Continuous time random walk with jump length correlated with waiting time

Liu, Jian; Bao, Jing‐Dong

doi:10.1016/j.physa.2012.10.019

Cited by 24 publications

(22 citation statements)

References 24 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…The original work of Montroll and Weiss [30] assumed there were no correlations between step size χ and and the waiting time τ , corresponding to a situation called a decoupled CTRW. The diffusion of atoms in optical lattices corresponds to a coupled spatial-temporal random walk theory first considered by Scher and Lax [63] (see [64][65][66] for recent developments). Define η s (x, t)dtdx as the probability that the particle crossed the momentum state p = 0 for the sth time in the time interval (t, t + dt) and that the particle's position was in the interval (x, x + dx).…”

Section: Montroll-weiss Equation For Fourier-laplace Transform Of P (mentioning

confidence: 99%

From the Area under the Bessel Excursion to Anomalous Diffusion of Cold Atoms

2014

View full text Add to dashboard Cite

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 87 Rb atoms and single 24 Mg þ 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 ∼ t 3=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.

show abstract

Section: Montroll-weiss Equation For Fourier-laplace Transform Of P (mentioning

confidence: 99%

From the Area under the Bessel Excursion to Anomalous Diffusion of Cold Atoms

2014

View full text Add to dashboard Cite

show abstract

“…In this paper we generally assume that waiting times and jump lengths are uncorrelated. For a discussion of the correlated case, we refer to [64][65][66][67][68][69][70]. A natural parametrisation of such a random walk is obtained in terms of the number n of jumps performed.…”

Section: Anomalous Processes With General Waiting Timesmentioning

confidence: 99%

Feynman–Kac equation for anomalous processes with space- and time-dependent forces

Cairoli¹,

Baule²

2017

J. Phys. A: Math. Theor.

View full text Add to dashboard Cite

Functionals of a stochastic process Y (t) model many physical time-extensive observables, for instance particle positions, local and occupation times or accumulated mechanical work. When Y (t) is a normal diffusive process, their statistics are obtained as the solution of the celebrated Feynman-Kac equation. This equation provides the crucial link between the expected values of diffusion processes and the solutions of deterministic second-order partial differential equations. When Y (t) is non-Brownian, e.g., an anomalous diffusive process, generalizations of the Feynman-Kac equation that incorporate power-law or more general waiting time distributions of the underlying random walk have recently been derived. A general representation of such waiting times is provided in terms of a Lévy process whose Laplace exponent is directly related to the memory kernel appearing in the generalized Feynman-Kac equation. The corresponding anomalous processes have been shown to capture nonlinear mean square displacements exhibiting crossovers between different scaling regimes, which have been observed in numerous experiments on biological systems like migrating cells or diffusing macromolecules in intracellular environments. However, the case where both spaceand time-dependent forces drive the dynamics of the generalized anomalous process has not been solved yet. Here, we present the missing derivation of the Feynman-Kac equation in such general case by using the subordination technique. Furthermore, we discuss its extension to functionals explicitly depending on time, which are of particular relevance for the stochastic thermodynamics of anomalous diffusive systems. Exact results on the work fluctuations of a simple non-equilibrium model are obtained. An additional aim of this paper is to provide a pedagogical introduction to Lévy processes, semimartingales and their associated stochastic calculus, which underlie the mathematical formulation of anomalous diffusion as a subordinated process.

show abstract

“…In the wait-first model the particle jumps at the end of the waiting time, whereas in the jump-first model the particle jumps and then rests for the corresponding waiting time. Such random walks were previously considered in the literature [18][19][20][21][22][23][24][25][26], and recently also with a method of infinite densities [27] in the sub-ballistic superdiffusive regime (flight or waiting times with finite mean, 1 < γ < 2). However, in general the exact analytical solutions describing the density of random walking particles are rare and can be obtained only for some particular values of γ [16,28].…”

Section: Introductionmentioning

confidence: 99%

Asymptotic densities of ballistic Lévy walks

et al. 2015

View full text Add to dashboard Cite

We propose an analytical method to determine the shape of density profiles in the asymptotic long time limit for a broad class of coupled continuous time random walks which operate in the ballistic regime. In particular, we show that different scenarios of performing a random walk step, via making an instantaneous jump penalized by a proper waiting time or via moving with a constant speed, dramatically effect the corresponding propagators, despite the fact that the end points of the steps are identical. Furthermore, if the speed during each step of the random walk is itself a random variable, its distribution gets clearly reflected in the asymptotic density of random walkers. These features are in contrast with more standard non-ballistic random walks.

show abstract

Continuous time random walk with jump length correlated with waiting time

Cited by 24 publications

References 24 publications

From the Area under the Bessel Excursion to Anomalous Diffusion of Cold Atoms

From the Area under the Bessel Excursion to Anomalous Diffusion of Cold Atoms

Feynman–Kac equation for anomalous processes with space- and time-dependent forces

Asymptotic densities of ballistic Lévy walks

Contact Info

Product

Resources

About