δV 0 in Equation (6.82) by the desired velocity distribution at stopping sites, Pstop(V ). In detail one has
" 1 X X 0 0 P (X, V, t + ∆t) = (1 − δXXi ) P (V |V )P (X − ∆tV, V , t) + i V 0=−1 1 1 # X X 00 0 00 0 +δXXi Pstop(V ) P (V |V )P (X − ∆tV ,V , t) . (6.85) V 00=−1 V 0=−1
It should be noted that still all information about a particle’s direction of travel before it was trapped is lost using this method. A delayed release from the stopping site can be realised in various ways. For one thing, one could simply reduce the probability of transitions leading away from that site. If one wishes a specific trapping time, one could add a delay line, that is a series of extra states that a particle has to run through after having been trapped and before being released again. This concept is illustrated in the tree diagram in Figure 6.10. How dead times may be introduced is also best explained using a tree diagram (cf. Figure 6.11). Instead of a simple delay line one now needs several copies of the vicinity of a stopping site, where positions are labelled by Y (j). The number and size of these copies is determined by the duration of the dead time. Each path through the graph that reaches a stopping site is passed on to one of these copies. There, evolution is just the same as in the original system except that transitions back to the stopping site are prohibited. When the dead time is over, the nodes in the copy are linked to the nodes in the original graph in the very same way as nodes within the latter a connected. A point to be aware of is the case if the dead time is long enough that neighbouring stopping sites may be reached within the copies.
6.4 A step towards bridging the gap back to reality
In Section 6.1, concerning the moments of the phase space pdf, it was possible to identify the solution of the two-state model with the approximation N2 (cf. Section 5.1.1) for cases with symmetric particles (B = 0) and at zero potential. One of the most pressing issues in this context is to check whether this relation can be extended to asymmetric objects as well by going to the three-state model and allowing for B 6= 0 in N2. If, indeed, a link between the motor’s shape in the “full” model and the transition rates in the three-state system can be established, the latter could also be used for quantitative predictions. So far, only a qualitative resemblance between plots of the solutions for comparable initial states can be asserted (cf. Figures 5.2 and 6.4), but this subject might be a paying starting point for further research. However, the more interesting connection actually is that of the models to real systems, which, at the current stage, can only be investigated for the minimal model. To give a first impression of the augmented three-state model’s performance, the results for the arbitrarily chosen test scenario are compared to the data from the monomeric version of the kinesin KIF1A (cf. Section 3.2). The authors of Reference [31] report that the spatial distribution of molecular motors fits a Gaussian curve, which drifts and spreads with time. In detail, the mean squared displacement can be described by
2 2 2 hX iexp = 2Dexpt + hV iexpt (6.86)
nm2 nm with Dexp = 44000 ± 1200 s and hV iexp = 140 ± 10 s . Similar behaviour is observed for the minimal model. Besides the fine structure due to the stopping site, P (X, t) also resembles a drifting and spreading Gauss distribution and in the stationary regime one finds hV˜ i = 0.036 and D˜ = 0.37, where the tilde is used to denote dimensionless quantities.
70 6.4 A step towards bridging the gap back to reality
Figure 6.11 Tree diagram illustrating the implementation of stopping sites with dead time. The upper part of the figure is a detail of the main graph of the process (cf. Figure 6.7 (a)) in the vicinity of a stopping site Xi. The lower part is a copy of this section, where the sites are labelled by Y (j) (j identifies the copy), and where transitions to the stopping site are forbidden. Here, after two time steps the binding site opens up again, wherefore the paths in the copy are fed back into the main graph. Note that only the links between nodes being crucial to explain the concept are depicted.
71 6 Minimal Models
If, for instance, one tries to match the distance between binding sites in experiment (δsexp = 8 nm) and model (δs˜ = 10), and the average velocities, one can deduce typical time and length scales
˜ δsexp hV i −4 tt := = 2.1 · 10 s, (6.87) hV iexp δs˜ δsexp Xt := = 0.8 nm. (6.88) δs˜ Having these at hand, the diffusion constant of the model yields in real units
X2 nm2 D := D˜ t = 112 , (6.89) tt s which differs by more than two orders from the experimental value. Even larger discrepancies are found when estimating the motor mass in the model. From the “vanishing stop” setup it is known that ˜ 2 1 2 kBT in equilibrium hV i = 2 . Assuming a temperature of 300 K in the experiment and using hV i = M , one has k T t2 M := B t = 1.7 · 10−8 kg. (6.90) 2 2 hV˜ i Xt This corresponds to an extremely heavy motor, as realistic values of motor proteins are in the range of tens to hundreds of ku or approximately 10−22 kg (cf. Reference [21]). However, these values were obtained for an arbitrary test case and there are several free parameters, which may probably be used to fix the inconsistencies. For instance, one finds from the stationary distribution C0 (cf. Equation (6.44)) of the three-state system that in equilibrium 2 hV˜ 2i = ≤ 1. (6.91) 2 + a W21 W12 As this value is bounded, but one, however, needs larger mean squared velocities to obtain smaller motor masses at given thermal energy, the only way out is to increase the real velocities attributed to the dimensionless velocities of ±1. If one wishes to keep the value of hV i unaffected by these rearrangements, a less asymmetric stationary velocity distribution is required, which presumably can be achieved by decreasing the asymmetry of the motor or rates, respectively. These smaller asymmetry ratios, too, are probably more realistic. Moreover, a different aspect is that the used model is generically unrealistic, as stopping is instan- taneous. In that sense one actually cuts out the times the motor is bound at a binding sites. Taking this into account would lead to smaller dimensionless velocities and, consequently, to a smaller typical time scale tt. This effect would also push D and M in the direction of more realistic values. It would be very important to check if a set of parameters can be found such that all the above quantities comply with the experimental observations.
