PHYSICAL REVIEW X 4, 021036 (2014)
From the Area under the Bessel Excursion to Anomalous Diffusion of Cold Atoms
E. Barkai, E. Aghion, and D. A. Kessler Department of Physics, Institute of Nanotechnology and Advanced Materials, Bar Ilan University, Ramat-Gan 52900, Israel (Received 4 July 2013; revised manuscript received 18 March 2014; published 28 May 2014) Lévy flights are random walks in which the probability distribution of the step sizes is fat-tailed. Lévy spatial diffusion has been observed for a collection of ultracold 87Rb atoms and single 24Mgþ ions in an optical lattice, a system which allows for a unique degree of control of the dynamics. Using the semiclassical theory of Sisyphus cooling, we formulate the problem as a coupled Lévy walk, with strong correlations between the length χ and duration τ of the excursions. Interestingly, the problem is related to the area under the Bessel and Brownian excursions. These are overdamped Langevin motions that start and end at the origin, constrained to remain positive, in the presence or absence of an external logarithmic potential, respectively. In the limit of a weak potential, i.e., shallow optical lattices, the celebrated Airy distribution describing the areal distribution of the Brownian excursion is found as a limiting case. Three distinct phases of the dynamics are investigated: normal diffusion, Lévy diffusion and, below a certain critical depth of the optical potential, x ∼ t3=2 scaling, which is related to Richardson’s diffusion from the field of turbulence. The main focus of the paper is the analytical calculation of the joint probability density function ψðχ; τÞ from a newly developed theory of the area under the Bessel excursion. The latter describes the spatiotemporal correlations in the problem and is the microscopic input needed to characterize the spatial diffusion of the atomic cloud. A modified Montroll-Weiss equation for the Fourier-Laplace transform of the density Pðx; tÞ is obtained, which depends on the statistics of velocity excursions and meanders. The meander, a random walk in velocity space, which starts at the origin and does not cross it, describes the last jump event χ in the sequence. In the anomalous phases, the statistics of meanders and excursions are essential for the calculation of the mean-square displacement, indicating that our correction to the Montroll-Weiss equation is crucial and pointing to the sensitivity of the transport on a single jump event. Our work provides general relations between the statistics of velocity excursions and meanders on the one hand and the diffusivity on the other, both for normal and anomalous processes.
DOI: 10.1103/PhysRevX.4.021036 Subject Areas: Atomic and Molecular Physics, Statistical Physics
I. INTRODUCTION the Green-Kubo formalism, namely, by the calculation of the stationary velocityR correlation function, which gives the The velocity vðtÞ of a particle interacting with a heat bath ∞ diffusivity K2 ¼ hvðtÞvð0Þidt. exhibits stochastic behavior, which in many cases is 0 An alternative approach is investigated in this work and difficult to evaluate. The position of the particle, assumed is based on the concept of excursions. We assume that the to start at the origin at time t ¼ 0, is the time integral over R random process vðtÞ is recurrent, and thus the velocity the fluctuating velocity xðtÞ¼ t vðt0Þdt0 and demands a 0 crosses the zero point, v ¼ 0, many times in the observation probabilistic approach to determine its statistical properties. window ð0;tÞ. We divide the path xðtÞ into a sum of Luckily, the central-limit theorem makes it possible, for increments, many processes, to predict a Gaussian shape for the diffusing packet. Then, the diffusion constant K2 character- Z Z Z t1 t2 tiþ1 izes the normal motion through its mean-square displace- xðtÞ¼ vðt0Þdt0 þ vðt0Þdt0 þ þ vðt0Þdt0 þ : 2 0 ment hx i¼2K2t. The remaining goal, for a given process t1 ti or model, is to compute K2 and other transport coefficients. (1) In normal cases, this can be done, at least in principle, via
Here, ft1;t2…g are the points in time of the velocity zero 0 crossings, vðtiÞ¼ . In the interval (ti, tiþ1), the velocity is either strictly positive or negative. The velocity in each Published by the American Physical Society under the terms of the Creative Commons Attribution 3.0 License. Further distri- interval is thus a stochastic process that starts and ends on bution of this work must maintain attribution to the author(s) and the origin without crossing it in between. Such a random the published article’s title, journal citation, and DOI. curve is called an excursion. The random spatial increment
2160-3308=14=4(2)=021036(34) 021036-1 Published by the American Physical Society E. BARKAI, E. AGHION, AND D. A. KESSLER PHYS. REV. X 4, 021036 (2014) R χ tiþ1 0 0 i ¼ ti vðt Þdt is the area under the excursion. The to a process vðtÞ described by a Langevin equation, with an position of the particle, according to Eq. (1), is the sum asymptotically logarithmic potential. We show how the of the random increments, namely, the sum of the signed random area under this Langevin excursion determines the areas under the velocity excursions, each having a random dynamics of cold atoms. We believe that Langevin excur- duration. The goal of this paper is to relate the statistics sions, or more generally, random excursions, are useful of the areas under these velocity excursions, and the tools in many areas of statistical physics; hence, their corresponding random time intervals between zero cross- investigation beyond the well-studied Brownian excursion ings, to the problem of spatial diffusion. This connection is is worthwhile. easy to find in the case in which the increments χi and If vðtÞ is a Brownian motion or, as we will proceed to τ − the duration of the excursions i ¼ tiþ1 ti are mutually show, the velocity of cold atoms in shallow lattices, the set uncorrelated, independent, and identically distributed ran- of points where vðtÞ¼0 has nontrivial properties, as it dom variables. Over a long measurement time, the number contains no isolated points and no interval. This is the case of excursions is t=hτi, where hτi is the average time for since we are dealing with a continuous path with power-law an excursion. Then, according to Eq. (1), the mean-square statistics of the crossing times (see details below). displacement is hx2i¼hχ2it=hτi, and hence we have Mathematicians have investigated the statistics of level- crossings, e.g., zero-crossings, of continuous paths in great hχ2i detail. The concept of local time, introduced by P. Lévy in K2 ¼ : (2) 2hτi the context of Brownian motion and Ito’s excursion theory for continuous paths, is a pillar in this field (see This equation is reminiscent of the famous Einstein formula Refs. [17,18] and references within). Here, we use a and shows that diffusion is related to the statistics of heuristic approach, with a modification of the original excursions. The original work of Einstein, discussed in process vðtÞ, by introducing a cutoff ϵ: the starting point many textbooks, is explicitly based on an underlying of the particle after each zero hitting, which is taken to zero random-walk picture, and so it does not involve the area at the end. This expediency makes it possible to use under random velocity excursions, nor zero crossings in renewal theory and continuous-time random walks velocity space. We will arrive at the simple equation (2) (CTRW), which are tools well studied in the physics only at the end of our work, in Sec. XI, while here it merely literature of discrete processes. In this sense, we differ serves as an appetizer to motivate the consideration of from rigorous mathematical approaches. Thus, we avoid velocity excursions in some detail and to suggest the the problem of continuous paths, for example, the infinite usefulness of developing tools for the calculation of hχ2i number of zero crossings, by replacing the original path and hτi (here, hχi¼0, by symmetry). Obviously, Eq. (2) is with an ϵ modified path for which the number of zero based on the assumption that the variance hχ2i is finite crossings is finite (when ϵ is finite). We show that physical and, as mentioned, that correlations are not important. The quantities characterizing the entire process have a finite major effort of this paper is directed at considering a more ϵ → 0 limit. For example, in Eq. (2), for Langevin 2 challenging case, namely, a physically relevant stochastic dynamics of vðtÞ, both hτi and hχ i approach zero as 2 process where the variance hχ i diverges, and more ϵ → 0; their ratio K2 approaches a finite limit. importantly, the process exhibits correlations between χ One might wonder why we wish to use excursions and and τ. The particular system we investigate is a model for their peculiar properties to evaluate the spreading of atoms diffusion of atoms in optical lattices, following the experi- or, more generally, other transport systems. The answer is ments in Refs. [1–3] and the semiclassical theory of that it turns into a useful strategy when the friction forces Sisyphus cooling [4–6]. are nonlinear. In particular, we will investigate laser cool- One type of excursion that has been thoroughly inves- ing, where within the semiclassical theory, the dimension- tigated is the Brownian excursion [7–9]. A Brownian less friction force is [5] (see details below) excursion is a conditioned one-dimensional Brownian 0 τ v motion vðtÞ over the time interval 021036-2 FROM THE AREA UNDER THE BESSEL EXCURSION TO … PHYS. REV. X 4, 021036 (2014) populations of fast particles [19]. In this case, as we show As is well known, the variance of the Lévy distribution later, the standard picture of diffusion breaks down. diverges, which implies that hx2i¼∞, which is unphys- More specifically, the problem of diffusion of cold ical. Indeed, as mentioned, Katori et al. experimentally atoms under Sisyphus cooling was partially treated by determined a finite mean-square displacement exhibiting Marksteiner et al. [6]. While clearly indicating the anoma- super-diffusion (see also Ref. [2]). We showed related lous nature of the diffusion process, the main tool used was numerical evidence that the Lévy distribution is cut off at the evaluation of the stationary velocity correlation function distances of the order x ∼ t3=2. This breakdown of Lévy of the process, followed by the use of the Green-Kubo statistics arises because of the importance of correlations formalism for the evaluation of K2. They showed that for a between jump lengths χ and jump duration τ, which are certain critical value of the depth of the optical lattice, the neglected in the derivation of the Lévy distribution. Our value of K2 diverges (see also Ref. [20]). Katori et al. [1] purpose here is to investigate these correlations in detail. As 2 2ξ measured the mean-square displacement hx i ∼ t and we proceed to show, the correlations are given by the recorded the onset of super-diffusion ξ > 1=2 below a statistical properties of the Langevin excursions discussed critical depth of the optical lattice [1]. Wickenbrock et al. above, for all except the last interval. Because the velocity [2], in the context of driven optical lattice experiments, is not constrained to be zero at the measurement time t, this demonstrated with Monte Carlo simulations an upper limit last interval is not described by an excursion but rather by a 2 3 on the spreading of the atoms hx i ≤ const × t . Sagi et al. Langevin meander, where the random walk vðtÞ begins at [3] showed that a packet of spreading Rb atoms can be the origin and does not return for the entire duration of the fitted with a Lévy distribution instead of the Gaussian walk. The properties of the excursions and meanders enter distribution found for normal diffusion. These findings in a modified Montroll-Weiss [30] equation for the Fourier- clearly indicate the breakdown of the usual strategy of Laplace transform of the density Pðx; tÞ that we derive. We treating normal diffusion and hence invite further theoreti- then use this equation to calculate a quantity that is cal investigations. The velocity correlation function is not sensitive to the correlations, namely, the mean-square stationary, and hence the Green-Kubo formalism must be displacement. The mean-square displacement exhibits replaced [21]. The moments exhibit multifractal behavior anomalous diffusion and is sensitive to the last jump event, [22]; there is an enhanced sensitivity to the initial prepa- i.e., to the statistics of the meander. Thus, our treatment ration of the system [23,24]; momentum fluctuations are modifies both the celebrated Montroll-Weiss equation, to described by an infinite covariant density [25]; and in include the last jump in the sequence (i.e., the meander), certain parameter regimes, the Lévy central-limit theorem and the existing theory of areas under Brownian meanders applies instead of the standard Gaussian version [6,26].In and excursions, to include the dissipative friction force this regime of shallow optical lattices, the power of the which is responsible for the cooling of the atoms. Our analysis of the area under Langevin excursions becomes analysis illuminates the rich physical behavior and provides essential, as we will show here. We note that the relation the needed set of mathematical tools beyond the decoupling of laser cooling with Lévy statistics is not limited to the approximation used to obtain the Lévy