72 7 Stochastic thermodynamics
Brownian or molecular motors in general differ a great deal from regular engines. They are small systems of rather few degrees of freedom and nowhere near a thermodynamic limit. Fluctuations dominate and operation takes place under highly non-equilibrium conditions. It is therefore clear that classical thermodynamics, which may well describe the functionality of any combustion engine, cannot be applied to the present model straightforwardly. In the last two decades, increasing endeavours (cf. References [39, 37, 15, 17, 25, 24, 7, 8, 16]) have been made to reformulate thermodynamics on a more detailed level to hold for microscopic systems as well and to be able of describing non-equilibrium processes more accurately than by just comparing them to the corresponding (quasi-) equilibrium processes by means of inequalities like W ≥ ∆F or ∆S ≥ 0.1 Key aspects of this novel framework of stochastic thermodynamics have been compiled by Udo Seifert in Reference [38], which is the main reference for the following sections. The word “stochas- tic” is motivated by the extension of the definitions of thermodynamic quantities such as work, heat or even entropy to the level of single phase space trajectories. As fluctuations are no longer negligible, the values of these quantities will differ for each realisation of a process, that is the variation of exter- nal control parameters according to some fixed protocol. Consequently, these quantities constitute stochastic variables with corresponding probability distributions. The laws of classical thermody- namics are recovered in form of averages over all possible individual trajectories, which is what one expects, since the seemingly deterministic statements of the classical theory are based on the fact that – in its scope – perceptible deviations from ensemble averages are extremely improbable. A very nice feature is that in case of non-equilibrium conditions, the inequalities stated above in fact derive from exact equalities, such as the Jarzynski relation. Moreover, the probability distributions are found to obey so-called fluctuation theorems, which hold for arbitrary protocols. The following analysis is concerned with the “full” model (cf. Section 4.2) again, that is with a two-dimensional convex body in an ideal gas. Additionally, this motor particle feels a potential U(X, k(t)), where the time dependence is now written in form of a control parameter k(t), which follows a given protocol. In Chapter 5, for instance, the control parameter would be the spring 1 2 constant of the harmonic potential U(X, t) = 2 k(t)X . In the present paper, the considerations are restricted to one control parameter, but can be generalised intuitively. In the other chapters, the expressions “system” and “model” are used somewhat synonymously. Here, a more precise terminology is appropriate. In this chapter, only the motor particle itself is referred to as system in the sense of a thermodynamic system which is in contact with an external heat bath. The role of this reservoir is taken over by the ideal gas, also referred to as medium.
7.1 Energy, work and heat
Following the lines of Reference [38] and transferring the results therein obtained for overdamped Langevin dynamics, it is fairly straightforward to identify expressions for the change of internal energy, performed work and exchanged heat for individual trajectories. Work is applied to the motor by changing the potential or the control parameter, respectively, whereas the transfer of kinetic energy through collisions with gas particles may be interpreted as heat flow. The (internal) energy of the system is 1 E(X, V, t) = MV 2 + U(X, k(t)). (7.1) 2 1As usual, W , F and S denote work, free energy and entropy, respectively.
73 7 Stochastic thermodynamics
Thus, ∂U ∂U dE = MV dV + dX + k˙ dt. (7.2) ∂X ∂k The last summand is the increment of work applied to the motor from outside the system:
∂U dw := k˙ dt. (7.3) ∂k
If one now identifies ∂U dq := − MV dV + dX (7.4) ∂X with the heat dissipated into the medium, one finds a first law like energy balance:
dE = dw − dq. (7.5)
The quantity dq is the change of kinetic energy less the part that stems from the force −∂X U, that is it is the change of kinetic energy due to collisions with gas particles. Hence, one may write it as well as X dq = δ(t − ti)∆qidt, (7.6) i
1 2 2 with ∆qi := − 2 M (Vi + ∆Vi) − Vi being the energy transferred to the medium at collision i. An important observation is that ∆q may also be expressed in terms of transition rates:
W (V + ∆V |V ) W (−(V + ∆V )| − V ) ∆q = k T ln = k T ln , (7.7) B W (−V | − (V + ∆V )) B W (V |V + ∆V ) as shown in Appendix A.2, which is also closely related to detailed balance (cf. Section 4.4.2). ∗ Integrated along a phase space trajectory Γ(t) of given length t0 ≤ t ≤ t , one furthermore obtains the expressions
t∗ Z ∂U(X(t), k) w[Γ(t)] = dt k˙ (t), (7.8) ∂k k=k(t) t0 t∗ Z ! ∂U(X, k(t)) q[Γ(t)] = − dt MV (t)V˙ (t) + V (t) ∂X X=X(t) t0 X = ∆qi (7.9) i and the first law
t∗ Z ∂U ∂U ∆E[Γ(t)] := w[Γ(t)] − q[Γ(t)] = dt k˙ + MV V˙ + V ∂k ∂X t0 1 1 = U(X(t∗), k(t∗)) − U(X(t ), k(t )) + MV (t∗)2 − MV (t )2 (7.10) 0 0 2 2 0 on the level of a single trajectory.
74 7.2 Entropy
7.2 Entropy
Dissipated heat is typically associated with changes in entropy, wherefore one is led to extent the concept of fluctuating thermodynamic quantities to entropy as well. Transferring the notions from classical thermodynamics, the increase of entropy of the medium is defined as q[Γ(t)] ∆s [Γ(t)] := . (7.11) m T As a trajectory dependent entropy of the system, the quantity
s(t) := −kB ln P (Γ(t), t) (7.12) was proposed (cf. Reference [37]), where the probability density function P (Γ, t) obtained by solving the time evolution equation (4.44) is evaluated along the trajectory Γ(t). The definition of s(t) is such that upon averaging the usual ensemble entropy Z dΓ S(t) := −kB P (Γ, t) ln P (Γ, t) = hs(t)i (7.13) 2π~/M is recovered. Moreover, in equilibrium with constant k(t), the well-known thermodynamic relation F = E(X(t),V (t)) − T s(t), (7.14)
R dΓ −1 with F := −kBT ln exp E(X,V ) , holds along an individual trajectory at any time. 2π~/M kBT Concerning entropy production rates, one has from Equations (7.11), (7.6) and (7.7) q˙(t) X W (Vi + ∆Vi|Vi) s˙ (t) = = k δ(t − t ) ln . (7.15) m T B i W (−V | − (V + ∆V )) i i i i It is noteworthy that this result slightly differs form the corresponding expression in References [38, 37], therein defined for general stochastic systems with an arbitrary set of states, as to the denominator of the quotient of transition rates is the rate of the time-reversed step −(Vi + ∆Vi) → −Vi rather than that of merely the backward step Vi + ∆Vi → Vi. In case of a symmetric motor both rates are identical, as the rate of the time-reversed step can as well be written as the rate of the backward step of a motor rotated by 180◦. Hence, it seems probable that the author of References [38, 37] simply overlooked this detail, as his analysis of mechanically driven systems is based on Langevin dynamics which do not account for the shape of the particle. Upon calculation of the time derivative of the system entropy s(t), the terms associated with deterministic motion are found to cancel and one is left with the stochastic contributions " ! ! ˙ ∂ ln P ((X,V (t)), t) ˙ ∂ ln P ((X(t),V ), t) s˙(t) = − kB X(t) + V (t) + ∂X ∂V X=X(t) sto V =V (t) sto ∂ ln P ((X(t),V (t)), t) + , (7.16) ∂t sto where ∂ ln P ((X,V ), t) X˙ (t) := 0, (7.17) ∂X sto ∂ ln P ((X,V ), t) X P ((X,Vi + ∆Vi), ti) V˙ (t) := δ(t − t ) ln , (7.18) ∂V i P ((X,V ), t ) sto i i i ∞ R dV 0 [W (V |V 0)P ((X,V 0), t) − W (V 0|V )P ((X,V ), t)] ∂ ln P ((X,V ), t) := −∞ . (7.19) ∂t sto P ((X,V ), t)
75 7 Stochastic thermodynamics
~ X(t) X(t) V(t)
t t 0 t* ~ V(t) t 0 t* t
Figure 7.1 Exemplary pairs of trajectory and anti-trajectory in position (X(t), X˜(t)) and velocity space (V (t), V˜ (t)), as defined in Equations (7.21) to (7.24).
Defining the total entropy production rates ˙tot(t) :=s ˙(t) +s ˙m(t), one finds
∞ −k Z s˙ (t) = B dV 0 [W (V (t),V 0)P ((X(t),V 0), t) − W (V 0|V (t))P ((X(t),V (t)), t)]− tot P ((X(t),V (t)), t) −∞ X P ((X(t),Vi + ∆Vi), ti)W (−Vi| − (Vi + ∆Vi)) − k δ(t − t ) ln . (7.20) B i P ((X(t),V ), t )W (V + ∆V |V ) i i i i i i Since the equilibrium distribution is a stationary solution of the master equation, the integral in the above equation vanishes in this particular case as well as the logarithms whose arguments become unity if detailed balance holds (cf. Section 4.4.2). Consequently, in equilibrium, the total entropy production is zero at any moment even on the level of a single trajectory. While under non-equilibrium conditions the entropy production along a single trajectory may as well be negative, it can be shown (cf. Equation (7.43)) that the average entropy change is always positive. Moreover, trajectories with negative entropy production are exponentially suppressed (cf. Equation (7.44)) and very unlikely to be observed in macroscopic systems.
7.3 Fluctuation theorems
In Reference [40], the Jarzynski-relation is proved for general stochastic processes and the way for obtaining further fluctuation theorems is outlined. The present model allows the verification of these relations by direct calculation and constitutes an illustrating example. Moreover, this fact is another hint for the correctness of the proposed probability density of specific phase space trajectories (cf. Equation (4.33)). The basis for the derivation of numerous fluctuation theorems is to compare the probability of the system to take a specific trajectory Γ(t) under a protocol k(t) to the probability of observing the so- ˜ ∗ called anti-trajectory Γ˜(t) under the time-reversed protocol k(t) := k(t +t0 −t). The anti-trajectory is defined such that the position variable X˜(t) runs through the path X(t) in opposite direction, that ∗ ∗ is it starts at X˜(t0) = X(t ) and ends at X˜(t ) = X(t0). Therefore, the sign of the velocity trajectory has to be inverted additionally. In summary, one has (cf. Figure 7.1)
∗ X˜(t) := X(t + t0 − t), (7.21) ∗ V˜ (t) := −V (t + t0 − t), (7.22)
∆V˜i := ∆VN+1−i, (7.23)
V˜i := −(VN+1−i + ∆VN+1−i), (7.24) where the tilde marks the quantities belonging to the anti-trajectory of the quantities without. As stated in Section 4.3.2, a trajectory starting at t0 at phase space position Γ0 is characterised by the series of collision-induced velocity-changes ∆V (t). The conditional probability density is (cf.
76 7.3 Fluctuation theorems
Equation (4.33))
∗ t ∞ N Z Z Y P [∆V (t)|Γ0, t0, k(t)] = exp − dt d∆V λ(∆V,V (t)) λ(∆Vi,Vi), (7.25) i=1 t0 −∞ where k(t) has been introduced into the expression to emphasize the dependence on the protocol. The corresponding probability density of the anti-trajectory under the time-reversed protocol thus reads t∗ ∞ Z Z N ˜ ˜ ˜ ˜ Y ˜ ˜ P [∆V (t)|Γ0, t0, k(t)] = exp − dt d∆V λ(∆V, V (t)) λ(∆Vi, Vi). (7.26) i=1 t0 −∞ Substituting the quantities with a tilde by Equations (7.21) to (7.24) yields
t∗ ∞ Z Z N ˜ ˜ ˜ Y P [∆V (t)|Γ0, t0, k(t)] = exp − dt d∆V λ(∆V, −V (t)) λ(∆Vi, −(Vi + ∆Vi)). (7.27) i=1 t0 −∞ One can now evaluate the ratio of both probabilities, which is done piecewise for clarity. Concerning the products of rates, one readily finds from Equation (7.7) that N Y λ(∆Vi,Vi) q[Γ(t)] ∆sm[Γ(t)] = exp = exp . (7.28) λ(∆V , −(V + ∆V )) k T k i=1 i i i B B The quotient of the exponentials yields after a short calculation
t∗ ∞ t∗ 2π Z Z Z Z exp − dt d∆V [λ(∆V,V (t)) − λ(∆V, −V (t))] = exp −ρ dt V (t) dθ SF (θ) sin θ .
t0 −∞ t0 0 (7.29) If θ, S and F (θ) refer to a closed curve, as it is the case if it is the outline of a body, the θ integral vanishes as stated in Section 4.4.4. In summary, the important relation P [∆V (t)|Γ , t , k(t)] q[Γ(t)] ∆s [Γ(t)] 0 0 = exp = exp m (7.30) ˜ P [∆V˜ (t)|Γ˜ 0, t0, k(t)] kBT kB is obtained, which may already be considered as a first fluctuation theorem, and which serves by the definition of the auxiliary quantity P [∆V (t)|Γ , t , k(t)]P (Γ ) ∆s [Γ(t)] P (Γ ) R := ln 0 0 0 0 = m + ln 0 0 (7.31) ˜ ˜ P [∆V˜ (t)|Γ˜ 0, t0, k(t)]P1(Γ˜ 0) kB P1(Γ0) as a starting point for the derivation of further fluctuation theorems (cf. Reference [38]). Here, P0 and P1 are some yet arbitrary initial distributions for the starting points of trajectories and anti-trajectories, the choice of which determines the fluctuation theorem to be deduced. It is noteworthy that R is defined in a way that he−Ri = 1, (7.32) where the angle brackets denote the average over all possible trajectories. The proof is simple: Z Z −R −R he i = D[∆V (t)] dΓ0 P [∆V (t)|Γ0, t0, k(t)]P0(Γ0)e = Z Z (7.31) ˜ = D[∆V (t)] dΓ0 P [∆V˜ (t)|Γ˜ 0, t0, k(t)]P1(Γ˜ 0)= Z Z ˜ = D[∆V˜ (t)] dΓ˜ 0 P [∆V˜ (t)|Γ˜ 0, t0, k(t)]P1(Γ˜ 0)=1. (7.33)
77 7 Stochastic thermodynamics
R R The symbol D[∆V (t)] dΓ0 denotes the integral over all trajectories as defined in Section 4.3.2. R R Since each trajectory is also an anti-trajectory (namely to its anti-trajectory), D[∆V (t)] dΓ0 R R might as well be replaced by the integral over all anti-trajectories D[∆V˜ (t)] dΓ˜ 0, which justifies the equality next to the last. One might worry that the phase space volume is not conserved due to heat flow in and out of the system, wherefore one may possibly not simply replace dΓ0 by dΓ˜ 0. To investigate this more closely, consider a bundle of phase space trajectories emerging from the volume element dΓ0 for a given series of velocity-changes ∆V (t). In between the velocity-jumps {∆Vi}, the propagation is deterministic and phase space volume is clearly conserved. On the other hand, when collisions occur, all trajectories of the bundle are merely shifted by the same amount in V direction, and the volume is left unchanged yet again. Hence, the Jacobian of the map dΓ0 7→ dΓ˜ 0 equals unity. Finally, the very last of the above equalities is simply due to normalisation. In the remainder of this section some prominent examples of fluctuation theorems will be briefly reviewed.