distribution in our case of Sisyphus cooling considered in this manuscript. previous work. For completeness, we present detailed Subrecoil laser cooling, a setup different from ours, also decoupled and coupled analyses, the latter being the main leads to fundamental relations between statistical physics of focus of the current work. rare events and laser-atom physics [27,28]. For a recent The paper is organized as follows. We start with a brief mini-review on the departure from Boltzmann-Gibbs stat- survey of the semiclassical theory of Sisyphus cooling [5,6] istical mechanics for cold atoms in optical lattices, and show the connection of the dynamics to Lévy walks see Ref. [29]. following Marksteiner et al. [6]. The importance of the correlations between τ and χ is emphasized, a theme which, A. Scope and organization of paper as mentioned, has not received its deserved attention. In The current work significantly extends the investigation Sec. IV, a simple scaling theory is presented, which yields of the properties of the spatial distribution of atoms the exponents describing the dynamics of the atomic undergoing Sisyphus cooling that began in Ref. [26]. packet. The main calculation of the distribution of the There we uncovered three phases of the motion which area under the Bessel excursion is found in Sec. V; the are controlled by the depth U0 of the optical lattice: a calculation of the area under the Bessel meander is given in Gaussian phase x ∼ t1=2, a Lévy phase x ∼ tξ and Appendix E. A new coupled continuous-time random-walk 1=2 < ξ < 3=2, and a Richardson phase x ∼ t3=2 (see theory in Sec. VI provides the connection between the Eq. (80) below). Within the intermediate phase, the density statistics of excursions and meanders, and the evolution of of particles, in the central region of the packet, is described the density profile. Asymptotic behaviors of the Fourier- by a symmetric Lévy distribution, similar to the fitting Laplace transform of the joint probability density function function used to analyze a recent Weizmann Institute (PDF) of jump lengths and waiting times is investigated in experiment [3]. However, this cannot be the whole story. Sec. VII. These, in turn, give us the asymptotic behaviors of 021036-3 E. BARKAI, E. AGHION, AND D. A. KESSLER PHYS. REV. X 4, 021036 (2014) the atomic density packet, the mean-square displacement, From the semiclassical treatment of the interaction of the and the different phases of the dynamics that are inves- atoms with the counterpropagating laser beams, we have tigated in Secs. VIII–XI. Derivation of the distribution of the time interval straddling time t for the Bessel process is D ¼ cER=U0; (7) carried out in Appendix F, which allows us to connect between our heuristic renewal approach and more rigorous where U0 is the depth of the optical potential, ER is the treatments [17,31] and further discuss the nontrivial fractal recoil energy, and the dimensionless parameter c [34] set of zero crossings. depends on the atomic transition involved [5,6,35]. U0 is a control parameter; hence, different values of D are II. SEMICLASSICAL DESCRIPTION OF COLD attainable in experiment, and exploration of different ATOMS—LANGEVIN DYNAMICS phases of the dynamics are within reach [1–3,19]. Equation (7) is rather intuitive since deep optical lattices, We briefly present the semiclassical picture for the i.e., large U0, imply small D, while large recoil energy dynamics of the atoms. TheR trajectory of a single particle leads to a correspondingly large value of D. t with mass m is xðtÞ¼ 0 pðtÞdt=m, where pðtÞ is its The behavior of the distribution for momentum only momentum. Within the standard picture [5,6,33] of (when x is integrated out from the Kramers equation, Sisyphus cooling, two competing mechanisms describe yielding the Fokker-Planck equation) is much simpler and the dynamics. The cooling force FðpÞ¼−α¯p=½1þ 2 has been presented in previous work [25]. The equilibrium ðp=pcÞ acts to restore the momentum to the minimum properties are governed by the effective potential in 0 energy state p ¼ . Momentum diffusion is governed by a momentum space, diffusion coefficient, which is momentum dependent, 2 Z D p D1 D2= 1 p=p p ð Þ¼ þ ½ þð cÞ . The latter describes 2 momentum fluctuations that lead to heating due to random VðpÞ¼− FðpÞdp ¼ð1=2Þ lnð1 þ p Þ; (8) 0 emission events that stochastically jolt the atom. We use → α¯ → dimensionless units, time t t , momentum p p=pc, which for large p is logarithmic, VðpÞ ∼ lnðpÞ. This large → α¯ and distance x xm =pc, and introduce the dimension- p behavior of VðpÞ is responsible for several unusual 2α¯ less momentum diffusion constant D ¼ D1=ðpcÞ .For equilibrium and nonequilibrium properties of the momen- simplicity, we set D2 ¼ 0 since it does not modify the tum distribution [1,19,23,25,36]. The equilibrium momen- → ∞ asymptotic jpj behavior of the diffusive heating term, tum distribution function is given by [19] nor that of the force, and therefore does not modify our main conclusions. 1 1 −VðpÞ=D 1 2 −1=2D The Langevin equations WeqðpÞ¼Z e ¼ Z ð þ p Þ : (9) dp pffiffiffiffiffiffiffi dx ¼ FðpÞþ 2DξðtÞ; ¼ p (4) Here, dt dt Z ∞ − pffiffiffi 1− 1 describe the dynamics in phase space. Here, the noise term Z VðpÞ πΓ D Γ (10) 0 ¼ exp dp ¼ = is Gaussian, has zero mean, and is white, hξðtÞξðt Þi ¼ −∞ D 2D 2D δðt − t0Þ. The now dimensionless cooling force is is the normalizing partition function. Equation (9) is − p Student’s t distribution, also sometimes called a Tsallis FðpÞ¼ 2 : (5) 1 þ p distribution [19]. Actually, the problem is, coincidentally, related to Tsallis statistics since, as mentioned, the equi- The force FðpÞ is a very peculiar nonlinear friction force. librium PDF is proportional to a Boltzmann-like factor, We note that friction forces that decrease with increasing W ðpÞ ∝ exp½−VðpÞ=D ,soD acts like a temperature. 1 eq velocity or momentum like =p are found also for some More importantly, the power-law tail of the equilibrium nanoscale devices, e.g., an atomic tip on a surface [32]. PDF, for sufficiently large D, implies a large population of One goal of the paper is to find the spatial distribution of fast particles, which in turn will spread in space faster than particles governed by Eq. (4). The stochastic equation (4) what one would expect using naive Gaussian central- gives the trajectories of the standard semiclassical model limit theorem arguments. For example, if 1=D < 3, the for the dynamics in an optical lattice, which in turn was ensemble-averaged kinetic energy in equilibrium diverges derived from microscopic considerations [5,6]. Denoting 2 ∞ 1 1 since hp ieq ¼ , while when =D < , the partition the joint PDF of ðx; pÞ by Wðx; p; tÞ, the Kramers equation function diverges and a steady-state equilibrium is never reads reached. A dramatic increase of the energy of atoms when ∂ ∂ ∂2 ∂ the optical lattice parameter U0 approaches a critical value W W − was found experimentally in Ref. [1], and a power-law þ p ¼ D 2 FðpÞ W: (6) ∂t ∂x ∂p ∂p momentum distribution was measured in Ref. [19].Of 021036-4 FROM THE AREA UNDER THE BESSEL EXCURSION TO … PHYS. REV. X 4, 021036 (2014) course, the kinetic energy of a physical system cannot be D → ∞ since then the process in momentum space is one infinite, and the momentum distribution must be treated as of pure diffusion (i.e., the force is negligible), and from a time-dependent object within the infinite covariant Polya’s theorem, we know that such one-dimensional walks density approach [25]. While much is known about the are recurrent. The cooling force, being attractive, clearly momentum distribution, both experimentally and theoreti- maintains this property. cally, the experiments [1–3] demand a theory for the spatial Let τ > 0 be the random time between one crossing spreading. We note that diffusion in logarithmic potentials event and the next, and let −∞ < χ < ∞ be the random has a vast number of applications (see Refs. [23–25,37–41] displacement for the corresponding τ. As shown schemati- and references therein), e.g., Manning condensation [42], cally in Fig. 2, the process starting with zero momentum is and unusual mathematical properties, including ergodicity defined by a sequence of jump durations fτ1; τ2; …g, breaking [36]. In general, logarithmic potentials play a with corresponding displacements fχ1; χ2; …g. These ran- special role in statistical physics [43–45]. dom waiting times and displacements generateR a Lévy τ1 0 0 walk R[6], as we explain presently. Here, χ1 ¼ 0 pðt Þdt , III. MAPPING THE PROBLEM TO A χ τ1þτ2 0 0 2 ¼ τ1 pðt Þdt , etc. Let the points on the time axis LéVY WALK PROCESS ft1;t2; …g denote times tn > 0 where the particle crosses 0 In principle, one may attempt to directly solve the the origin of momentum p ¼ (seeP Fig. 2). These times k τ Kramers equation (6) to find the joint PDF of the random are relatedR to the waiting times: tk ¼ i¼1 i. We see that χ ¼ tk pðt0Þdt0 is the area under the random momentum variables x; p at a given time t. In Fig. 1, we plot a k tk−1 ð Þ 0 histogram of the phase space obtained numerically. We see curve constrained in such a way that pðt Þ in the time a complicated structure: Roughly along the diagonal, clear interval (tk−1, tk) does not cross the origin, while it started correlations between x and p are visible; on the other hand, and ended there. The total displacement, namely, the along the x and p axes, decoupling between momentum position of the particleP at time t, is a sum of the individual n χ χ and position is evident, together with broad (i.e., non- displacements x ¼ i¼1 i þ . Here, n is the random Gaussian) distributions of x and p. At least to the naked number of crossings of zero momentum in the time interval 0 χ eye, no simple scaling structure is found in x − p space, ð ;tÞ, and is the displacement made in the last interval and hence, we shall turn to different, microscopic, variables ðtn;tÞ. By definition, in the time interval ðtn;tÞ, no zero τ − that do exhibit simple scaling. This leads to an analysis crossing was recorded. The time ¼ t tn is sometimes centered on the mapping of the Langevin dynamics to a called the backward recurrenceP time [47]. The measure- n τ τ Lévy walk scheme [6,46] and the statistics of areas under ment time is clearly t ¼ i¼1 i þ . In standard transport random excursions. problems, the effect of the last displacement χ on the Starting at the origin, p ¼ 0, the particle along its position of particle x is negligible, and similarly, one ≃ stochastic path in momentum space crosses p ¼ 0 many usually assumes t tn when t is large. However, for times (see Fig. 2). In other words, the random walk in anomalous diffusion of cold atoms, where the distributions momentum space is recurrent; this is the case even when 15 τ τ τ τ 1 2 3 4 10 χ 2 5 0 χ p 4 max -5 χ -10 3 χ 1 -15 -20 t1 t2 t3 t4 t max FIG. 2. Schematic presentation of the momentum of the particle versus time. The times between consecutive zero crossings are 6 FIG. 1. The joint density of x and p for D ¼ 2=3, 10 particles called the waiting times τ, and the shaded area under each 6 and t ¼ 10 . This scatter plot of phase space shows correlation excursion are the random flight displacements χ. The τ’s and the between x and p and rare events with large fluctuations both for x χ’s are correlated since a long waiting time typically implies a and p. large displacement. 021036-5 E. BARKAI, E. AGHION, AND D. A. KESSLER PHYS. REV. X 4, 021036 (2014) of displacements and jump durations are wide, these last from the analysis of ψðχ; τÞ, and then from this, we use the events cannot be ignored. Lévy walk scheme to relate the single-excursion informa- One goal is to find the long-time behavior of Pðx; tÞ, the tion to the properties of the entire walk and, in particular, normalized PDF of the spatial position of a particle that Pðx; tÞ for large t. started at the origin x ¼ 0, p ¼ 0 at t ¼ 0. It is physically There exists a strong correlation between the excursion clear that, in the long-time limit, our results will not be duration τ and the displacement length χ, which is encoded changed if instead we consider narrow initial conditions, in ψðχ; τÞ. Let the PDFs of χ and τ be denoted qðχÞ and for example, Gaussian PDFs of initial position and momen- gðτÞ, respectively. Weak correlations would imply the tum. Initial conditions with power-law tails will likely lead decoupled scheme ψðχ; τÞ ≃ qðχÞgðτÞ, which is far easier to very different behaviors [23,24]. Once we find Pðx; tÞ, to analyze; however, as we shall see, this decoupling is we have the semiclassical approximation for the spatial generally not true, and it leads to wrong results for at least density of particles. The latter can be compared with the some observables of interest. Properties of qðχÞ and gðτÞ Weizmann Sisyphus cooling experiments [3], provided that were investigated in Refs. [6,35] and are also studied below. collisions among the atoms are negligible. The problem becomes more interesting and challenging The Lévy walk process under investigation is a renewal because of these correlations. As we show, they are process [47] in the sense that once the particle crosses responsible for the finiteness of the moments of Pðx; tÞ, the momentum origin, the process is renewed (since the in particular, the mean-square displacement hx2i, and for Langevin dynamics is Markovian). This is crucial in our the existence of a rapidly decaying tail of Pðx; tÞ. This, in treatment, and it implies that the waiting times τi are turn, is related to the Lévy flight versus Lévy walk dilemma statistically independent, identically distributed random [46], to multifractality [22], and to the physical meaning variables, as are the χi. However, as we will soon discuss, of the fractional diffusion equation [48] used as a fitting the pairs fτi; χig are correlated. tool for the Weizmann experiment [3] [see Eq. (105) and Since the underlying Langevin dynamics is continuous, discussion therein]. As we show below, above the critical we need a refined definition of the Lévy walk process. Both value D ¼ 1, the correlations can never be ignored, and the τi’s and the χi’s are infinitesimal; however, the number they govern the behavior of the entire packet, not only the of renewal events, n, diverges for any finite measurement large x tails of Pðx; tÞ. Physically, the correlations are time t, in such a way that the total displacement x is finite. obvious since long durations of flight τ involve large In this sense, the Lévy walk process under investigation is momenta p, which in turn induce large displacements χ. different from previous works where the number of renew- As an example of these correlations, we plot in Fig. 3 the als, for finite measurement time, is finite. One way to treat displacement jχj versus the corresponding τ obtained from the problem is to discretize the dynamics, as is necessary in computer simulation. The figure clearly demonstrates the χ ∝ τ3=2 any event to perform a computer simulation, and then χi correlations, and it also shows a j j scaling, which χ − τ and τi are, of course, finite. In our analytical treatment, we now investigate. Note how much simpler the following Marksteiner et al. [6], we consider the first distribution is, compared to the x − p distribution in Fig. 1. passage time problem for a particle starting with momen- Simulations presented in Fig. 3 were performed on a tum pi and reaching pf 021036-6 FROM THE AREA UNDER THE BESSEL EXCURSION TO … PHYS. REV. X 4, 021036 (2014) − 1 1 γ discrete lattice in momentum space, starting on the first qðχÞ ∼ constχ ð þ2ηþηÞ: (16) lattice point (see Appendix D for details). Comparing with Eq. (11), we get a simple equation for — the unknown exponent η,whichis1 þ 1=ð2ηÞþγ=η ¼ IV. SCALING THEORY RELATION 4 3 β η 3 2 BETWEEN EXPONENTS = þ ,sousingEq.(12),wefind ¼ = .Thisis precisely the scaling behavior χ ∼ τ3=2 that we observe As shown by Marksteiner et al. [6] and Lutz [35], the in our simulation, Fig. 3. Hence, the natural scaling PDFs of the excursion durations and displacements satisfy solution of the problem is the asymptotic laws 1 χ χ τ ∼ ffiffiffiffi ffiffiffiffi τ ∼ τ−ð3=2þγÞ χ ∼ χ −ð4=3þβÞ pð j Þ p 3 2 B p 3 2 : (17) gð Þ g ;qð Þ q j j ; (11) Dτ = Dτ = with In hindsight, this result is related to Brownian scaling. For a particle free of the cooling force, its momentum 1 1 ∼ τ 1=2 γ ¼ ; β ¼ : (12) will exhibit Brownian scaling, p ðD Þ , and hence 2D 3D the excursions that are integrals of the velocity scale pffiffiffiffi as χ ∼ Dτ3=2. The crucial point is that this simple In Appendixes A and B, we study these PDFs using Brownian scaling is maintained even in the presence of backward Fokker-Planck equations. In particular, we find the cooling force for all D.Thisisbecauseofthe g q the amplitudes and , and the relevant moments. The marginal nature of the weak 1=p friction force for large exponents β and γ are the most crucial aspects of these → ∞ γ β 0 p, which leaves the Brownian scaling intact but changes PDFs. When D , we get ¼ ¼ , and Eq. (11) gives the scaling function. While the 3=2 scaling is simple and “ ” D familiar limits. In the high-temperature limit of large , D independent, the shape of Bð·Þ is sensitive to this the cooling force is negligible, and the Langevin equation control parameter. In the next section, we investigate reduces to Brownian motion in momentum space. Then, Bð·Þ, and for this, we must go beyond simple scaling τ ∝ τ−3=2 gð Þ , which is the well-known asymptotic behavior arguments. of the PDF of first passage times for unbounded one- dimensional Brownian motion [49,50]. Less well known is −4=3 V. AREA UNDER THE BESSEL EXCURSION limD→∞qðχÞ ∝ jχj , which describes the distribution of the area under a Brownian motion until its first passage A natural generalization of the Brownian excursion is a time (see Refs. [51,52], in which this PDF is given Langevin excursion. Such a stochastic curve is the path explicitly). Notice that the power-law behavior Eq. (11) pðt0Þ, given by the Langevin equation, in the time interval 0 ≤ 0 ≤ τ ϵ yields a diverging second moment of the displacement χ for t , such that it starts and ends at pi ¼ pf ¼ but is 0 D>1=5, which in turn gives rise to anomalous statistics constrained to remain positive in between. Here, pi ¼ pð Þ τ for x. is the initial, and pf ¼ pð Þ is the final location in The correlations between χ and τ are now related to momentum space. For our application, the path is consid- the asymptotic behaviors of their PDFs, Eq. (11).We ered in the limit ϵ → 0. Since the path neverR crosses the χ τ 0 0 rewrite the joint PDF origin, the area under such a curve is ¼ 0 pðt Þdt , and hence the PDF of χ for fixed τ yields the sought-after ψðχ; τÞ¼gðτÞpðχjτÞ; (13) conditional PDF pðχjτÞ. Obviously, χ is the integral over the constrained path pðt0Þ; hence, by definition, it is the where pðχjτÞ is the conditional PDF to find the jump area under the excursion. The meander will describe the length χ for a given jump duration τ. We introduce a last jump χ since at the measurement time the particle’s scaling ansatz, which is expected to be valid at large τ: velocity is generally nonzero. For now, we will discuss only χ τ excursions, find pð j Þ, and return to the meander later. χ Here, we focus our attention on a specific excursion χ τ ∼ τ−η pð j Þ B τη : (14) we call the Bessel excursion, corresponding to the case FðpÞ¼−1=p: R Since g τ ∞ ψ χ; τ dχ, we have the normalization ð Þ¼R −∞ ð Þ dp pffiffiffiffiffiffiffi condition ∞ BðzÞdz ¼ 1. By definition, ¼ −1=p þ 2DξðtÞ; (18) −∞ dt Z Z ∞ ∞ χ −3=2−γ−η so that the effective potential is the nonregularized loga- qðχÞ¼ ψðχ; τÞdτ ∝ dττ B η : (15) 0 0 τ rithm, VðpÞ¼lnðpÞ and p>0. Since the scaling approach is valid for long times, where excursions are long, the Changing variables to z ¼ χ=τη,weget typical momentum p is large, and the details of the force 021036-7 E. BARKAI, E. AGHION, AND D. A. KESSLER PHYS. REV. X 4, 021036 (2014) D = ∞ D = 2/3 D = 2/5 400 400 400 300 300 300 p(t) 200 200 200 100 100 100 0 0 0 0 20000 40000 60000 80000 0 20000 40000 60000 80000 0 20000 40000 60000 80000 t FIG. 4. We show three typical Bessel excursions for D ¼ ∞, 2=3, 2=5 (left to right panel) and the mean (red curve). The random excursions are constrained Langevin paths, with a −1=p force, that do not cross the origin in the observation time t and that start and end on ϵ → 0. For simulations, we use the regularized force field, which only alters the dynamics when p ≃ 1 and is negligible in the long- time limit (see Appendix D). When D → ∞, we get a Brownian excursion. We see that as D is decreased, the excursions are further pushed from the origin p ¼ 0 since small D effectively implies large attractive forces; hence, to avoid the zero crossing, the particles must drift further away from the origin. Thus, the attractive force repels particles, which at first might sound counterintuitive and might seem to be an unexplored property of excursions. ˆ field close to the origin are negligible for the purpose of the formalism to find Gτðs; pjpiÞ (see details below). The calculation of the scaling function Bð·Þ. We will check this constraint that the path pðt0Þ is always positive enters as an assumption with numerical simulations, which of course absorbing boundary condition at p ¼ 0 [41]. use a regularized form of the force law. Some sample paths (ii) The second step is to consider pi ¼ p ¼ ϵ → 0 and of Bessel excursion are presented in Fig. 4. The name obtain the Laplace transform Bessel excursion stems from the fact that Langevin Z ∞ ˆ dynamics in the nonregularized friction force field 1=p Gτðs; ϵjϵÞ pˆ ðsjτÞ¼ pðχjτÞe−sDχdχ ¼ lim : (20) corresponds to a well-known stochastic process called ϵ→0 ˆ ϵ ϵ 0 Gτðs; j Þjs 0 the Bessel process [41,53]. More information on the ¼ regularized and nonregularized processes is given in The denominator in the above equation ensures the Appendixes A and B. ˆ τ 1 normalization since we must have pðsj Þjs¼0 ¼ . From symmetry, pðχjτÞ¼pð−χjτÞ, and so we may (iii) Finally, we invert Eq. (20) back to χ space using the ϵ 0 χ 0 restrict our discussion to pi ¼ > and > . Later, inverse Laplace transform, sD → χ, which yields pðχjτÞ. we will use this symmetry, which implies that for −∞ < From there, we can also immediately find the scaling χ < ∞ we have pðχjτÞ¼pðjχj; τÞ=2, where pðjχj; τÞ is function Bð·Þ, Eq. (14). theR PDF of positive excursions, normalized according We now implement these steps to solve the problem. ∞ χ τ χ 1 to 0 pðj j; Þdj j¼ . The FK formalism [52] treats functionals of Brownian χ Let Gτð ;pjpiÞ be theR joint PDF of the random variables motion. Here, we use a modified version of the FK equation χ χ τ 0 0 χ 0 and p. Since ¼ 0 pðt Þdt ,wehave ¼ at time Rto treat overdamped Langevin paths [54]. Let A ¼ τ 0 χ δ − δ χ τ 0 0 ¼ , so initially Gτ¼0ð ;pjpiÞ¼ ðp piÞ ð Þ. Later, 0 U½pðt Þ dt be a functional of the Langevin path and we will take p to be the final momentum pf, which assume Uð·Þ > 0 so that we are treating positive func- ϵ → 0 similarly to pi, will be set to the value (for the sake tionals. Here, GτðA; pjp Þ is the joint PDF of A and p, and ˆ i of notational brevity, we omit the subscript in pf). The Gτðs; pjpiÞ is the corresponding Laplace A → sD trans- calculation of pðχjτÞ follows three steps. For the Brownian form. The generalized FK equation reads case FðpÞ¼0, this method was successfully applied by Majumdar and Comtet [8]. ∂ ˆ Gτðs; pjpiÞ ˆ − ˆ (i) We find the Laplace χ → sD transform of ∂τ ¼½Lfp sDUðpÞ Gτðs; pjpiÞ: (21) Gτðχ;pjpiÞ, ˆ ∂ 2 −∂ Z Here, Lfp ¼ Dð pÞ pFðpÞ is the Fokker-Planck oper- ∞ −sDχ ˆ ator. When FðpÞ¼0, Eq. (21) is the celebrated FK e Gτðχ;pjpiÞdχ ≡ Gτðs; pjpiÞ: (19) 0 equation, which is an imaginary-time Schrödinger equation, and −sDUðpÞ is the potential of the correspond- The reason why we multiply s by D in the Laplace ing quantum problem. The constraint on positive excur- transform will become clear soon. Since χ is a functional sions, namely, p>0, gives the boundary condition 0 ˆ of the path pðt Þ, we will use the Feynman-Kac (FK) Gτðs; 0jpiÞ¼0. In the quantum language, this is an infinite 021036-8 FROM THE AREA UNDER THE BESSEL EXCURSION TO … PHYS. REV. X 4, 021036 (2014) potential barrier for p<0. This formalism can be used, in where p~ ¼ s1=3p. Notice that when D → ∞, the second principle, to obtain the area under Langevin excursions for term vanishes and we get the Airy equation. For this all forms of FðpÞ. technical reason, we have chosen in Eq. (19) the Laplace For the Bessel excursion under investigationR here, the variable as sD not s. Also, it should be noted that the fk are τ 0 0 functional U½pðtÞ ¼ pðtÞ is linear since χ ¼ 0 pðt Þdt . orthonormal in the variable p~ . Quantum mechanically, this gives a linear potential It follows from Eq. (27) and the absorbing boundary and hence the connection to the Airy function found condition that for FðpÞ¼0 in Refs. [7,8]. With the force field −1 ˆ ∂ 2 ∂ −1 ~ 1þ1=ð2DÞ FðpÞ¼ =p,wehaveLfp ¼ Dð pÞ þ pp , and fkðp~ Þ ∼ dkp~ ; p~ → 0; (28) hence we find, using Eqs. (18), (19), and (21) (the former ~ ~ gives the sD term), with dk a k-dependent coefficient. The dk are evaluated Rfrom solutions of Eq. (27) with the normalization condition ˆ 2 ∞ 2 ∂Gτ s; p p ∂ ∂ 1 ~ ~ 1 ð j iÞ − ˆ 0 ½fkðpÞ dp ¼ . For the initial and final conditions ¼ D 2 þ sDp Gτðs; pjpiÞ: ∂τ ∂p ∂p p under investigation, pi ¼ p ¼ ϵ, we have (22) X 2 2=3 ˆ νþ2=3 1þ2α ~ −Dλks τ Gτðs; ϵjϵÞ ∼ s ϵ ðdkÞ e ; (29) The solution of this equation is found using the separation k ansatz where the exponents ν and α will turn out to be useful, X∞ − τ ˆ ψ DEk Gτðs; pjpiÞ¼ ak kðpÞe ; (23) 1 þ D 1 þ D ν ¼ ; α ¼ : (30) k¼0 3D 2D which yields the time-independent equation Note that classification of boundaries for a nonregularized ∂2 1 ∂ ψ Bessel process was carried out in Ref. [41] and is discussed ψ k − ψ − ψ 2 k þ sp k ¼ Ek k: (24) here in Sec. XII. For an absorbing boundary condition, both ∂p D ∂p p the sign of the probability current and the usual condition of the vanishing of the probability on the absorbing point must We now switch to the more familiar one-dimensional be taken into consideration [41]. Schrödinger equation via the similarity transformation ˆ According to Eq. (20), we need Gτðs ¼ 0; ϵjϵÞ, in order ϕ 1=ð2DÞψ k ¼ p k, which gives to normalize the solution. This s ¼ 0 propagator can be 2 found exactly. The eigenvalue problem now reads ∂ 1 1 −2 − ϕk þ þ 1 p þ sp ϕk ¼ Ekϕk: (25) ∂p2 2D 2D ∂2 1 1 ϕ0 − ϕ0 þ þ 1 k ¼ E0ϕ0: (31) ∂ 2 k 2D 2D 2 k k This Schrödinger equation has a binding potential, which p p yields discrete eigenvalues, with an effective repulsive −2 The superscript 0 indicates the s ¼ 0 case. Since s ¼ 0, the potential with a p divergence for p → 0 and a binding linear field in Eq. (25) is now absent, so the “particle” is not linear potential for large p provided s ≠ 0. Equation (25) bounded; hence, one finds a continuous spectrum E0 ¼ k2. describes a three-dimensional