Crooks fluctuation theorem and Jarzynski-relation: Here, P0 and P1 are set to the equilibrium ∗ distributions corresponding to the values of k(t0) and k(t ), respectively: F (k(t0)) − E(Γ, k(t0)) P0(Γ) = exp , (7.34) kBT F (k(t∗)) − E(Γ, k(t∗)) P1(Γ) = exp . (7.35) kBT
1 F is the free and E the total energy. The auxiliary quantity R is then given by R = k T (w[Γ(t)]−∆F ), ∗ B where the first law (7.10) and ∆F := F (k(t )) − F (k(t0)) have been used. The difference between the changes in free energy and work is also often referred to as dissipated work, as it is the amount of work “lost” in a non-equilibrium process in comparison to a quasi-static reversible process. If one takes into account that the work done along an anti-trajectory is minus the work along the corresponding trajectory, and if one defines Z Z P [W, k(t)] := D[∆V (t)] dΓ0 P [∆V (t)|Γ0, t0, k(t)]P0(Γ0)δ(W − w[Γ(t)]) (7.36) as the probability of measuring an amount of work W at one realisation of the protocol k(t), one finds by similar reasoning as above the Crooks fluctuation theorem (cf. Reference [7])
P [W, k(t)] W − ∆F = exp . (7.37) P [−W, k˜(t)] kBT
The latter has already been experimentally applied to determine free energy changes (an equilibrium quantity) out of non-equilibrium work measurements (cf. Reference [6]). Furthermore, from Equation (7.32) one easily obtains the Jarzynski-relation (cf. References [25, 24])
−w[Γ(t)] −∆F exp = exp (7.38) kBT kBT and by Jensen’s inequality (cf. Reference [34] p. 62, for instance)
hw[Γ(t)]i ≥ ∆F, (7.39) which is a refinement of the statement from classical thermodynamics that the maximum amount of work gained from a process connecting two states is given by their difference in free energy. Note that the sign of w[Γ(t)] is negative if work is extracted from the system, as is the sign of ∆F when descending from higher to lower free energy.
78 7.3 Fluctuation theorems
Integral fluctuation theorem and second law: If one chooses P1 to be the probability distribution obtained by letting the system evolve from the initial distribution P0, that is P1 is the solution of the time evolution Equation (4.44), one may write both in terms of the trajectory dependent entropy s(t) := −kB ln P (Γ, t) as −s(t0) P0(Γ) =exp , (7.40) kB −s(t∗) P1(Γ) =exp . (7.41) kB
1 ∆stot Consequently, R = (∆sm + ∆s) = , from which follows immediately the integral fluctuation kB kB theorem (cf. References [37, 38]) ∆s exp tot = 1. (7.42) kB Applying Jensen’s inequality, one has h∆stoti ≥ 0, (7.43) which constitutes a refined version of the second law of thermodynamics. Moreover, analogous to the Crooks fluctuation theorem one may obtain the relation
P [Γ(t), k(t)] P [∆s , k(t)] ∆s = tot = exp tot , (7.44) ˜ ˜ P [Γ˜(t), k(t)] P [−∆stot, k(t)] kB which quantifies nicely how irreversibility may arise from microscopically time-reversible equations of motion. The greater the entropy production along a specific trajectory, the unlikelier is its anti- trajectory in the reversed process. Turning to a well known picture, k(t) could, for instance, be the position of a separating plate in a box, one part of which is initially filled with a gas. If one removes the plate, but reinserts it after a short while, it is very unlikely to find the gas molecules all gathered solely in the original part of the chamber again.
79 7 Stochastic thermodynamics
80 8 Conclusion and Outlook
The physical perspective Asymmetric particles have the ability to perform biased Brownian motion under non-equilibrium conditions. Concerning the velocity distribution, such systems usually relax from even symmetric non-equilibrium states to the again symmetric equilibrium through a sequence of asymmetric intermediate states, causing a net motion (cf. Chapters 5 and 6). However, there also seem to be differences to symmetric particles even in thermal equilibrium, as indicated by the velocity auto-correlation function of the minimal model in Section 6.2. For this system, the transition to a symmetric particle is characterised by the collapse of complex relaxation processes with two time constants to simple exponential decays with only one time scale. Moreover, the distinction between odd and even detailed balance (i.e., detailed balance which accounts for the behaviour of velocities under time reversal or not) vanishes. In general, no principle differences between systems featuring odd or even detailed balance were observed, as far as the ability for asymmetric relaxation into equilibrium is concerned. Regarding the characterisation of non-equilibrium, the dominant quantity was found to be the deviation of the mean squared velocity, hV 2i, from its equilibrium value (see Section 5.1.1). A biologically inspired way to influence the velocity distribution is to stop or slow down the particle at binding sites, which can be modelled as time-dependent potential wells as demonstrated in Chapter 5. Upon releasing the particle from the well by gradually decreasing the potential strength, the velocity distribution is narrowed similar to the cooling of an expanding gas. However, this effect is bounded, as hV 2i has to be always positive, and, in fact, spreading the velocity distribution instead of narrowing would allow for greater deviations from equilibrium. A point to be aware of is the correct ordering of the involved time scales. The typical (initial) cycle time of oscillations in the well should preferably be smaller than the time it takes to turn off the potential. Else, the particle will perform hardly any work during this process and lose only little average kinetic energy (cf. Section 5.2). Additionally, the decay of the potential has to be faster than the velocity relaxation in the medium. If not, any deviation of the velocity distribution from its equilibrium form will almost instantaneously be compensated for, and one ends up with a quasi static equilibrium process. This property could be clearly observed in the examples of Section 5.3.2. In general, the efficiency η := |hXi/phX2i − hXi2| (cf. Section 5.1.2) of a singular release turned out to be rather low, meaning that the motion is still rather undirected and the width of the distribu- tions of positions is much larger than its overall shift. After having reached a maximum, which for the tested case was of the order of 10−2, η decays for lager times like t−1/2, since the net motion ceases and unbiased diffusion takes over. The reason is, of course, that the system is allowed to equilibrate, wherefore one of the prerequisites of directed Brownian motion is lost. In the limit of small ratios of p m motor to particle mass, δ := M , one finds that the height of the efficiency maximum increases with δ and the deviation of hV 2i from its equilibrium value, and is moreover dependent on a geometric factor, which – roughly speaking – contains how asymmetric and how streamlined the motor is. Using a minimal model in Section 6.3, it could be demonstrated how an effective molecular motor can be constructed by applying periodic stopping sites which open up again after having released the particle. This result is in accordance with molecular dynamics simulations reported in References [42, 20]. What does the trick is that the ability to bind a once released particle again enables the stopping sites to prevent the system from equilibration, especially if they are distributed over the whole space such that the particle can never diffuse freely over large distances. In fact, concerning the distribution of velocities, a stationary non-equilibrium state was found to establish. In contrast to the decaying efficiency after a singular release, now in the steady state, an increase of η with t1/2 could be observed, corresponding to the linear behaviour of both the mean position and its
81 8 Conclusion and Outlook variance. Thus, the distribution of positions is drifting faster than it spreads. This behaviour is in accordance with experimental data from the monomeric kinesin KIF1A. A quantitative comparison of an arbitrarily constructed test scenario to the experiment shows profound discrepancies, which, however, can possibly be overcome by finding a correct set of parameters. From a thermodynamic viewpoint, an appropriate description of the processes is possible in the framework of stochastic thermodynamics introduced in Chapter 7. This concept allows to discuss energy transfer and even entropy production on the level of single trajectories. It is therefore able to cope with two central properties of the model, large thermal fluctuations and non-equilibrium, which are hardly accessible within classical thermodynamics. Due to the fluctuations, the quantities mentioned above no longer behave deterministically, but as stochastic variables. For their probability distributions one can find so-called fluctuation theorems connecting forward an backward processes. With respect to experimental realisations, stochastic thermodynamics provides powerful tools to understand and analyse data.
The technical perspective One of the major achievements, concerning technical aspects, was the inclusion of positions, X, and time-dependent potentials, U(X, t), into the formalism. This was done in Chapter 4 by modelling the movement of the motor as a Poisson type stochastic process. Although the normalisation of the proposed conditional probability density for an individual trajectory (cf. Equation (4.33)) could not be proved in general, there are numerous evidence for its correctness. These include the verification for a simpler, but equally designed process, the successful direct derivation of the fluctuation theorems, which are already known to hold from very general considerations, and the corresponding time evolution equation of the phase space pdf, which has a very intuitive and reasonable form (being a combination of Liouville and master equation) and from which the expression used in References [42, 41, 10] follow as a special case. Based on the time evolution equation of the phase space pdf, a method of obtaining approximate equations for its moments was proposed and applied in Chapter 5, containing an expansion in the p m ratio of masses of motor and gas particles, δ = M . The zeroth order (N0) corresponds to a freely moving particle, which only feels U(X, t). In order δ1 (N1), linear damping or friction terms arise, 0,2 which are proportional to (W1 (K))XX , the XX element of the first Minkowski tensor of rank two of the motor. This term is insensitive to asymmetry. The first order where asymmetry comes in to play 2 0,3 is δ (N2), where terms proportional to (W1 (K)XXX appear. Concerning the first five moments of the phase space pdf, the inclusion of higher orders did bring no qualitative and only very small quantitative changes. The area of validity of this statement, however, does not extend to cases where parametric resonance occurs. In case of well separated time scales, one may treat the release process and subsequent relaxation independently of each other. First, one considers the motor in the changing potential, neglecting the gas particles. Then, when the potential has been switched off, one turns to the collision driven evolution. This method was applied to obtain analytical approximations in Section 5.3.3. Moreover, if one is only interested in the larger time scale (i.e., the one of Brownian motion), one could treat the releases as instantaneous changes of the velocity distribution, retrieving to some extend the picture of stopping sites used in References [42, 41]. The second important issue are the reduced models developed in Chapter 6. Especially the system with three velocity states constitutes a generic minimal model of an asymmetric Brownian particle in one dimension. At least three states are needed to allow for a symmetric equilibrium distribution while introducing an asymmetry via the choice of transition rates. To resemble the more realis- tic model, detailed balance in the stationary state is demanded, which adds further conditions. In contrast to the complex system, this model is fully analytical solvable, while exhibiting the same characteristic asymmetric relaxation when brought out of equilibrium. Consequently, it is ideally suited for addressing fundamental questions concerning asymmetry or detailed balance. Moreover, it can easily be augmented by a discrete position space to investigate the effect of stopping sites. Although this extended model has only been solved numerically so far, it still has advantages com-
82 pared to other techniques such as MD simulations, as it provides complete information (concerning the phase space pdf) at relatively low computational cost.
Ideas for further studies The minimal models are promising candidates for investigating further properties of asymmetric Brownian particles or motors, respectively. Section 6.1 showed that a two- state model precisely corresponds to approximation N1 (or N2 without asymmetry) of the more realistic system in case of zero potential. The single non-trivial eigenvalue of the transition matrix of the two-state system can be identified with the linear damping term in the complex model. The question arises whether a similar correspondence can be established between a three-state model and the approximation N2. In both cases a new quantity comes into play – a second non-trivial eigenvalue on the one side, and a geometry factor reflecting asymmetry on the other side. Regarding the equations for the moments of the phase space pdf, both additional components are found to vanish again for symmetric motors. Considering the augmented three-state model, it might be possible to find analytical descriptions of the numerical results. For instance, the probability distribution of positions resembled in its overall shape a travelling Gaussian wave packet with dispersion. Moreover, the influence of numerous variations and extensions of this model can be easily tested. It would be interesting to study the dependence of the motor’s behaviour on the distance between stopping sites (bearing in mind that the efficiency of a singular release has a maximum after a specific time), or to investigate the effect of stopping sites with delay or dead times. A way for their implementation was suggested in Section 6.3.3. Another thing one could do with these minimal models is to apply stochastic thermodynamics to them. This promises to be especially interesting in the context of entropy and entropy production, since there the complete knowledge of the phase space pdf is necessary, which was not accessible in the “full” model. Due to the exponentials in the fluctuation theorems, large deviation properties of thermodynamic variables such as work or entropy are put into focus. For instance, take the Jarzynski relation (7.38), where the exponential average is dominated by small values of w[Γ(t)]. This fact imposes great difficulties to its experimental applicability, as large deviations are rarely sampled. To address this problem, a method to analytically determine the asymptotics of the work distribution of driven Langevin systems is devised in Reference [14]. The procedure builds on the method of optimal fluc- tuation, which in their case corresponds to a saddle-point approximation in a functional integral over stochastic trajectories. As the basis for a path integral representation of the asymmetric Brownian particle is already laid, one could try to transfer this technique to the presented type of stochastic process. A step towards finding an approximation for the whole phase space pdf of the “full” model, rather than only for its moments, could be the following idea. The n’th Kramers-Moyal coefficient was found to be of order δn−1 (cf. Equation 4.60). If one expands to the lowest order where asymmetry comes into play, δ2 that is, the Kramers-Moyal expansion (4.57) terminates after the third summand, and one arrives at a partial differential equation of third order in V . By treating the δ2 terms as small perturbations, one could now try to transform this equation to an effective second order standard Fokker-Planck equation. This approach would essentially consist in trying to find the unperturbed solution and plugging it into the perturbation terms to estimate their contributions. A good quantity, which may be used to characterise and compare the present system either to experiments or to other models, could be the randomness r := lim (hX2i(t) − hXi(t)2)/hXi(t) (cf. t←∞ References [43, 35]). Obviously, r is closely linked to the efficiency η, but has the advantage that it adopts a single characteristic value for motors as the augmented three-state model with periodic persistent stopping sites. Last but not least a point of again somewhat broader perspective is the question what really characterises the asymmetry of the system. Is it only the shape of the motor, or is the anisotropy of its degree of freedom not a second equally important component? To put things differently, one could
83 8 Conclusion and Outlook also ask how the Brownian motion of a free asymmetric particle would look like, where free means allowing translations in all spatial direction as well as rotations. Presumably, the freedom to rotate will have an inhibitory effect on the rectification properties of an initially oriented non-equilibrium ensemble of particles.