nonrelativistic quantum par- k −2 The wave functions consistent with the boundary condition ticle in a linear potential [55,56], the p part correspond- are ing to an angular-momentum term. ϕ ffiffiffiffi R As usual, kðpÞ yields a complete orthonormal basis ϕ0 p ∞ kðpÞ¼Bk pJαðkpÞ; (32) 0 ϕkðpÞϕmðpÞdp ¼ δkm, and the formal solution of the problem is where Jαð·Þ is the Bessel function of the first kind. The X 1=ð2DÞ second solution with J−αðkpÞ is unphysical because of the ˆ pi −DE τ Gτðs; pjp Þ¼ ϕ ðp Þϕ ðpÞe k : (26) boundary condition [41]. The normalization condition is i p k i k k Z L 1 2 2 We can scale out the Laplace variable s from the time- ¼ðBkÞ p½JαðkpÞ dp; (33) 1=6 1=3 0 independent problem, defining ϕkðpÞ¼s fkðs pÞ and 2=3λ Ek ¼ s k. Using Eq. (25), we find where L → ∞ is the “box” size that will eventually drop ∂2 1 1 out ofp theffiffiffiffiffiffiffiffiffiffiffi calculation. Solving the integral in Eq. (33) gives − 1 ~ −2 ~ λ B ¼ πk=L. For initial and final conditions p ¼ p ¼ ϵ, 2 fk þ 2 2 þ p þ p fk ¼ kfk; (27) k f i ∂p~ D D the s ¼ 0 propagator then reads 021036-9 E. BARKAI, E. AGHION, AND D. A. KESSLER PHYS. REV. X 4, 021036 (2014) X∞ ~ πϵ 2 TABLE I. Table of the first five eigenvalues and coefficients d , ˆ 2 −Dk τ k Gτðs ¼ 0; ϵjϵÞ¼ k½JαðkϵÞ e : (34) 2 5 L for D ¼ = . k¼0 ~ k dk λk Using the small z expansion of the Bessel function ∼ 2 α Γ 1 α 0 0.41584 3.5930 JαðzÞ ðz= Þ = ð þ Þ, one finds 1 −0.56973 5.0753 πϵ1þ2α X 2 0.68170 6.3754 ˆ 2αþ1 −Dk2τ Gτ s 0; ϵ ϵ ∼ k e : 3 −0.77287 7.5582 ð ¼ j Þ 4αΓ2 1 α (35) L ð þ Þ k 4 0.85117 8.6567 Notice that replacing Jαð·Þ with J−αð·Þ leads to 1−2α Gτðs ¼ 0; ϵjϵÞ ∼ ϵ , which diverges since α > 1=2;as It is easy to tabulate the eigenvalues λk and the mentioned, this unphysical solution is rejected because of coefficients d~ using Eq. (27) and standard, numerically the absorbing boundary conditions [41]. The usual density k exact techniques. In Table I, we tabulate the first few slopes of states calculation follows the quantization rule of a ~ dk and eigenvalues for D ¼ 2=5. For large k, i.e., large particle in a box extending between 0 and L, which gives 2 P R energy, the 1=p part of the potential is irrelevant, and the kL ¼ nπ with integer n. Hence, → ∞ dk L , and k 0 π eigenvalues λ converge to the eigenvalues of the Airy one finds, after a change of variable to x ¼ k2, k Z problem considered previously. Similarly, we expect 2α 1 1 ϵ þ ∞ dx limk→∞jdkj¼ for any finite D, since in the Airy problem ˆ 0 ϵ ϵ α −Dxτ ~ Gτðs ¼ ; j Þ¼ 2α 2 x e limit, i.e., the D → ∞ case [7,8], jd j is unity for all k. 2 ½Γð1 þ αÞ 0 2 k To complete the calculation, we need to perform an ϵ2αþ1 ¼ ðDτÞ−ð1þαÞ: (36) inverse Laplace transform, namely, invert from sD back to 22αΓð1 þ αÞ χ > 0 (and multiply the PDF by 1=2 if interested in both positive and negative excursions). In Eq. (37),wehave Inserting Eqs. (29) and (36) into Eq. (20), we find our first terms with the structure main result: X∞ X∞ − n νþ2ðnþ1Þ=3 2α 1 3 2 ν 2 3 ~ 2 − λ 2=3τ ν 2 3 2 3 ð ckÞ s pˆ s τ 2 þ Γ 1 α s Dτ = þ = d e D ks : þ = − = ð j Þ¼ ð þ Þ½ ð Þ ð kÞ s expð cks Þ¼ ! : (38) 0 0 n k¼ (37) n¼ ~ In the limit D → ∞, we find jdkj → 1, and the λk’s are Each term on the right-hand side can be inverse trans- the energy eigenvalues of the Airy equation, related to the formed separately since the inverse Laplace transform of zeros of the Airy function; up to a rescaling of s, we get the ðDsÞγ is χ−γ−1=Γð−γÞ. After term-by-term transformation, result obtained by Darling [10] and Louchard [11] for we sum the infinite series (summation over n) using Maple. the area under the Brownian excursion, namely, the Laplace Thus, we arrive at our first destination; the conditional PDF transform of the Airy distribution [8]. We can show that for χ > 0 is found in terms of generalized hypergeometric pˆ ðs ¼ 0jτÞ¼1, as is expected. functions Γ 1 α 4 1=3τ αþ1X 5 2 3ν 4 ν 5 ν 1 2 4 λ3τ3 ð þ Þ D ~ 2 þ D k pðχjτÞ¼− ½d Γ þ ν sin π 2F2 þ ; þ ; ; ; − 2πχ ðχÞ2=3 k 3 3 3 2 6 2 3 3 27χ2 k 1=3λ τ 7 4 3ν 7 ν 5 ν 2 4 4 λ3τ3 D k þ D k − Γ þ ν sin π 2F2 þ ; þ ; ; ; − ðχÞ2=3 3 3 6 2 3 2 3 3 27χ2 1 D1=3λ τ 2 ν 3 ν 4 5 4Dλ3τ3 þ k Γð3 þ νÞ sin ðπνÞ F 2 þ ; þ ; ; ; − k ; (39) 2 χ2=3 2 2 2 2 2 3 3 27χ2 where the summation is over the eigenvalues. In Fig. 5,we representation of the scaling function for 0 < χ < ∞ and 2 5 → ∞ χ plot the solution for D ¼ = and Dpffiffiffiffi corresponding to hence, by symmetry, for all . The scaling variable is the Brownian case. Notice the χ ∼ Dτ3=2 scaling, which χ proves the scaling hypothesis Eq. (17). The sum in Eq. (39) ffiffiffiffi v3=2 ¼ p ; (40) converges quickly as long as χ is not too large, and so we Dτ3=2 can use it to construct a plot of pðχjτÞ. For example, for Fig. 5, only k ¼ 0; …; 4 is needed to obtain excellent and the scaling function Bðv3=2Þ in Eq. (17) can be read convergence. We stress that Eq. (39) gives an explicit directly from Eq. (39). 021036-10 FROM THE AREA UNDER THE BESSEL EXCURSION TO … PHYS. REV. X 4, 021036 (2014) 2 VI. MONTROLL-WEISS EQUATION FOR Brownian Excursion FOURIER-LAPLACE TRANSFORM OF P x;t Simulation: D = ∞ ð Þ 1.5 Bessel Excursion Simulation: D = 0.4 We now use tools developed in the random-walk community [58–61] to relate the joint PDF of a single ( χ | τ ) excursion, Eqs. (13) and (14), p 1 1/2 ) 3 −1 2 −3 2 1 2 3 2 τ ψðχ; τÞ¼gðτÞD = τ = Bðχ=ðD = τ = ÞÞ; (42) D (2 0.5 to the probability density Pðx; tÞ for the entire walk. We find a modified Montroll-Weiss [30,48,62] type of equation 0 -2 -1 0 1 2 for the Fourier-Laplace transform of Pðx; tÞ, χ / (2 τ3)1/2 Z Z D ∞ ∞ Pˆ ðk; uÞ¼ dxeikx dτe−utPðx; tÞ; (43) FIG. 5. The conditional probability pðχjτÞ is described by the Airy −∞ 0 distribution when D → ∞ and otherwise by the distribution derived here based on the area under the Bessel process theory, Eq. (39).The in terms of the Fourier-Laplace transform of ψðχ; τÞ, which distributions describe areas under constrained Brownian (or Bessel) will be denoted ψˆ ðk; uÞ. One modification is that we excursions when the cooling force FðpÞ¼0 [or FðpÞ ≠ 0]. include here the correct treatment of the last jump. Usually, the CTRW model [48] has as an input a single joint PDF of jump lengths and times, while in our case, we have essentially two such functions describing the excur- A. The Bessel meander sions [i.e. Bð:Þ] and the meander [i.e., BMð:Þ]. Generally, As noted, the last zero crossing of the momentum Montroll-Weiss equations are the starting point for deri- process pðtÞ takes place at a random time tn, and hence vation of fractional diffusion equations and the asymptotic at time t, the particle is unlikely to be on the origin of behaviors of the underlying random walks [48]. The momentum. Since the particle, by definition, does not cross original work of Montroll and Weiss [30] assumed that the origin in ðtn;tÞ, its momentum remains either positive there were no correlations between step size χ and the or negative in the backward recurrence time interval waiting time τ, corresponding to a situation called a τ − ¼ t tn (with equal probability). This means that, in decoupled CTRW. The diffusion of atoms in optical lattices this time interval, the motion is described by a meander corresponds to a coupled spatial-temporal random-walk not an excursion. The area under the Brownian meander theory first considered by Scher and Lax [63] (see was investigated previously [8,9,57]. For our purposes, we Refs. [64–66] for recent developments). need to investigate the Bessel meander. This is a Langevin We define ηsðx; tÞdtdx as the probability that the particle path described by Eq. (18) constrained to remain positive, crossed the momentum state p ¼ 0 for the sth time in the which starts at the origin but is free to end with p>0. time interval (t, t þ dt) and that the particle’s position was 3=2 More specifically, the diffusive scaling of χ ∼ ðτ Þ still in the interval (x, x þ dx). This probability is related to the holds. Then, as for the pairs ðχ; τÞ, we have the conditional probability of the previous crossing according to PDF Z Z ∞ t η χ τη − χ − τ sðx; tÞ¼ d d s−1ðx ;t Þ χ −∞ 0 −1=2 −3=2 ffiffiffiffi pMðχ jτ Þ ∼ D ðτ Þ BM p : (41) 1 χ Dðτ Þ3=2 × pffiffiffiffi B pffiffiffiffi gðτÞ; (44) Dτ3=2 Dτ3=2 where we have used Eq. (42). We change variables Here, the subscript M stands for a meander. The scaling 1=2 3=2 according to χ ¼ v3=2D τ and obtain function BMð:Þ is different from Bð:Þ, though clearly both Z Z are symmetric, with mean equal to 0 (since positive and ∞ ∞ pffiffiffiffiffiffiffiffi 3 negative meanders and excursions are equally probable). ηsðx; tÞ¼ dv3=2 dτηs−1ðx − v3=2 Dτ ;t− τÞ −∞ 0 The calculation of BMð:Þ runs parallel to that for the excursion and is presented in Appendix E. Soon we will × Bðv3=2ÞgðτÞ: (45) demonstrate the importance of the meander for specific observables of interest. Having explicit information on Bð:Þ The process is now described by a sequence of waiting and gðτÞ, Eq. (11) (see Appendix A for details), and the times τ1; τ2; … and the corresponding generalized velo- scaling form of BMð:Þ (see Appendix E), we can now cities v3=2ð1Þ;v3=2ð2Þ; …. The displacement in the sth investigate the packet Pðx; tÞ. interval is 021036-11 E. BARKAI, E. AGHION, AND D. A. KESSLER PHYS. REV. X 4, 021036 (2014) pffiffiffiffi 3=2 ˆ χ ¼ v3 2ðsÞ Dðτ Þ : (46) Ψ ðk; uÞ s = s Pˆ ðk; uÞ¼ M : (52) 1 − ψˆ ðk; uÞ The advantage of this representation of the problem in terms of the pair of microscopic stochastic variables τ, v3 2 Ψˆ = Here, Mðk; uÞ pisffiffiffiffi the Fourier-Laplace transform of (instead of the correlated pair τ; χ) is clear from Eq. (45): D−1=2τ−3=2B ðχ= Dτ3=2ÞWðτÞ. Equation (52) relates sta- τ M We may treat v3=2 and as independent random variables tistics of velocity excursions and meanders to the Fourier- τ whose corresponding PDFs are gð Þ and Bðv3=2Þ, Laplace transform of the particle density. The approach is τ 0 −∞ ∞ respectively. Here, > and Here, ηˆsðk; uÞ is the Fourier-Laplace transform of ηsðx; tÞ, In this section, we investigate the small k and small u ˆ ψˆ B the Fourier transform of Bðv3=2Þ, behaviors of ðk; uÞ, which in the next sections will turn Z out to be important in the determination of the long-time pffiffiffiffiffiffiffiffi ∞ pffiffiffiffiffiffiffiffi ˆ 3 3 behavior of Pðx; tÞ. The small k (u) limit corresponds to B k Dτ ¼ exp ikv3=2 Dτ Bðv3=2Þdv3=2; −∞ large distance (time), as is well known [48]. (49) A. k 0 and small u Lˆ Lˆ τ ¼ R andR is the Laplace transform operator ½hð Þ ¼ ∞ ∞ Usingthenormalizationcondition Bðv3 2Þdv3 2 ¼ 1, 0 expð−uτÞhðτÞdτ. By definition, the Fourier-Laplaceffiffiffiffiffiffiffiffi −∞ = = ˆ p 3 transform of ψðχ; τÞ is ψˆ ðk; uÞ¼L½Bˆ ðk Dτ ÞgðτÞ , and it is easy to verify that hence ψ ˆ ðk; uÞjk¼0 ¼ gðuÞ; (54) ηˆsðk; uÞ¼ψˆ ðk; uÞηˆs−1ðk; uÞ: (50) the Laplace transform of thewaiting-time PDF.This waiting- This implies that time PDF is investigated analytically in Appendix A. The s small u behavior of gˆðuÞ differs depending on the value of D. ηˆsðk; uÞ¼½ψˆ ðk; uÞ ; (51) According to Eq. (11),ifD<1, the average waiting time hτi reflecting the renewal property of the underlying random is finite and so walk. Summing the Fourier-Laplace transform of Eq. (47), applying again the convolution theorem for Fourier and gˆðuÞ ∼ 1 − hτiu when D<1: (55) Laplace transforms and using Eq. (51), we find a Montroll- R ∞ Weiss type of equation for the Fourier-Laplace transform Here, hτi¼ 0 τgðτÞdτ is given by a well-known formula of Pðx; tÞ: for the mean first passage time [50]: 021036-12 FROM THE AREA UNDER THE BESSEL EXCURSION TO … PHYS. REV. X 4, 021036 (2014) Z Z ϵ ∞ ˆ 1 − Using BðkÞ, the Fourier transform of Bðv3 2Þ, Eq. (49),we hτi¼ dyeVðyÞ=D e VðzÞ=Ddz; (56) = D 0 y can rewrite Eq. (61) as Z ∞ pffiffiffiffi where VðpÞ is the effective momentum potential, Eq. (8). ψˆ 1 − τ 1 − ˆ τ3=2 τ ðk; uÞju¼0 ¼ d ½ Bðk D Þ gð Þ: (62) Equation (56) reflects the absorbing boundary condition at 0 the origin and a reflecting boundary at infinity. Notice that limϵ→0hτi¼0, as is expected, and the leading-order Taylor This expression is the starting point for the small k expansion yields expansion carried out in Appendix C, which gives for ν < 2,(D>1=5), Z hτi ∼ ϵ: (57) 2 νπ pffiffiffiffi 2D ψˆ ∼ 1 − ν −Γ −ν ν ðk; uÞju¼0 g 3 hjv3=2j ið ð ÞÞ cos 2 j Dkj : Here, Z is the normalizing partition function, Eq. (10). Since (63) Z diverges when D → 1 from below, we see that the average waiting time diverges in that limit. The nonanalytical character of this expansion is responsible For D>1, the mean waiting time is infinite, and so for the anomalous diffusion of the atom’s position. The first Eq. (55) does not hold and instead, as we show in term on the right-hand side is the normalizationffiffiffiffi condition, p ν Appendix A, and the second, proportional to j Dkj , is consistent with the fat-tailed PDF of jump lengths qðχÞ ∝ jχj−ð1þνÞ, α gˆðuÞ ∼ 1 − g jΓð−αÞju ;1=2 < α < 1; (58) Eq. (11), namely, an excursion length whose variance diverges since ν < 2. As expected, this behavior is in full with agreement with Eq. (11) since 4=3 þ β ¼ 1 þ ν.In Eq. (63), we see the noninteger moment of the scaling 2α g ¼ pffiffiffiffi ϵ: (59) function Bð:Þ, 2 2αΓ α Z ð DÞ ð Þ ∞ ν ν hjv3=2j i¼ jv3=2j Bðv3=2Þdv3=2: (64) For large τ, this yields the power-law behavior in Eq. (11), −∞ gðτÞ ∼ g τ−ð1þαÞ, which in fact also describes the tail for Given the formidable structure of the scaling function α > 1. The prefactor g vanishes as ϵ → 0 and, importantly, Bðv3 2Þ, we do not describe here [67] the direct method does not depend on the full shape of the effective potential = to obtain noninteger moments like Eq. (64). Instead, we VðpÞ but rather only on the value of the dimensionless ν present a method that gives hjv3 2j i indirectly. In a future parameter D. The case D ¼ 1 contains logarithmic correc- = publication [67], we will discuss this and other moments tions and will not be discussed here. Using Eqs. (58) and of Bðv3=2Þ. (59), we find nontrivial distributions of the time-interval − 1 ν We can use Eq. (60) together with qðχÞ ∼ q jχj ð þ Þ to straddling time t, and the backward and forward recurrence find, in Fourier space, times, which were previously treated by mathematicians ϵ without the trick. This connection between renewal πq jkjν theory and the statistics of zero-crossings of the Bessel q~ðkÞ ∼ 1 − (65) sinðπνÞΓð1 þ νÞ process is discussed in Appendix F. 2 for ν < 2. In Appendix B, we investigate the area under a B. The u ¼ 0, small k limit Bessel excursion regularized at the origin using a backward Similarly to Eq. (54), by definition, Fokker-Planck equation [6], which gives the amplitude of jump lengths ψˆ ~ ðk; uÞju¼0 ¼ qðkÞ; (60) ν ð3ν − 1Þν q ∼ ϵ : (66) where q~ðkÞ is the Fourier transform of the symmetric jump 32ν−1 2ΓðνÞ length distribution qðχÞ. Alternatively, we can use Eq. (53) 0 Comparing with Eq. (63), we arrive at the following simple withpuffiffiffiffi¼ , and then, upon changing variables according to 3=2 χ=ð Dτ Þ¼v3=2 and using the normalization of the relation: scaling function B v3 2 , we find ð = Þ 3 ν q Z Z hjv3 2j i¼lim : (67) ∞ ∞ pffiffiffiffiffiffi = ϵ→0 ν=2 τ3 D g ψˆ 1 − τ 1 − ikv3=2 D τ ðk; uÞju¼0 ¼ d dv3=2ð e Þgð Þ 0 −∞ Thus, we may use Eqs. (59) and (66) to find the desired νth (61) × Bðv3=2Þ: moment of the scaling function Bð:Þ, 021036-13 E. BARKAI, E. AGHION, AND D. A. KESSLER PHYS. REV. X 4, 021036 (2014) 2α−1 ν 2 ΓðαÞ VIII. THE LÉVY PHASE hjv3=2j i¼ 2ν−1 Γ ν : (68) 3 ð Þ We now explain why Lévy statistics, and hence the generalized central-limit theorem, describes the central In the limit D → ∞,wehaveν ¼ 1=3, and then part of the diffusion profile Pðx; tÞ at long times for 1=5 2 5 83 hðv3 2Þ i ≃ þ : (70) where we have used Eqs. (55) and (65). Here, from Eq. (65) = 6 270D c0 ¼ π=½sinðπν=2ÞΓð1 þ νÞ . Equation (72) corresponds to a decoupling scheme, ψˆ ðk; uÞ ≃ gˆðuÞq~ðkÞ, which, accord- Similarly, for the meander, we find ing to arguments in Ref. [60], is exact in the long-time limit in the regime under investigation. Notice that 1=5 5 τ ϵ As mentioned, both h i and q vanish as approaches zero. 4.5 Exact, Excursion Rearranging, we have Linear Approx., Excursion 4 Simulation, Meander ˆ 1 3.5 Linear Approx., Meander Pðk; uÞ ∼ ν ; (74) u Kν k 〉 þ j j 2 3 3/2 v 〈 2.5 where 2 c0q Kν ¼ lim ; (75) 1.5 ϵ→0 hτi 1 012345and from Eqs. (57) and (66) [26], 1/D 1 π 3ν − 1 ν−1 c0q ð Þ FIG. 6. The variance of the area under the Bessel excursion Kν ¼ lim ¼ : (76) ϵ→0 τ Z πν 32ν−1 Γ ν 2 2 2 1 h i sinð 2 Þ j ð Þj and meander, hðv3=2Þ i and hðv3=2Þ iM, versus =D. The exact solution for the excursion [67] is well approximated by Eq. (70) and the simulation for the meander with Eq. (71) (averages over Kν is called the anomalous diffusion coefficient. When 2 × 105 particles and τ ¼ 105). returning to physical units, we get 021036-14 FROM THE AREA UNDER THE BESSEL EXCURSION TO … PHYS. REV. X 4, 021036 (2014) ν 16 ~ pc 10 D=15 Kν ¼ Kν; (77) να¯ ν−1 D=5 m 14 10 D=3 ν D=0.7 which has units cm = sec. An equivalent expression is 12 10 D=0.5 ν ν=2 ˆ Kν ¼ c0hjv3=2j iD limϵ→0g =3hτi. Pðk; uÞ as given in D=0.3 〉 2 10 Eq. (74) is, in fact, precisely the symmetric Lévy PDF x 10 〈 in Laplace-Fourier space, whose ðx; tÞ presentation [see 8 10 Eq. (B17) of [62]]is 6 10 1 x 4 ∼ 10 Pðx; tÞ 1=ν Lν;0 1=ν (78) ðKνtÞ ðKνtÞ 2 3 4 5 10 10 10 10 t for 2=3 < ν < 2. The properties of the Lévy function FIG. 8. Simulations and theory for the mean-square displace- Lν;0ð:Þ are well known. The Fourier transform of this ν ment versus time nicely match without fitting. The D>1 solution is expð−Kνtjkj Þ for 2=3 < ν < 2, which can x2 ∝ t3 D 15 serve as the working definition of the solution via the Obukhov phase yields h i scaling; hence, for ¼ ,5, 3 the curves are parallel. In the Lévy phase 1=5 0 displacement can be derived from a more rigorous first- 10 principle approach, to which we will turn in Sec. IX. -2 10 A. The diffusion exponent t = 125 Wickenbrock et al. [2] investigated the additional P(x,t) = 1250 -4 t 1/ν 10 effect of a high-frequency (HF) oscillating force F ¼ ) t = 12500 HF t ν ´ ω ϕ K Levy AHF sin ð HFt þ 0Þ on the dynamics of the atoms, where ( ω -6 the frequency HF is much larger than other frequencies in 10 the system. According to Ref. [2], in the limit of a strong drive, the depth of the optical lattice potential is renormal- -8 10 ized, U0 → U0jJ0ð2krÞj, where J0ð:Þ is the Bessel function 0.01 0.1 1 10 100 2 1/ν of the first kind, r A = mω , and k is the laser-field |x| / (K t) ¼ HF ð HFÞ ν wave vector. This elegant setup allows the control of the 1=ν 1=ν transport via the renormalization of the optical depth U0 FIG. 7. ðKνtÞ Pðx; tÞ versus x=ðKνtÞ for ν ¼ 7=6 (D ¼ 2=5). The Lévy PDF, Eq. (78),withKν, Eq. (76), perfectly and, according to Eq. (7), the control of the dimensionless matches simulations without fitting. Notice the cutoff for large x, parameter D. For example, in the vicinity of the zeros of the which is due to the coupling between jump lengths and waiting Bessel function J0, we clearly find an effective shallow times, making x>t3=2 unphysical. lattice, which, according to the theory, corresponds to the 021036-15 E. BARKAI, E. AGHION, AND D. A. KESSLER PHYS. REV. X 4, 021036 (2014) 3.5 can be based on a super-aging velocity-velocity correlation 3 function approach [21,36]. To derive Eq. (78), we assumed in Eq. (74) that u and 2.5 ν Kνjkj are of the same order of magnitude. We now consider a different small k, u limit of Pˆ ðk; uÞ. We first 2 ˆ ξ 2 expand the numerator and denominator of Pðk; uÞ in the 1.5 small parameter k to second order. We leave u fixed and 1 find 0.5 1 2 ˆ 2 1 1 − D v3 2 f1 u k ˆ ∼ 2 hð = Þ iM ð Þ Pðk; uÞ 1 2 ˆ 2 : (81) 0 u 1 þ Dhðv3 2Þ if2ðuÞk 01234 2 = kr This second-order expansion contains the second-orderR 2 ∼ 2ξ 2 ∞ 2 FIG. 9. The anomalous diffusion exponent hx i t versus the moment of the scaling function hðv3=2Þ i¼ −∞ðv3=2Þ high-frequency control parameter kr defined in the text. The 2 Bðv3=2Þdv3=2 and similarly for hðv3=2Þ iM. Thus, the triangles and diamonds are simulation results from Fig. 1 in numerator (denominator) in Eq. (81) contains a term 200 Ref. [2] with U0 ¼ Er (with varying photon scattering rate; describing the meander (excursions) contribution. Hence, see details in Ref. [2]). The curve is the theory, Eqs. (7) and (80), from symmetry hv3=2i¼0, the expansion does not contain with a renormalized U0 as explained in the text with c ¼ 34.7. The transition between normal 2ξ ¼ 1, Lévy 1 < 2ξ < 3, and linear terms. Here, Richardson 2ξ ¼ 3 behaviors is clearly visible, and as shown by 1 3 Wickenbrock et al., it is controlled by the high-frequency field. ˆ − d ˆ f1ðuÞ¼ 3 WðuÞ; (82) Wˆ ðuÞ du Richardson phase. In the experiment [2], resonances in the ˆ transport are observed close to the zeros of the Bessel where WðuÞ¼½1 − gˆðuÞ =u is the Laplace transform of functions, namely, an enhanced spreading of the atoms. the survival probability WðτÞ and However, as pointed out in Ref. [2], many superdiffusing d3 atoms are lost, which leads to an underestimate of the 3 ˆ ˆ du gðuÞ f2ðuÞ¼− : (83) diffusion exponent. In Ref. [2], the diffusion exponent was 1 − gˆðuÞ found using Monte Carlo simulation (see Fig. 1 of that reference). To demonstrate the predictive power of our The third-order derivative with respect to u is clearly theory, at least for exponents, we compare between theory related to the χ ∝ τ3=2 scaling we have found and to the and the numerics [2]. As mentioned in our summary, we second-order expansion. While the k2 expansion in postpone a comparison of experiments to theory until the Eq. (81) works fine for small k and finite u,when losses become insignificant. u → 0 we get divergences. For example, when α < 1 From Eq. (80) and within the Lévy phase, we find α 2 2ξ we have gˆðuÞ ∼ 1 − Gu þ , and hence the third-order superdiffusion and x ∝ t with 2ξ 4 − 3ν=2 −3 h i ¼ ¼ derivative of gˆðuÞ diverges as u as u → 0.For 7=2 − 1=ð2DÞ for 1=5 021036-16 FROM THE AREA UNDER THE BESSEL EXCURSION TO … PHYS. REV. X 4, 021036 (2014) 1 2 D 2 ˆ 2 ˆ hxˆ ðuÞi ¼ lim ½hðv3 2Þ i f1ðuÞþhðv3 2Þ if2ðuÞ : ϵ→0 u = M = 0.8 (85) ] 3 The second moment is finite because of the observed fast Dt 0.6 decay of the scaling function Bðv3=2Þ for v3=2 ≫ 1, ensur- 2 2 /[(2/3) 〉 ing that hðv3=2Þ i and hðv3=2Þ iM are both finite. 2 0.4 x 〈 A. Obukhov-Richardson diffusion 0.2 When α < 1 (D>1), we have gˆðuÞ ∼ 1 − Guα þ , Γ −α ˆ ∼ 0 where G ¼ g j ð Þj. Using Eqs. (82) and (83), f1ðuÞ 0 0.2 0.4 0.6 0.8 1 −3 ˆ ∼ −3 c1u and f2 c2u for small u with 1/D c1 ¼ð1 − αÞð2 − αÞð3 − αÞ; FIG. 10. Scaled mean-square displacement (filled circles) for α 1 − α 2 − α (86) the Richardson-Obukhov phase. Simulations and theory (solid c2 ¼ ð Þð Þ: line) nicely match without fitting. Close to the transition to the → 1 ˆ Lévy phase, i.e., D , the finite time simulations slowly The small k, u expansion of Pðk; uÞ, Eq. (81),is converge to the asymptotic limit, as might be expected. In the simulations, we used t ¼ 105 and averaged over 105 particles. 1 2 −3 2 1 1 − D v3 2 c1u k ˆ ∼ 2 hð = Þ iM Pðk; uÞ 1 2 −3 2 ; (87) u 1 þ 2 Dhðv3=2Þ ic2u k excursions is large. Notice, however, that for the calculation ϵ which is g and independent. The mean-square displace- of the Lévy density Pðx; tÞ, Eq. (78), the statistics of the ment in the small u limit is meander do not enter. Hence, depending on the observable of interest, and the value of D, the meander may be either a ˆ2 ∼ 2 2 −4 hx ðuÞi D½c1hðv3=2Þ iM þ c2hðv3=2Þ i u : (88) relevant part of the theory or not. Converting to the time domain, B. Superdiffusion 3 1 α 2 1 3 1 2 ∼ 2 2 Dt For < < ( = 021036-17 E. BARKAI, E. AGHION, AND D. A. KESSLER PHYS. REV. X 4, 021036 (2014) 14 When α > 3, we have the expansion gˆðuÞ ∼ 1 − uhτiþ u2hτ2i=2 − u3hτ3i=6 þ , where the first three integer 12 moments of the waiting-time PDF are finite (see ˆ 10 Appendix A). Then, the function f1ðuÞ is negligible when ˆ 3 α u → 0, while f2ðuÞ ∼ hτ i=uhτi. We find 4- t 8 / 〉 2 x 3 〈 6 2 2 hτ i hx i ∼ Dhðv3 2Þ ilim t: (94) = ϵ→0 τ 4 h i 2 Thus, as stated above, α ¼ 3 (or D ¼ 1=5) marks the transition between the anomalous superdiffusive phase and 0 12345 the normal diffusion phase. Furthermore, in this case, the 1/D meander is of no importance. However, Eq. (94) is correct only if our assumptions FIG. 11. Scaled mean-square displacement in the Lévy phase. about scaling are valid. By definition, In the simulations (filled circles), we used t ¼ 105 and averaged 5 Z Z over 10 particles; to reduce fluctuations in the vicinity of the ∞ ∞ transition to the Gaussian phase, we averaged over trajectory hχ2i¼ ψðχ; τÞχ2dχdτ; (95) times in the range 103–105 (last two points). The solid line is the 0 −∞ prediction of Eq. (93). and if the scaling hypothesis holds, Z Z pffiffiffiffi ∞ ∞ χ τ3=2 The same result is valid for 2 < α < 3. This behavior χ2 τ Bð =ffiffiffiffiD Þ χ2 χ τ h i¼ gð Þ p 3 2 d d ; (96) depends on the normalizing partition function Z; 0 −∞ Dτ = namely, the shape of the potential VðpÞ in the vicinity of the momentum origin becomes important, unlike the which gives Richardson-Obukhov phase, where only the large p behav- χ2 2 τ3 ior of VðpÞ is important in the long-time limit [i.e., the case h i¼Dhðv3=2Þ ih i: (97) α < 1, Eq. (89)]. Notice that jc1j in Eq. (93) tends to zero when α → 3 from below. This means that the meander Inserting Eq. (97) into (94), we get the expected result becomes irrelevant when we approach the normal diffusion discussed in the Introduction, phase x2 ∼ t. Figure 11 compares simulation and theory 2 4−α hχ2i and demonstrates that hx i=t diverges as the transition to hx2i ∼ lim t: (98) the Gaussian phase D<1=5 is approached. ϵ→0 hτi C. Breakdown of scaling This simple result is the correct one, even though the assumption—normal diffusion scaling assumption is not valid in this regime. Sometimes Our starting point was the scaling hypothesis χ2 ∼ τ3, Eq. (14) and Fig. 3. Indeed, we have shown that for large χ and τ, the scaling function Bð:Þ describes the conditional probability density pðχjτÞ for all values of D. However, this does not imply that the scaling solution Bð:Þ is always relevant for the calculation of the particle density Pðx; tÞ. Roughly speaking, so far we have assumed that large jumps χ and long waiting times τ dominate the underlying process. No one guarantees, however, that this large χ limit (long jumps) is important while small χ (small jumps) can be neglected. Indeed, when we switch over to the normal diffusion phase D<1=5, the small-scale aspects of the process become important [meaning that the regulari- zation of the potential VðpÞ when p → 0 becomes crucial]. This is similar to the Lévy versus Gaussian central-limit 2 3 FIG. 12. We show 2Dlimϵ→0hðv3=2Þ i=hτ i − 1 versus 1=D, theorem arguments, where the former is controlled by the which is zero if scaling holds, and D<1=5. We see that scaling tails of the distribution while the latter is controlled by the holds only when D → 1=5, which marks the transition from variance. Let us see this breakdown of scaling in more normal to the Lévy type of diffusion. For D>1=5, scaling holds detail. since then hχ2i and hτ3i diverge. 