84 A Derivations and Proofs
A.1 Coefficients of the differential Chapman-Kolmogorov equation
For the calculation of the quantities W (Γ|Γ0, t0), Ai and Bij, defined in Section 4.1.2, it is apt to rewrite the conditional probability from Section 4.3.2 in the form
∞ X P (Γ, t0 + ∆t|Γ0, t0) = PN (Γ, t0 + ∆t|Γ0, t0), (A.1) N=0 with
t +∆t ∞ 0Z Z P0(Γ, t0 + ∆t|Γ0, t0) := exp − dt d∆V λ(∆V,V (t)) δ(Γ − Γ(t0 + ∆t)), (A.2)
t0 −∞ t +∆t ∞ 0Z Z P1(Γ, t0 + ∆t|Γ0, t0) := dt1 d∆V1 λ(∆V1,V1)×
t0 −∞ t +∆t ∞ 0Z Z × exp − dt d∆V λ(∆V,V (t)) δ(Γ − Γ(t0 + ∆t)), (A.3)
t0 −∞ t +∆t t +∆t ∞ ∞ 0Z 0Z Z Z P2(Γ, t0 + ∆t|Γ0, t0) := dt1 dt2 d∆V1 d∆V2 λ(∆V1,V1)λ(∆V2,V2)×
t0 t1 −∞ −∞ t +∆t ∞ 0Z Z × exp − dt d∆V λ(∆V,V (t)) δ(Γ − Γ(t0 + ∆t)), (A.4)
t0 −∞ . . that is, to divide it into parts for trajectories with one, two, three, etc. collisions. Moreover, the exponentials will be abbreviated by exp(...) as their limit as ∆t → 0 is just 1. Quantities like Γ(t) will be written as Γ(t, t1, t2,...) to clarify their dependence not only on the time t they are measured at, but also on the times where velocity-changes happened.
A.1.1 W (Γ|Γ0, t0) The transitions rates W were defined by
lim P (Γ, t0 + ∆t|Γ0, t0)/∆t = W (Γ|Γ0, t0), (A.5) ∆t→0 where |Γ − Γ0| ≥ . Since P (Γ, t0 + ∆t|Γ0, t0) implies the δ-distribution, it is appropriate to treat the desired quantity as a functional and regard the limit in the sense of weak convergence. If one defines
85 A Derivations and Proofs the functional
1 Z F : φ 7→ F [φ] := dΓ P (Γ, t + ∆t|Γ , t )φ(Γ) ∆t ∆t ∆t 0 0 0 |Γ−Γ0|≥ ∞ X 1 Z = dΓ P (Γ, t + ∆t|Γ , t )φ(Γ), (A.6) ∆t N 0 0 0 N=0 |Γ−Γ0|≥ | {z } N =:F∆t[φ] the sequence F∆t is said to converge weakly against F as ∆t → 0 if F∆t[φ] → F [φ] for any test function φ. The limits are now considered independently for each number of collisions N.
N = 0:
1 Z F 0 [φ] = dΓ exp(...)δ(Γ − Γ(t + ∆t))φ(Γ) ∆t ∆t 0 |Γ−Γ0|≥ 1 = exp(...)Θ(|Γ(t + ∆t) − Γ | − )φ(Γ(t + ∆t)) (A.7) ∆t 0 0 0
This clearly goes to zero, as Γ(t0 + ∆t) goes to Γ0 continuously if no collisions occur, wherefore for ∗ ∗ 0 any > 0 there exists a ∆t such that |Γ(t0 + ∆t) − Γ0| < for all ∆t < ∆t . Hence, F∆t → 0.
N = 1:
t0+∆t ∞ Z Z 1 F 1 [φ] = dt d∆V λ(∆V ,V (t )) exp(...)Θ(|Γ(t + ∆t, t ) − Γ | − )φ(Γ(t + ∆t, t )) ∆t 1 1 1 1 1 ∆t 0 1 0 0 1 t0 −∞ (A.8) If φ is assumed to be a continuous function, the integrand of the first integral is continuous with respect to t1. As there are no collisions before t1, V1(t1) := lim V (t) is clearly a continous function t%t1 of t1. The same is true for Γ(t0 + ∆t, t1), but not for the Heaviside Θ. However, the effect of the latter essentially consists in clipping the range of the integral over ∆V1, which again takes place in a continuous way as t1 is varied. As a consequence, the first mean value theorem for integrals applies, ∗ which states that there is a time t0 ≤ t1 ≤ t0 + ∆t such that
∞ t0+∆t Z 1 Z F 1 [φ] = d∆V λ(∆V ,V (t∗)) exp(...)Θ(|Γ(t + ∆t, t∗) − Γ | − )φ(Γ(t + ∆t, t∗)) dt ∆t 1 1 1 1 ∆t 0 1 0 0 1 1 −∞ t0 ∞ Z ∗ ∗ ∗ = d∆V1 λ(∆V1,V1(t1) exp(...)Θ(|Γ(t0 + ∆t, t1) − Γ0| − )φ(Γ(t0 + ∆t, t1)) (A.9) −∞
∗ In the limit ∆t & 0: t1 → t0, V1 → V0, X(t0 + ∆t) → X0, V (t0 + ∆t) → V0 + ∆V1 and exp(...) → 1,
86 A.1 Coefficients of the differential Chapman-Kolmogorov equation wherefore
∞ Z 1 F∆t[φ] → d∆V1 λ(∆V1,V0)Θ(|∆V1| − )φ((X0,V0 + ∆V1)) −∞ ∞ ∞ Z Z = dX dV λ(V − V0,V0)Θ(|V − V0| − )φ((X,V ))δ(X − X0) −∞ −∞ Z = dΓ λ(V − V0,V0)δ(X − X0)φ(Γ), (A.10)
|Γ−Γ0|≥
1 where the substitution V := V0 + ∆V1 was used. Thus, F∆t → λ(V − V0,V0)δ(X − X0).