021036-18 FROM THE AREA UNDER THE BESSEL EXCURSION TO … PHYS. REV. X 4, 021036 (2014) we reach truthful conclusions, even though the assumptions Kolmogorov’s theory of 1941 [see Eq. (3) in Ref. [69]] and along the way are invalid. To see the breakdown of scaling, to Richardson’s diffusion, hx2i ∼ t3 [70]. Equation (99) is χ2 2 τ3 −1 we plot, in Fig. 12, Ydev ¼ limϵ→0h i=½Dhðv3=2Þ ih i valid when the optical potential depth is small since versus 1=D, which should be zero in the normal phase if D → ∞ when U0 → 0. This limit should be taken with the scaling hypothesis holds (where hχ2i and hτ3i are care, as the observation time must be made large before finite). The calculations of hχ2i and hτ3i are given in considering the limit of weak potential. In the opposite Appendixes A and B. We see that at the transition point, scenario, U0 → 0 before t → ∞, we expect ballistic motion 1 5 0 ≠ 0 D ¼ = , Ydev ¼ , but otherwise Ydev . Hence, the jxj ∼ t since in this case the optical lattice has not had time scaling hypothesis does not work in the normal phase to make itself felt [3]. Physically, the atoms in this phase D<1=5. This means we need another approach for normal perform a random walk in momentum space due to random diffusion, which, luckily, is easy to handle. For D>1=5, emission events, which in turn give the x ∼ t3=2 scaling. For long jumps dominate since hχ2i diverges, and then our shallow lattices, the Sisyphus cooling mechanism breaks scaling theory works fine. down, in the sense that transitions from maximum to minimum of the potential field created by the laser fields X. THE RICHARDSON-OBUKHOV PHASE are not preferred over the inverse transitions. Thus, the deterministic dissipation is not effective, and In Sec. VIII, we discussed the Lévy phase that is found we are left with Brownian scaling in momentum for 1=5 P (x,t) t = 128000 0.4 variance of the jump lengths. This variance is investigated in Appendix B using a backward Fokker-Planck 0.2 equation. In the small k, u limit, ψˆ ðk; uÞ ∼ 1 − uhτiþ k2hχ2i=2 þ , and the Montroll-Weiss equation (52) is 0 -3 -2 -1 0 1 2 3 3/2 1 x / t ˆ ∼ Pðk; uÞ 2 : (100) u þ K2k FIG. 13. Pðx; tÞ versus x=t3=2 obtained from numerical simu- lation for D ¼ 20=3. Increasing time, the solution converges to a scaling function. Such a x ∼ t3=2 Richardson-Obukhov scaling is This is the expected Gaussian behavior for the position found for D>1. probability density 021036-19 E. BARKAI, E. AGHION, AND D. A. KESSLER PHYS. REV. X 4, 021036 (2014) 1 x2 coupled joint distribution of waiting-time and jump length Pðx; tÞ ∼ pffiffiffiffiffiffiffiffiffiffiffiffiffi exp − ; (101) 4πK2t 4K2t is the microscopic ingredient for a coupled continuous-time random walk [48,58–63]. These distributions, however, are and difficult to obtain from first-principle calculations and are usually treated in a simplified manner. For example, hχ2i postulating ψðχ; τÞ¼gðτÞδðjχj − τÞ=2 is common in sto- K2 ¼ lim (102) ϵ→0 2hτi chastic theories. Hence, we provide an important pillar for the foundation of this widely applied approach. is the diffusion constant, namely, Our work gives the relation between velocity excursions and random-walk theory and hence diffusion phenomena. 2 hx ðtÞi ∼ 2K2t: (103) Since zero crossing in velocity space is obviously a very general feature of physical paths of random processes, we Equation (102) relates the statistics of the excursions, expect that the areal distributions of excursions and 2 i.e., the variance of the area under the excursion hχ i meanders will play an important role in other systems, and its average duration hτi to the diffusion constant (here, at least as a tool for the calculation of transport and hχi¼0). As noted in the Introduction, the equation has diffusion (e.g., in principle, our approach can also treat the structure of the famous Einstein relation, relating the the case of a constant applied force, where a mean net flow variance of jump size and the time between jumps to the is induced). The celebrated Montroll-Weiss equation needs diffusion. two important modifications. First, since the underlying 2 While hτi [Eq. (57)] and hχ i [Eq. (B22)] approach zero process is continuous, we regularized the process, such that when ϵ → 0, their ratio remains finite and gives the excursions and meander start at ϵ → 0. Second, the Z Z statistics of the last jump event, i.e., the meander, must be 1 ∞ ∞ 2 VðpÞ=D 0 −Vðp0Þ=D 0 treated with care. In contrast, usually it is assumed that only K2 ¼ dpe dp e p : (104) DZ −∞ p ψðχ; τÞ is needed for a microscopical description of a continuous-time random-walk model. This and the corre- This equation was derived previously using different lations between χ and τ make the problem challenging, and approaches [20,36]. The diffusion constant K2 diverges as usual approaches to diffusion fail. With our approach, the D → 1=5 from below, indicating the transition to the super- behavior of the packet is mapped onto a problem of the diffusive phase. A sharp increase in the diffusion constant K2 calculation of areas under Bessel excursions and meanders. as the intensity of the laser reaches a critical value was For example, we derived the relation between the mean- demonstrated experimentally [20]. Equation (104) can be square displacement of the atoms and the areas under both derived using the Green-Kubo formalism [6], so in the the Bessel excursion and meander, Eqs. (85), (89), and (93). normal phase, the analysis of the statistics of excursions A decoupled scheme that neglects the correlations is an alternative to the usual methods. It seems that in the ψðχ; τÞ ≃ gðτÞqðχÞ gives a diverging result for ν < 2, Lévy phase, the analysis of statistics of excursions is vital. which is unphysical. Specifically, the usual Green-Kubo formalism breaks down In the regime 1=5 021036-20 FROM THE AREA UNDER THE BESSEL EXCURSION TO … PHYS. REV. X 4, 021036 (2014) derivative. The fractional space derivative ∇ν is a Weyl- and B) of the nonregularized process, i.e., a logarithmic Reitz fractional derivative [48]. To see the connection potential for any p, some (but not all) of the ϵ cancellation between our results and the fractional equation, we reex- will not take place [see Eq. (67) where both q ∝ ϵ and press Eq. (74) as g ∝ ϵ for the regularized process, which allows for the cancellation]. For example, both for the regularized and the ˆ ν ˆ τ ∼ τ−ð1þαÞ uPðk; uÞ − 1 ¼ −Kνjkj Pðk; uÞ; (106) nonregularized processes, gð Þ g (so the expo- nent α is identical in both cases); however, g ∝ ϵ for the 2α which is the Fourier-Laplace transform of Eq. (105). Here, regularized process, while g ∝ ϵ for the nonregularized we recall [48] that the Fourier-space representation of ∇ν is case, Eq. (A7). The cancellation of ϵ is further discussed −jkjν and that uPˆ ðk; uÞ − 1 is the representation of the time in Appendix F for three types of random durations. That derivative in Laplace-Fourier space of a δ-function initial appendix shows that, while the first passage time PDF gðτÞ ϵ condition centered on the origin. We see that Pðx; tÞ for depends on (see Appendix A), the time interval straddling 2=3 < ν < 2 satisfies the fractional diffusion equation. In time t and the backward and forward recurrence times other words, for initial conditions starting on the origin, the are insensitive to this parameter. Thus, one upshot of solution of Eq. (105) is the Lévy PDF, Eq. (78). Appendix F is a further demonstration that this intuitive ϵ However, the use of the fractional diffusion equation renewal trick actually works. A rigorous mathematical must be performed with care. It predicts a seemingly treatment of the problem will lead to a stronger foundation unphysical behavior: The mean-square displacement is of our results. apparently infinite. Although, as mentioned in the Regularization of the process is crucially important in the 1 Introduction, the mean-square displacement was shown Lévy and Gaussian phase D< where observables depend to exhibit superdiffusion, experimentally by Katori et al. [1] on the details of the potential VðpÞ. This is related to work and in simulations in [2], it is nevertheless finite at all finite of Martin et al. [41] on the classification of boundaries for times. In fact, the coupling between excursion duration τ the nonregularized Bessel process. They showed that, using and extent χ analyzed herein implies that the fractional our notation, for D<1, p ¼ 0 is an exit boundary while 3 2 1 0 equation is valid only in a scaling region, jxj 021036-21 E. BARKAI, E. AGHION, AND D. A. KESSLER PHYS. REV. X 4, 021036 (2014) the system to one of its boundaries. Previous theoretical the depth of the optical lattice approaches zero, since work on first passage times for Lévy walks and flights clearly in the absence of an optical lattice, the fastest might give a useful first insight [73–77]; however, it is left particles are ballistic. for future work to compare these simplified models with (ii) While our theory correctly predicts a Lévy phase in the microscopic semiclassical picture of the underlying agreement with experiments, in Ref. [3] two exponents dynamics. Evaporation is an important ingredient of cool- were used to fit the data. In contrast, the theory we ing (beyond the Sisyphus cooling) since it gets rid of the developed suggests a single Lévy exponent ν. This could very energetic particles; hence, this line of research has be due to the leakage of particles and also to the t3=2 cutoff practical applications. Theoretically, understanding the of the Lévy density, which are due to the correlations boundary conditions for fractional diffusion equations, between χ and τ investigated in this manuscript. Avoiding needed for the calculation of statistics of first passage the leakage of particles is an experimental challenge, and times, is still an open challenge. Yet another challenge is to once this is accomplished, a more informed comparison of characterize the joint PDF Wðx; p; tÞ and, in particular, the our theoretical prediction on the Lévy phase with a single correlations between position and momentum, which are characteristic exponent ν could be made. Success on this expected to be nontrivial. Again, some elegant ideas and front would provide an elegant demonstration of Lévy’s tools were recently developed [78] within the Lévy walk central-limit theorem, with the characteristic exponent framework; however, more work is needed for direct controlled by the depth of the optical potential. comparison with the physical picture under consideration. (iii) As pointed out in Ref. [2], losses of atoms lead to an underestimate of the diffusion exponent, as many super- B. Cold-atom experiments diffusing atoms are lost. Characterization of these losses both Experiments on ultracold diffusing particles in optical theoretically and experimentally could advance the field, lattices have been performed on a single 24Mgþ ion [1] and since this yields insight on the underlying processes and an ensemble of Rb atoms [2,3,20]. Experiments on the leads us towards better control of the particles. Specifically, spreading of the density of atoms yield ensemble averages we do not know how many particles are kept in the system, like the mean-square displacement and the density Pðx; tÞ. versus time, for varying strength of the optical potential. Analysis of individual trajectories yields, in principle, (iv) In the Lévy phase, measurements of noninteger deep insight on paths, for example, information on the moments hjxjqi with q<ν < 2 will, according to theory, first passage time in velocity space, i.e., gðτÞ, zero crossing exhibit Lévy scaling hjxjqi ∼ tq=ν. In contrast, the main times, statistics of excursions, meanders, etc. For a com- focus of experiments so far was the second moment q ¼ 2, prehensive understanding of the dynamics of the particles, which is difficult to determine statistically since, as we have both types of experiments are needed. The theory presented shown here, it depends on the tails of Pðx; tÞ and very fast herein provides statistical information on the stochastic particles. Thus, ideally, a wide spectrum of moments hjxjqi trajectories, e.g., zero crossing events, which in turn are should be recorded in experiment, with the low-order related to the spreading of the packet of particles. Thus, moments q<ν giving information on the center part of τ from single trajectories, one may estimate gð Þ, Bðv3=2Þ, the density and the higher-order moments giving informa- and ψðχ; τÞ. From gðτÞ, one may then obtain the exponent tion on the correlations and tails. α, which can be used to predict the qualitative features of In the case where experiment and theory do not reconcile, the spreading of the ensemble of particles. For example, we will have a strong indication that the current semiclassical with an estimate of α, we can determine the phase of theory is not sufficient; then, we will be forced to investigate motion, be it Richardson, Lévy, or Gauss. Of course, a at least four other aspects of the problem: more quantitative investigation is now possible since we (a) Effects of collisions on anomalous diffusion of have analytically related α, the mean-square displacement, the atoms. and Kν with microscopical parameters like the optical (b) Effect of higher dimensions. potential depth U0. In principle, statistics of Bessel excur- (c) Quantum effects beyond the semiclassical approach sions, so far of great interest mainly in the mathematical used here. In particular, it would be very interesting to literature, could be detected in single-particle experiments. simulate this system with full quantum Monte Carlo These single-ion experiments are also ideal in the sense simulations [6], to compare the semiclassical theory with that collisions/interparticle interactions do not play a role, quantum dynamics. We note that the Richardson phase, and they also provide insights on ergodicity. Three specific which, as mentioned, was not observed in experiments, is unsolved issues are as follows: actually a heating phase, and quantum simulations become (i) Can the Richardson phase be experimentally dem- difficult because the numerical lattice introduces a cutoff on onstrated? So far, this phase was obtained in our theory and velocities which induces artificial ballistic motion. Monte Carlo simulations [2], while experiments exhibit (d) Other cutoff effects that modify the anomalous ballistic diffusion as the upper limit [1–3]. This might be diffusion. For example. at high enough velocities, related to the subtle limit of taking time to infinity before Doppler cooling is expected to diminish the fast particles. 