N > 1: If one proceeds similar to the case of N = 1, one finds that the additional integrals over N t1, t2, etc. vanish in the limit of ∆t → 0, which is why F∆t → 0 for N > 1.
Summary: The recombination of the results yields W (Γ|Γ0, t0) = λ(V − V0,V0)δ(X − X0).
A.1.2 Ai(Γ0, t0) and Bij(Γ0, t0)
Let A := (A1,A2). According to Section 4.1.2 it is defined as the part of
1 Z lim dΓ (Γ − Γ0)P (Γ, t0 + ∆t|Γ0, t0) = ∆t→0 ∆t |Γ−Γ0|< ∞ X 1 Z = lim dΓ (Γ − Γ0)PN (Γ, t0 + ∆t|Γ0, t0) (A.11) ∆t→0 ∆t N=0 |Γ−Γ0|< that is independent of . Again it is convenient to treat the summands one after the other.
N = 0:
1 Z lim dΓ (Γ − Γ0)P0(Γ, t0 + ∆t|Γ0, t0) = ∆t→0 ∆t |Γ−Γ0|< 1 Z = lim dΓ (Γ − Γ0) exp(...)δ(Γ − Γ(t0 + ∆t)) = ∆t→0 ∆t |Γ−Γ0|<
Γ(t0 + ∆t) − Γ0 = lim exp(...) = Γ˙ (t0) (A.12) ∆t→0 ∆t
The fact that the integral is limited to a region around Γ0 is no hindrance, since again Γ(t0 + ∆t) comes arbitrarily close to it in the limit process. However, as is taken to zero, jumps are excluded ˙ 1 ∂U(X,t) and it is more precise to replace Γ by the expression V, − M ∂X .
87 A Derivations and Proofs
N = 1: 1 Z lim dΓ (Γ − Γ0)P1(Γ, t0 + ∆t|Γ0, t0) = ∆t→0 ∆t |Γ−Γ0|<
t0+∆t ∞ 1 Z Z Z = lim dΓ dt1 d∆V1 (Γ − Γ0) exp(...)λ(∆V1,V1(t1))δ(Γ − Γ(t0 + ∆t, t1)) ∆t→0 ∆t |Γ−Γ0|< t0 −∞ (A.13)
This resembles Equation A.8 and an analogue proceeding, that is integration over Γ with the intro- duction of a Heaviside Θ and application of the mean value theorem, leads to 1 Z lim dΓ (Γ − Γ0)P1(Γ, t0 + ∆t|Γ0, t0) = ∆t→0 ∆t |Γ−Γ0|< Z 0 = d∆V1 λ(∆V1,V0), (A.14) ∆V1 − which clearly depends on .
N > 1: 1 Z lim dΓ (Γ − Γ0)PN (Γ, t0 + ∆t|Γ0, t0) = 0, (A.15) ∆t→0 ∆t |Γ−Γ0|< for N > 1 again due to the additional time integrals, which cannot be compensated for by 1/∆t anymore.
Summary: As one might expect, only the term N = 0 (no collisions) contributes to A, which was 1 ∂U(X,t) already attributed to continuous motion. One finds A(Γ, t) = V, − M ∂X . In an analogous way it is possible to show that Bij(Γ0, t0) = 0, which reflects the fact that the continuous parts of the trajectory are fully deterministic and posses no diffusive properties of their own.
A.2 Relation of dissipated heat and transition rates
The aim of this section is to establish the relation W (V + ∆V |V ) W (−(V + ∆V )| − V ) ∆q = k T ln = k T ln , (A.16) B W (−V | − (V + ∆V )) B W (V |V + ∆V ) stated in Sections 4.4.2 and 7.1, between the heat dissipated in the step V → V + ∆V , 1 ∆q := − M (V + ∆V )2 − V 2 , (A.17) 2 and associated transition rates. By using Equations (4.39) and (4.30) on pages 25 and 22, respectively, one finds
W (−V | − (V + ∆V )) =λ(∆V, −(V + ∆V )) = 2π Z ∆V ∆V = dθ a Θ − exp −b(∆V c − (V + ∆V ))2 , (A.18) α|α| α 0
88 A.3 Expansion of the Kramers-Moyal coefficients
m q m m 2 1 2 M sin θ where a := −ρSF (θ) , b := sin θ, c := and α := m 2 . The exponent is 2πkBT 2kBT α sin θ 1+ M sin θ rearranged to yield
−b(∆V c − (V + ∆V ))2 = − b(∆V c + V )2 − b(1 − 2c) (V + ∆V )2 − V 2 ∆q = − b(∆V c + V )2 − . (A.19) kBT Plugged in again one immediately has
∆q W (−V | − (V + ∆V )) = exp − W (V + ∆V |V ). (A.20) kBT
The proof of the second equality in Equation (A.16) is analogous.
A.3 Expansion of the Kramers-Moyal coefficients
In this section the crucial steps of the expansion of the Kramers-Moyal coefficients αn(V ) in powers of δ := pm/M are outlined. According to Equation (4.59), one page 27 one has
2π 0 !2 Z sin θ n Z r M δ α (V ) ∝ dθ F (θ) dRRn+1exp − R + √ V sin θ . (A.21) n 2 2 1 + δ sin θ kBT 2 0 | {z }−∞ (i) | {z } (ii) | {z } (iii)
Part (i) is formally equivalent to (1/(1 + x))n and one finds by expanding in a Taylor series
n ∞ 1 X (−1)k(n + k)!nxk = , (A.22) 1 + x k!n!(n + k) k=0 since k n −n (−1) (n + k)!n ∂ (1 + x) = . (A.23) x x=0 n!(n + k) Part (ii) is of the same form as exp −(x + y)2, which can also be expanded to yield
" ∞ ∞ n+m n n+2m # 2 2 X X (−1) (2x) y e−(x+y) = e−x . (A.24) n!m! n=0 m=0
After some reordering one finds
" ∞ " k 2k−m 2k−2m k 2k+1−m 2k+1−2m ## 2 2 X X (−1) (2x) X (−1) (2x) e−(x+y) = e−x y2k + y2k+1 . (2k − 2m)!m! (2k + 1 − 2m)!m! k=0 m=0 m=0 (A.25) R 0 j 2 If one plugs expression (ii) into (iii), integrals of the type −∞ dx x exp(−x ) appear, which yield for i ≥ 0 (cf. Ref. [1] p. 76)
0 ∞ Z j Z j j −x2 (−1) j−1 −y (−1) 1 + j dx x e = dy y 2 e = Γ , (A.26) 2 2 2 −∞ 0
89 A Derivations and Proofs where the substitution y := x2 was used. Finally, after combining all parts and some repeated reordering, one arrives at expression (4.60)
2π r n+1 Z n 3n−1 kBT ρS n−1 n α (V ) = (−1) 2 2 √ δ dθ F (θ) sin θ× n M π 0 ∞ " l k l−k−m k−2m n r 2k X X X (−1) 2 (n − 1 + l − k)!Γ( + 1 + k − m) M × δ2l sin2l θ 2 V 2k+ (l − k)!(n − 1)!(2k − 2m)!m! kBT l=0 k=0 m=0 l k l−k−m k−2m+ 1 n+3 r 2k+1 # X X (−1) 2 2 (n − 1 + l − k)!Γ( + k − m) M +δ2l+1 sin2l+1 θ 2 V 2k+1 , (l − k)!(n − 1)!(2k − 2m + 1)!m! kBT k=0 m=0 (A.27) stated on page 28.
A.4 Limit of the mean squared velocity of the time-dependent undamped harmonic oscillator
According to Equation (5.55) on page 40, the mean squared velocity is given by 2 kBT 2 1 2 hV i(t) = M21(t, t0) 2 + M22(t, t0) . (A.28) M ω(t0)
By inserting the definitions of M21 and M22 (cf. Equations (5.45) and (5.46) on page 38), one obtains
k T x2 h i hV 2i = B π2 (J (x )Y (x) − J (x)Y (x ))2 + (J (x)J (x ) − J (x )Y (x))2 (A.29) M 4 1 0 1 1 1 0 1 0 0 0 0 1 if one substitutes
x0 = 2τω(t0), (A.30) x = 2τω(t). (A.31)
As ω(t) decays exponentially, the limit t → ∞ corresponds to the limit x → 0. The asymptotic forms of the used Bessel functions for small x are given by (cf. Ref. [1] p. 104)
1 xα J (x) ≈ , (A.32) α Γ(α + 1) 2 −1 2 α Y (x) ≈ , (A.33) 1 π x wherefore one finds k T x2 4 x2 hV 2i ≈ B π2 J (x )2 + J (x )2 + Y (x )2 + Y (x )2 + M 4 π2x2 1 0 0 0 4 1 0 0 0 2 + (J (x )Y (x ) + J (x )Y (x )) . (A.34) π 1 0 1 0 0 0 0 0
Consequently, 2 kBT 2 2 lim hV i = J1(x0) + J0(x0) . (A.35) t→∞ M
90 A.5 Moments of the phase space distribution of an harmonic oscillator in equilibrium
For large values of ω(t0) or x0, respectively, one may use the asymptotic form r 2 απ π J (x) ≈ cos x − − (A.36) α πx 2 4 (cf. Ref. [1] p. 108) and finds that k T 2 lim hV 2i → B (A.37) t→∞ M πx0 in this case.
A.5 Moments of the phase space distribution of an harmonic oscillator in equilibrium
In equilibrium at temperature T , the moments of the phase space distribution of an harmonic oscil- lator with frequency ω and mass M are given by
∞ Z i j dXdV i j ~ω M 2 2 2 hX V ieq = X V exp − V + ω X . (A.38) 2π~/M kBT 2kBT −∞ The main task is therefore to evaluate integrals of the type
∞ Z 2 dx xie−ax , (A.39)
−∞ where i ∈ N and a > 0. If i is an odd number, this integral is clearly zero, as one integrates the product of a symmetric and an antisymmetric function. In case of i = 2n with n ∈ N, one can substitute y := ax2 yielding (cf. Ref. [1] p. 76)
∞ ∞ Z Z i −ax2 i − i+1 i−1 −y i − i+1 i + 1 dx x e = 1 + (−1) a 2 dy y 2 e = 1 + (−1) a 2 Γ . (A.40) 2 −∞ 0 Having this identity at hand, it is easy to arrive at expression (5.58)
i+j 1 (i+j) r 2 2 i + 1 j + 1 k T hXiV ji = Γ Γ 1 + (−1)i 1 + (−1)j ω−i B , (A.41) eq 4π 2 2 M stated on page 41.
A.6 Properties of the spectrum of the transition matrix A.6.1 The leading eigenvalue and its relation to other eigenvalues If transition rates equal to zero are excluded, then for any transition matrix W one can find a real number κ such that A := (aij) = W + κI (A.42) is positive, that is aij > 0 for all i, j. The symbol I denotes the identity matrix. The Perron- Frobenius theorem (cf. Reference [18] p. 398) then states that there exists a real eigenvalue r0 > 0 of A, and that all other eigenvalues ri are strictly smaller than r0, meaning |ri| < r0. Moreover, X X min aij ≤ r0 ≤ max aij (A.43) i i j j
91 A Derivations and Proofs
(cf. p. 408 in Ref. [18]). As the sum of the elements of any column of W is zero to conserve the total probability, this statement is equivalent to r0 = κ. Shifting back the eigenvalues ri of A to the eigenvalues µi = ri − κ of W, this implies that
µ0 = 0 and (A.44)
<(µ1/2) < 0. (A.45)
A.6.2 Real eigenvalues for even detailed balance According to Section 6.2.3 the transition matrix of the three state system with even detailed balance reads −(W13 + W21) W12 W13 0 0 0 W = W21 −2W12 W21 + (a − 1) 0 −W12 W21 , (A.46) W13 W12 −(W13 + W21) 0 W12 −W21 where all variables are greater than or equal to zero. The eigenvalues are then given by √ −(W + W )(a + 1) − W ± K µ = 12 21 13 , (A.47) 1/2 2 with 2 2 2 2 2 K := (W21 + 2W21W12)(a − 1) + 4W13 − 4W13W12(a + 1) + W12(a + 1) . (A.48) The first summand of K is clearly positive. The remaining three summands can be rewritten as
4x2 − 4xy + y2, (A.49) where x := W13 ≥ 0 and y := W12(a + 1) ≥ 0. If y = 0, this is obviously greater than or equal to zero, else one can substitute z := x/y, yielding
(A.49) = y2[4(z − 1)z + 1], (A.50)
1 which has a global minimum at z = 2 with a value of zero. Hence, expression (A.49) is allways greater than or equal to zero, wherefore the eigenvalues are real.
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Acknowledgement
I am grateful to Professor Klaus Mecke for giving me the opportunity to write this thesis, for inspi- ration and his constant support. I thank Professor Thomas Franosch for fruitful discussions, Susan Nachtrab for patiently answering my questions on her diploma thesis and everyone else from the theoretical physics group for their help and friendship.
97
Erkl¨arung
Ich habe diese Arbeit selbst¨andig verfasst und keine anderen als die angegebenen Quellen und Hilfs- mittel verwendet.
Martin Reichelsdorfer
Erlangen, Mai 2010
99