021036-22 FROM THE AREA UNDER THE BESSEL EXCURSION TO … PHYS. REV. X 4, 021036 (2014) In Ref. [79], we simulated the effect of Doppler friction namely, the probability that a particle initially at pi does and showed that the anomalous character of the diffusion is not cross the boundary pf ACKNOWLEDGMENTS 2 2 1 ðp Þ − ðp Þ 1 hτ i¼ i f : (A5) This work was supported by the Israel Science B 4 α − 1 D Foundation. We thank Yoav Sagi and Nir Davidson for discussions on the experiment [3] and Andreas Dechant Notice that hτBi diverges when α → 1, corresponding to and Eric Lutz for collaboration on related problems. D → 1. Not surprisingly, the average time in Eq. (A5) is generally different from the expression for the average waiting time for the regularized process, Eqs. (56) and (57). APPENDIX A: THE WAITING-TIME PDF Expanding Eq. (A4) for small u for the case 0 < α < 1, The waiting time is the time it takes the particle starting we get with momentum p > 0 to reach p 021036-23 E. BARKAI, E. AGHION, AND D. A. KESSLER PHYS. REV. X 4, 021036 (2014) rffiffiffiffi One can show that this fat-tail behavior is also valid for 2 1 u α α 1 C3ðuÞ ≈ (A13) the regime > . The procedure involves expansion of ΓðαÞ 2 D Eq. (A4) beyond the first two leading terms; for example, α for 1 < α < 2, one finds gˆ ðuÞ¼1 − uhτ iþCu …:, and −1=2 B B to leading order in u.For1 ≪ pi ≪ u , our two the third term gives the tail of the PDF. Note that the approximations must agree, and so we find, using the τ PDF gBð Þ yields the same exponent as in Eq. (11). second-order expansion of KαðzÞ, However, to calculate the amplitude g , we must consider rffiffiffiffi the regularized process. Γ 1−α 2α 1=Dþ1 ð Þ u pi 1− p ≈ 1þC1ðuÞ ; (A14) Γð1þαÞ i D 1=Dþ1 2. First passage time for the regularized process For the optical lattice problem, we need to treat the and recognizing that ð1=D þ 1Þ¼2α, 2 regularized force FðpÞ¼−p=ð1 þ p Þ. Our aim is to find the asymptotic behavior in Eq. (11), while the average 2αΓð1 − αÞ u α C1ðuÞ ≈ − : (A15) waiting time is given already in Eqs. (56) and (57). From Γð1 þ αÞ 4D these, we know that for α < 1 the average first passage time from pi to pf is infinite (the derivation can be Thus, using Eq. (A11), easily generalized for arbitrary initial and final states). Z Furthermore, for long times, the large-momentum behavior Γ 1 − α 1 α p ð Þ i 0 α of F p plays the crucial role, and hence we expect, for gˆðuÞ ≈ 1 − 2 eVðp Þ=Ddp0 u : ð Þ ΓðαÞ 4D small u, pf (A16) gˆðuÞ ≈ 1 − Guα;0< α < 1; (A8) Using the Tauberian theorem, which implies α → τ−ð1þαÞ Γ −α τ where we must determine G, which then gives the ampli- u = ð Þ, we find the large behavior tude g introduced in Eq. (11). The equation for gˆðuÞ is the R 0 same as Eq. (A3) but with the regularized force 2α pi eVðp Þ=Ddp0 τ ∼ pf τ−ð1þαÞ gð Þ Γ α 4 α : (A17) p ð Þ ð DÞ Dð∂ Þ2gˆðuÞ − i ∂ gˆðuÞ − ugˆðuÞ¼0: (A9) pi 1 þðp Þ2 pi i It can be shown, with additional work, that this result is also valid for α > 1. To conclude, we find p ¼ 0 for the final We need to solve for gˆðuÞ for small u, so to leading order, f state and p ϵ ≪ 1 for the initial state, we drop the last term and find i ¼ Z 2α ϵ 2αϵ ∂ 2 ˆ ∂ ˆ 0 Vðp0Þ=D 0 Dð p Þ gðuÞþFðpiÞ p gðuÞ¼ : (A10) g ¼ α e dp ≈ α : (A18) i i ð4DÞ ΓðαÞ 0 ð4DÞ ΓðαÞ Hence, This, of course, vanishes as ϵ → 0, in this case linearly, as Z → ϵ 0 p opposed to the pi , pf ¼ limit of Eq. (A7), where the i Vðp0Þ=D 0 gˆðuÞ¼C1ðuÞ e dp þ C2ðuÞ; (A11) dependence is of higher order. As shown in this paper, for pf the purpose of calculation of the asymptotic behavior of Pðx; tÞ, all we need is ∂ϵg when ϵ → 0, which is a finite 1 2 1=2 ˆ 1 where VðpÞ¼lnð þp Þ . Since gðuÞ¼ when pi ¼ pf, constant that depends on D only. The full shape of VðpÞ is 1 C2ðuÞ¼ . This approximation breaks down, however, unimportant for the calculation of g , indicating a degree ˆ ∝ 1þ1=D when pi is too large since then gðuÞ pi , and the last of universality. In contrast, hτi depends on Z and so is term in Eq. (A9) is comparable in size to the first two when nonuniversal. Since g is a measure of the long-time ∼ −1=2 ≫ 1 pi u . In the large pi regime, however, pi ,sowe behavior, the detailed shape of the potential is not impor- ≈ −1 can approximate FðpiÞ by its Bessel form, FðpiÞ =pi, tant, provided it is regularized. and the solution is, as above, rffiffiffiffi 3. Moments of gðτÞ ˆ ≈ u α gðuÞ C3ðuÞKα pi p ; (A12) τ τ3 D i In the text, we show that h i and h i are relevant when D<1=5, so these averages do not diverge. The moments hτi and hτ3i are found using where we cannot use the pf → pi limit to normalize gˆðuÞ since pf is not necessarily large. Nevertheless, gˆðu ¼ 0Þ¼1 −α ˆ 1 − τ 2 τ2 2 − 3 τ3 6 and KαðzÞ ∼ ðΓðαÞ=2Þðz=2Þ for z → 0, implying that gðuÞ¼ uh iþu h i= u h i= þ ; (A19) 021036-24 FROM THE AREA UNDER THE BESSEL EXCURSION TO … PHYS. REV. X 4, 021036 (2014) where gˆðuÞ is the Laplace transform of the waiting-time 1. PDF of jump lengths for the Bessel process τ PDF gð Þ. From As mentioned, for the Bessel process, FðpÞ¼−1=p. The PDF qBðχÞ, which depends of course on the start and p ∂ 2 − ∂ − ˆ 0 end points pi and pf, satisfies the following backward Dð pÞ 2 p u gðuÞ¼ ; (A20) 1 þ p equation [50]: we get ∂ 2 − 1 ∂ − ∂ 0 Dð pi Þ qB ð =piÞ pi qB pi χqB ¼ ; (B1) p where the subscript B again stands for Bessel. Since χ > 0, Dð∂ Þ2 − ∂ hτi¼−1; (A21) p 1 þ p2 p we define the Laplace transform Z ∞ p ˆ −χ χ χ Dð∂ Þ2 − ∂ hτ2i¼−2hτi; (A22) qBðsÞ¼ expð sÞqBð Þd : (B2) p 1 þ p2 p 0 → χ → δ χ χ 0 When pf pi, we have qBð Þ ð Þ and qBð Þχ¼0 ¼ ∂ 2 − p ∂ τ3 −3 τ2 if p ≠ p since it takes time for the particle to reach the Dð pÞ 2 p h i¼ h i: (A23) i f 1 þ p boundary, and hence we cannot get an excursion whose size is zero. Of course, in the opposite limit of large χ, Solving Eq. (A21), χ 0 limχ→∞qBð Þ¼ . In Laplace space, Eq. (B1) yields Z Z p ∞ 2 1 − ð∂ Þ qˆ ðsÞ − ð1=Dp Þ∂ qˆ ðsÞ − ðp s=DÞqˆ ðsÞ¼0: hτi¼ dyeVðyÞ=D e VðxÞ=Ddx; (A24) pi B i pi B i B D 0 y (B3) so hτi¼0 when p ¼ 0, as expected for the first passage It is easy to verify that the appropriate solution is from p to the origin. For the second and third moments, pffiffiffi 3ν=2 2 s 3=2 ðp Þ Kνð ðp Þ Þ Z Z ˆ i 3 pDffiffiffi i ∞ qBðsÞ¼ 3ν=2 2 s 3=2 ; (B4) 2 p ðp Þ Kνð ðp Þ Þ hτ2i¼ dyeVðyÞ=D e−VðxÞ=DhτðxÞidx (A25) f 3 D f D 0 y with ν ¼ð1 þ DÞ=3D, and as in the previous appendix, and Kνð·Þ is the modified Bessel function of the second kind. Our goal is to find the large χ behavior of qBðχÞ,sowe Z Z 0 ν 1 p ∞ expand Eq. (B4) in the small s limit, finding, for < < , 3 3 − 2 hτ i¼ dyeVðyÞ=D e VðxÞ=Dhτ ðxÞidx: (A26) D 0 y 1 2ν Γ 1−ν ˆ ∼1− 3ν − 3ν ð Þ 3ν−1 ν ν qBðsÞ 3 ½ðpiÞ ðpfÞ Γ 1 ν ð Þ s : ϵ → 0 ð þ Þ Thus, in the p ¼ limit, (B5) Z 3ϵ ∞ hτ3i ∼ e−VðzÞ=Dhτ2ðzÞidz: (A27) The first term on the left-hand side is simply the normali- D 0 zation, and the sν term indicates that the average excursion length diverges since ν < 1. Passing from s to χ, we get for large χ, 2ν 3ν − 3ν 1 ðpiÞ ðpfÞ ν −1−ν APPENDIX B: THE JUMP LENGTH PDF qðχÞ q ðχÞ ∼ ð3ν − 1Þ χ : (B6) B 3 ΓðνÞ The excursion length χ is the distance the particle travels from its beginning with momentum pi > 0 until it reaches With a similar method, one can show that Eq. (B6) also the momentum origin pf ¼ 0 for the first time. Here, we holds for 1 < ν < 2. There, the expansion, Eq. (B5), investigate its PDF qðχÞ, which in the limit of pi → 0 and contains three terms; the additional term yields the average for the regularized force field, gives the jump length PDF. of χ, which is now finite. In Ref. [6], a more complicated Previously, this PDF was investigated for the −1=p force method was used to investigate the same problem, and a field in Ref. [6]. Note that, in what follows, pi > 0 so similar result was found, although with a typographical χ > 0. As elsewhere in this paper, from symmetry, positive error. That reference reports ð3ν þ 1Þν in the prefactor, and negative χ are equally probable, so we may restrict our whereas we find ð3ν − 1Þν, a difference with some impor- attention to χ > 0. tance since the prefactor found here goes to zero when 021036-25 E. BARKAI, E. AGHION, AND D. A. KESSLER PHYS. REV. X 4, 021036 (2014) D → ∞, which is necessary to obtain reasonable physical 3. The variance of χ results. Here, we obtain the finite hχ2i for the case ν > 2 for the regularized force FðpÞ¼−p=ð1 þ p2Þ. Using the Laplace 2. PDF of χ for the regularized process χ → s transform of Eq. (B7), we find the following back- ˆ To obtain the large χ behavior of qðχÞ for the regularized ward equation for qðsÞ: force FðpÞ¼−p=ð1 þ p2Þ, we follow the same steps ∂2 ∂ carried out in Appendix A2. The equation to solve is ˆ ˆ − ˆ 0 D 2 qðsÞþFðpiÞ ∂ qðsÞ pisqðsÞ¼ : (B14) ∂ðpiÞ pi 2 Dð∂ Þ q þ Fðp Þ∂ q − p ∂χq ¼ 0: (B7) pi i pi i χ 0 Here, we used qð Þjχ¼0 ¼ since the particle starting with 0 For ν < 1, we switch to Laplace space χ → s, i.e., pi > cannot reach zero momentum without traveling ∂ 2 ˆ ∂ ˆ − ˆ 0 some finite distance. The Laplace transform qˆðsÞ is Dð pi Þ q þ FðpiÞ pi q pisq ¼ , and drop the last term for small s, yielding expanded in s: Z Z ∞ p − χ 2 2 i 0 ˆ s χ χ 1 − χ χ 2 ˆ ≈ Vðp Þ=D 0 qðsÞ¼ e qð Þd ¼ sh iþs h i= þ : qðsÞ C1ðsÞ e dp (B8) 0 p f (B15) after applying the boundary condition qˆj → ¼ 1. Again, pf pi Here, hχi is the average jump size, for positive excursions, for p ≫ 1, we can approximate Fðp Þ by its large p i i i i.e., those that start with p > 0, and hχ2i is the second approximation, yielding i moment. Note that in the original model, we have both rffiffiffiffiffiffiffiffi positive and negative excursions, and so we have p ¼þϵ 2 3 i 3ν=2 spi or pi ¼ −ϵ with probability 1=2; hence, for that case qˆðsÞ ≈ C3ðsÞp Kν : (B9) i 3 D −∞ < χ < ∞ and from symmetry hχi¼0. The variance of the original process is hχ2i because of symmetry. As we 0 χ 0 χ In the small s limit, qðs → 0Þ¼1 implies restrict ourselves to pi > , > and h i in Eq. (B15) is finite. rffiffiffiffi 2 1 s ν Inserting Eq. (B15) into Eq. (B14), we find C3ðsÞ ≈ (B10) ΓðνÞ 3 D ∂2 hχ2i D −shχiþs2 þ ∂ðp Þ2 2 to leading order. These two approximations must agree for i −1=3 2 1 ≪ pi ≪ s , yielding p ∂ hχ i − i −shχiþs2 þ 1 þðp Þ2 ∂p 2 i i Γð1 − νÞ 2 ≈ 31−2ν 3ν − 1 ν ν hχ i C1ðsÞ ð Þ Γ ν s : (B11) − 1 − χ 2 0 (B16) ð Þ pis sh iþs 2 þ ¼ : This implies that 1 The s terms give Z ν p 1 ð3ν − 1Þ i 0 2 χ ≈ ν Vðp Þ=D 0 χ−1−ν ∂ p ∂ qð Þ 2ν−1 e dp : (B12) − χ i χ − 0 3 Γ ν D 2 h iþ 2 h i pi ¼ ; (B17) ð Þ pf ∂ ∂ðpiÞ 1 þðpiÞ pi This calculation was done assuming pi, pf are positive. while the s2 terms are Allowing also for negative momenta, and so negative χ,we get an additional prefactor of 1=2: ∂2 1 ∂ D χ2 − pi χ2 χ 0 Z 2 ∂ 2 h i 21 2 ∂ h iþpih i¼ : (B18) ν p ðpiÞ þðpiÞ pi 1 ð3ν − 1Þ i 0 qðχÞ ≈ ν eVðp Þ=Ddp0 jχj−1−ν: 32ν−1 2ΓðνÞ pf This means that the first and then the second moment can (B13) be found by repeated integration. The boundary conditions are that both the first and second moments are zero if One can show that this result is valid in the whole domain pi ¼ 0, while they diverge if pi ¼ ∞. Using a reflecting of interest 1=3 < ν < 2, i.e., the domain where the variance boundary at pi ¼ ∞, we find (see, e.g., Ref. [50], of χ is infinite. Chapter 5) 021036-26 FROM THE AREA UNDER THE BESSEL EXCURSION TO … PHYS. REV. X 4, 021036 (2014) Z Z ∞ 1 pi − VðyÞ=D −VðzÞ=D From symmetry, Bðv3=2Þ¼Bð v3=2Þ, and hence the hχðpiÞi ¼ dye dze z; (B19) ~ D 0 y Fourier transform BðkÞ is a real function. Thus, Z ~ where the lower bound is the momentum origin and the ∞ 1 − BðkÞ ~ Iν ¼ Re dk ; (C4) potential is given in Eq. (8). Integrating Eq. (B18), we find ~ 1þν kc ðkÞ Z Z p ∞ 2 2 i − hχ i¼ dyeVðyÞ=D dze VðzÞ=DzhχðzÞi: (B20) and kc will eventually be taken to zero. Using the definition D 0 y of the Fourier transform, −ν Z Z As explained in the text, we consider only the limit when ∞ ∞ ikv3=2 I kc − 2 e p ¼ ϵ is small, so we have ν ¼ Re dv3=2Bðv3=2Þ 1 ν dk : i ν 0 k þ Z kc 2ϵ ∞ (C5) hχ2i ∼ dze−VðzÞ=DzhχðzÞi; (B21) D 0 Let iv3=2 ¼ −x, and then using integration by parts, since Vð0Þ¼0. Using Eq. (B19) and integrating by parts, Z ∞ ikv3=2 −ν e − ðkcÞ ν we get Re ¼ Re e kcx − x Γð1 − νÞ=ν : k1þν ν Z Z kc 2ϵ ∞ ∞ 2 χ2 ∼ VðzÞ=D −VðzÞ=D (C6) h i 2 dze dzze : (B22) D 0 z Inserting the last expression in Eq. (C5), the diverging −ν RNote that from symmetry,RVðpÞ¼Vð−pÞ,sowehave ðkcÞ terms cancel each other, and we find ∞ − − −z − z dzz exp½ VðzÞ=D ¼ −∞ dzz exp½ VðzÞ=D . Z 2Γ 1 − ν ∞ I ð Þ ν ν ¼ ν Re dv3=2Bðv3=2Þx : (C7) APPENDIX C: EXPANSION OF EQ. (62) 0 τ ∼ τ−3=2−γ τ γ 1 2 Since gð Þ g for large , with ¼ =ð DÞ,we Using x ¼ −iv3=2, we get assume that there exists some t0 such that gðτÞ¼ −ð3=2þγÞ g τ for τ >t0. Then, using Eq. (59) and ν πð1 þ νÞ I ν ¼ −hjv3 2j i sin Γð−νÞ; (C8) ν ¼ 1=3 þ 2γ=3, = 2 Z pffiffiffiffiffiffiffiffi R t0 ν ∞ ν ψˆ 1− τ 1− ˆ τ3 τ where hjv3=2j i¼ −∞ jv3=2j Bðv3=2Þdv3=2. Inserting ðk;uÞju¼0 ¼ d ½ Bðk D Þ gð Þ 0 Eq. (C8) into (C2) and using (C3), we get Eq. (63). Z ∞ pffiffiffiffiffiffiffiffi τ 1− ~ τ3 τ−3=2−γ þ d ½ Bðk D Þ g : (C1) t0 APPENDIX D: ON THE SIMULATIONS The first term on the right-hand side yields the normali- 1. The simulation zation condition ψˆ k; u 1; the second term is zero ð Þjk¼u¼0 ¼ In this appendix, we briefly discuss the simulations for k 0, and because of the symmetry of the jumps, ¼ presented in the paper to test our predictions. We have two B v3 2 B −v3 2 , this term, when expanded in k,gives ð = Þ¼ ð = Þ classes of simulations: One generates Bessel excursions, a k2 term. As we will show, the last term gives a k ν term, j j and the other solves the Langevin equations. The first is so as long as ν < 2, we can neglect the second term. Then, pffiffiffiffiffiffiffiffi based on a discrete random-walk treatment of the excur- changing variables to k~ k Dτ3, we find ¼ sion, wherein the particle takes a biased walk to the right Z or left in momentum space every time step. The degree of 2 ∞ 1 − B~ k~ ψˆ 1 − 2 ν=2 ð Þ ~ bias varies with p in accordance with the force FðpÞ.Our ðk; uÞju¼0 ¼ g ðDk Þ ~ 1 ν dk: (C2) 3 0 ðkÞ þ dimensionless control parameter D is the ratio of the coefficient of the diffusion term to the coefficient of the Here, we have taken the small k limit in such a way that 1=p large-p behavior of the force. Since for our simple 3=2 kðt0Þ , which appears in the lower limit of an integral, is random walk the diffusion constant is 1=2, the parameter D negligible. enters via the strength of the bias. The probability to move From Eq. (C2), we must investigate the integral right or left is then given by Z ∞ 1 − B~ ðkÞ 1 1 I ~ 1 Δ ν ¼ ~ 1 ν dk: (C3) P ðpÞ¼ FðpÞ p : (D1) 0 ðkÞ þ 2 2D 021036-27 E. BARKAI, E. AGHION, AND D. A. KESSLER PHYS. REV. X 4, 021036 (2014) The continuum limit is approached, as usual, when YN ; Δp → 0. Given that, in our units, FðpÞ varies on the scale F ¼ fi i¼1 of unity, it is sufficient to take Δp ≪ 1. In practice, we take Δ Δ 0 1 FðpiÞ p p ¼ . . We also monitor the position, adding p to the 1 þ Step i is a right step: 2D (D2) position every time the particle takes a step, starting at fi ¼ Δ 1 − FðpiÞ p momentum p. 2D Step i is a left step: To simulate paths of the Langevin dynamics, we used both a straightforward Euler-Maruyama integration of the This works well as long as D is not too small. For very Langevin equation and one based on the random-walk small D, the convergence is quite slow since small areas picture above. Both methods give equivalent results. have anomalously large weight unless N is very large. As long as one is interested in the large-N behavior, better 2. The Bessel excursion convergence is achieved by taking FðpÞ¼θðp − 1Þ=p. An efficient way to generate the set of Bessel excursions is to generate all discrete Brownian excursions with equal APPENDIX E: AREAL DISTRIBUTION probability and then to weight each excursion by its OF THE BESSEL MEANDER appropriate weight. The task of generating the ensemble OurR goal is to find the conditional PDF pMðχ jτ Þ, where of Brownian excursions is, at first sight, a nontrivial χ ¼ τ pðt0Þdt0 is the area under the Bessel meander. The problem since the constraint that the random walk not 0 starting point for the calculation of p ðχ jτ Þ is a modi- cross the origin is nonlocal in the individual left/right steps M fication of Eq. (20). As for the area under the Bessel constituting the walk. However, it is easily accomplished excursion, we consider only the positive meander where ’ using Callan s proof [81] of the famed Chung-Feller 0 < χ < ∞, and later we use the symmetry of the process theorem [82]. Callan gives an explicit mapping of any N to find the areas under both positive and negative meanders. left-step, N right-step random walks to a unique walk that Since, contrary to the condition on the excursion, the does not cross the origin. This mapping maps exactly meander is not bound to return to the origin, we now keep 1 2N ˆ N þ of the ð N Þ N left, N right walks to each of the the end point free and integrate the propagator Gτ ðs; pjpiÞ 2N 1 nonzero crossing walks [of which there are ð N Þ=ðN þ Þ]. over all possible values of p. Hence, Since every Brownian excursion of N þ 2 steps consists R ∞ ˆ of an initial right step, a non-zero-crossing N left-step, N Gτ ðs; pjϵÞdp pˆ ðsjτ Þ¼lim R 0 ; (E1) right-step walk and a final left step back to the origin, M ϵ→0 ∞ ˆ Gτ ðs ¼ 0;pjϵÞdp ’ 2 0 Callan s mapping allows us to generate an N þ -step R Brownian excursion by generating a random N left-step, N with pˆ ðsjτ Þ¼ ∞ p ðχ jτ Þ expð−sDχ Þdχ . M 0 M ˆ right-step walk, applying the map, and prepending and We expand the propagator Gτ ðs; pjpiÞ in a complete postpending the appropriate steps. In this way, every orthonormal basis, using the same approach as in the main excursion is generated with exactly the same probability. text, while accounting for the boundary conditions of the Callan’s mapping is as follows. For a given N left-step, N meander. We thus rewrite Eq. (20) for the case of the right-step walk, one accepts it as is if it does not cross the meander and integrate over p, origin. Otherwise, one finds the leftmost point reached by Z Z ∞ ∞ X the walk (i.e., the minimal x of the path), which we denote ˆ 1=2D 1=3 −1=2D 1=3 Gτ ðs; pjϵÞdp ¼ ϵ s dpp fkðs ϵÞ 0 0 xn, where n is the number of steps to reach this point. If this k point is visited more than once, then one takes the first visit. 1 3 − τ = DEk (E2) One then constructs a new walk based on the original walk. × fkðs pÞe ; We divide the original walk into two segments, the first n − 1 steps and the remaining part. The first part of the new where the functions fkð:Þ, as before, are the solutions of the 2=3λ walk consists of the second segment, shifted by −x such time-independent equation (27) and Ek ¼ s k. Changing n variables to p~ ¼ s1=3p, and using the small p behavior, that the new walk starts on the origin. In the original walk, 1=3 ~ 1=3þ1=6D 1þ1=2D Eq. (28), which gives fkðs ϵÞ ∼ dks ϵ ,we the step before reaching xn is obviously a left step. One appends this left step at the end of the walk under find Z Z construction and then attaches the first segment (of the ∞ X ∞ ~ 2α ν − 2=3λ τ fkðpÞ ˆ ϵ ϵ ~ Ds k ~ original walk). This new walk clearly does not cross the Gτ ðs; pj Þdp ¼ s dk e 1 2 dp; 0 0 ~ = D origin, while preserving the number of left and right steps; k p hence, it starts and ends on the origin. (E3) To account for the bias, i.e., to switch from Brownian to Bessel excursions, it is sufficient to weight each excursion for the numerator in Eq. (E1). We now define a new by the weight factor F: k-dependent coefficient 021036-28 FROM THE AREA UNDER THE BESSEL EXCURSION TO … PHYS. REV. X 4, 021036 (2014) Z ∞ ~ 2 ≡ fkðpÞ ~ with Ek ¼ k in this continuum limit. To solve Eq. (E9), ak 1 2 dp: (E4) 0 p~ = D define q ¼ kp, and so Z The a ’s, similar to the d~ ’s, are evaluated from a numeri- ∞ k k ˆ 0 ϵ cally exact solution of Eq. (27). Rewriting in terms of the dpGτ ðs ¼ ;pj Þ 0 Z Z new coefficients, ϵ2α ∞ ∞ 1=D 1=2−1=2D −Dk2τ Z ¼ dq dkk q JαðqÞe : ∞ X 2αΓ 1 α 2=3 ð þ Þ 0 0 ˆ 2α ν ~ −Ds λkτ Gτ ðs; pjϵÞdp ¼ ϵ s dk ake : (E5) (E10) 0 k 2 For the denominator in Eq. (E1), we expand the propagator Then, inserting x ¼ k , we get s 0 Z in the orthonormal basis for the ¼ case, ∞ ˆ 0 ϵ X 1 2 Gτ ðs ¼ ;pj Þdp ϵ = D 0 ˆ 0 0 −DE τ Gτ ðs ¼ 0;pjϵÞ¼ ϕ ðϵÞϕ ðpÞe k : (E6) Z Z k k ϵ2α ∞ ∞ k p α−1 −Dxτ 1=2−1=2D ¼ α 1 dxx e dqq JαðqÞ: 2 þ Γð1þαÞ 0 0 Integrating over p, as required for the meander, we get (E11) Z ∞ ˆ dpGτ ðs ¼ 0;pjϵÞ The two integrals on the right-hand side of Eq. (E11) can be 0 Z evaluated analytically, and the final result for the denom- ∞ X ϵ 1=2D 0 0 − τ inator of Eq. (E1) is ϕ ϵ ϕ DEk (E7) ¼ dp kð Þ kðpÞe : 0 k p Z ∞ ϵ2α ˆ −α Gτ s 0;pϵ p Dτ : ϕ0 ϵ ϕ0 ð ¼ j Þd ¼ 22αΓ 1 α ð Þ (E12) Pluggingpffiffiffiffiffiffiffiffiffiffiffi in kð Þ and kðpÞ using Eq. (32), with 0 ð þ Þ Bk ¼ πk=L, we find Z Plugging Eqs. (E3), (E4), and (E12) into Eq. (E1), we reach ∞ ˆ our first main result for the area under the Bessel meander, dpGτ ðs¼0;pjϵÞ 0 Z X 2α 3 2 ν − 2=3λ τ π ∞ X ffiffiffiffi ˆ τ 2 Γ 1 α τ = ~ Ds k 1=2þ1=2D −1=2D p −DE τ pMðsj Þ¼ ð þ Þ½sðD Þ dk ake : ¼ϵ dp p kJαðkϵÞ pJαðkpÞe k : 0 k L k (E8) (E13) ˆ PTransformingR the sum into an integral over k : The inverse of pMðsjτ Þ, i.e., sD → χ ,is → L ∞ k π 0 dk, and using the small p behavior of the α X ϵ ∼ ðkϵÞ χ τ 22αΓ 1 α ν=2 τ 3=2 ν ~ Bessel function for Jαðk Þ 2αΓð1þαÞ, we get pMð j Þ¼ ð þ ÞD ½ð Þ dk ak Z k ∞ X∞ − n χ −ðνþ2n=3þ1Þ ˆ 0 ϵ ð ckÞ ð Þ dpGτ ðs ¼ ;pj Þ × ; (E14) 0 Γ −ν − 2n=3 n! Z Z n¼0 ð Þ ϵ2α ∞ ∞ 1− 1 1þα −DE τ ¼ dpp2 2D dkk JαðkpÞe k ; 2αΓ 1 α 1=3 ð þ Þ 0 0 with ck ¼ D λkτ . And finally, summing over n (E9) using Maple, 43=2 1=2 τ 3=2 ν 1 X ν ν 1 1 2 −4 1=3λ τ 3 D ð Þ ~ ðD k Þ p ðχ jτ Þ¼Γð1 þ αÞ − d a × ½Γð1 þ νÞ sinðπνÞ2F2 þ 1; þ ; ; ; M χ πχ k k 2 2 2 3 3 27ðχ Þ2 k 1=3 1=3 3 D λkτ 5 2 þ 3ν ν 4 ν 5 2 4 −4ðD λkτ Þ − Γ þ ν sin π 2F2 þ ; þ ; ; ; ðχ Þ2=3 3 3 2 3 2 6 3 3 27ðχ Þ2 1=3 2 1=3 3 1 D λkτ 7 4 þ ν ν 7 ν 5 4 5 −4ðD λkτ Þ þ Γ þ ν sin π 2F2 þ ; þ ; ; ; : (E15) 2 ðχ Þ2=3 3 3 2 6 2 3 3 3 27ðχ Þ2 021036-29 E. BARKAI, E. AGHION, AND D. A. KESSLER PHYS. REV. X 4, 021036 (2014) 2 axis will be clustered, and while their number is infinite, Brownian Meander visualizing these dots with a simulation, we will observe a ∞ 1.5 Simulation: D = fractal dust. ) ∗ Bessel Meander τ In this appendix, we investigate the statistics of duration | | ∗ Simulation: D=0.4 t |χ of the excursion straddling time , deriving some known p( 1 results along the way. Our approach is based on renewal 1/2 ) 3 ∗ theory, using methods given by Godreche and Luck [47]. τ The previous mathematical approaches are based on direct (2D 0.5 analysis of Brownian and Bessel motion, while we use the ϵ trick introduced in the main text. As mentioned, we replace the continuous regularized Bessel process or Brownian path 0 0123with a noncontinuous one, with jumps of size ϵ after zero ∗ ∗3 1/2 |χ |/(2Dτ ) crossing, and at the end, ϵ is taken to zero. Consider the PDF of waiting times gðτÞ with 0 < τ < ∞, FIG. 14. The conditional PDF of the area under the Bessel with the long-time limiting behavior (D ¼ 2=5) and Brownian (D → ∞) meanders, Eq. (E15). For 2 5 D ¼ = , we used five-term summation in Eq. (E15), with gðτÞ ∼ g τ−ð1þαÞ: (F1) a1 ¼ 0.849, a2 ¼ −0.314, a3 ¼ 0.535, a4 ¼ −0.3072, a5 ¼ 0.441 ~ and the values of dk in Table I. The simulation method is outlined 5 α 1 2 in Appendix D, and we averaged over 2 × 10 samples with While ¼ð þ DÞ= D for the problem in the main τ ¼ 105. The theoretical curve was plotted with Maple. manuscript (hence, α ≥ 1=2), here we will assume that 0 < α < 1. The case α ¼ 1=2 is the Brownian case since a particle starting on ϵ will return for the first time to the This is the main result of this appendix. In Fig. 14,we origin according to the well-known law gðτÞ ∝ τ−3=2. In our plot this areal distribution, comparing it with a histogram physical problem, the ϵ trick means that g is ϵ dependent, obtained from finite time simulations. Eq. (59). As in the main text, we denote ft1;t2; …;tk; …g as times of the renewal events (i.e., in our problem, zero APPENDIX F: THE TIME INTERVAL crossing events). This is a finite set of times if ϵ is finite. STRADDLING t The waiting times fτ1; τ2; …g are independent, identically The velocity process vðt0Þ restricted to the zero-free distributed random variables because of the MarkovianP τ k τ interval containing a fixed observation time t is called property of the underlying paths, and t1 ¼ 1, tk ¼ i¼1 i. the excursion process straddling t, and the portion of it up We call t the observation time, and the number of renewals 0 to t is the meandering process ending at t. For Brownian in ð ;tÞ is denoted by n; thus, by definition, tn 021036-30 FROM THE AREA UNDER THE BESSEL EXCURSION TO … PHYS. REV. X 4, 021036 (2014) 8 and compute < 1 1−ð1−Δ~ Þ1=2 Δ~ 1 π Δ~ 3=2 < d~ðΔ~ Þ¼ ð Þ (F10) Z Z : 1 ~ ∞ ∞ 3 2 Δ > 1. −st −uΔδ Δ− − Δ ðΔ~ Þ = e e ½ ðtnþ1 tnÞ Iðtn τ −τ When gð Þ¼expð Þ, we find, in the limit of large t, where we usedR the small R and u limit and, as usual, Δ → Δ −Δ ∞ −uΔe dtð Þ expð Þ, so the minimum of this PDF is at hexpð−uΔeÞi ¼ 0 e deðΔeÞdΔe, where deðΔeÞ is the Δ ¼ 0 and Δ ¼ ∞, which is expected. For our more PDF of Δ . This is the known result for the Bessel process α e interesting case, gˆðuÞ ∼ 1 − g jΓð−αÞju for small u; when scaled properly, i.e., R ¼ 1 in Ref. [31], where the hence, we get, in the small u and s limits, with an arbitrary inverse Laplace transform of Eq. (F11) is also given, thus ratio between them, providing an explicit formula for the PDF deðΔeÞ. To see the connection between renewal theory and ðu þ sÞα − uα statistics of the duration of Bessel excursion straddling d ðuÞ ∼ : (F8) s s1þα an exponential time, notice that in Ref. 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