First-passage and first-exit times of a Bessel-like stochastic process

Martin, Edgar; Behn, Ulrich; Germano, Guido

doi:10.1103/physreve.83.051115

Cited by 47 publications

(55 citation statements)

References 68 publications

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“…field 1/p corresponds to a well-known stochastic process called the Bessel process [41,53]. More information on the regularized and non-regularized processes is given in Appendixes A,B.…”

Section: Area Under the Bessel Excursionmentioning

confidence: 99%

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

2014

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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.

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“…field 1/p corresponds to a well-known stochastic process called the Bessel process [41,53]. More information on the regularized and non-regularized processes is given in Appendixes A,B.…”

Section: Area Under the Bessel Excursionmentioning

confidence: 99%

“…Since χ is a functional of the path p(t ) we will use the Feynman-Kac (FK) formalism to find G τ (s, p|p i ) (see details below). The constraint that the path p(t ) is always positive enters as an absorbing boundary condition at p = 0 [41].…”

Section: Area Under the Bessel Excursionmentioning

confidence: 99%

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

2014

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“…Generally, it is called first passage time when the random variable reaches a certain level for the first time, and first exit time when leaving a certain interval for the first time [10]. The example of first passage time naturally reaching our mind is the decision of an investor to buy or sell stock when its fluctuating prices reach a certain threshold.…”

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Mean exit time and escape probability for the anomalous processes with the tempered power-law waiting times

Deng¹,

Wu²,

Wang³

2017

EPL

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-The mean first exit (passage) time characterizes the average time of a stochastic process never leaving a fixed region in the state space, while the escape probability describes the likelihood of a transition from one region to another for a stochastic system driven by discontinuous (with jumps) Lévy motion. This paper discusses the two deterministic quantities, mean first exit time and escape probability, for the anomalous processes having the tempered Lévy stable waiting times with the tempering index µ > 0 and the stability index 0 < α ≤ 1; as for the distribution of jump lengths (in the CTRW framework) or the type of the noises driving the system (in the Langevin picture), two cases are considered, i.e., Gaussian white noise and non-Gaussian (tempered) β-stable (0 < β < 2) Lévy noise. Firstly, we derive the nonlocal elliptic partial differential equations (PDEs) governing the mean first exit time and escape probability. Based on the derived PDEs, it is observed that the mean first exit time depends strongly on the domain size and the values of α, β and µ; when µ is close to zero, the mean first exit time tends to ∞. In particular, we also find an interesting result that the escape probability of a particle with (tempered) power-law jumping length distribution has no relation with the distribution of waiting times for the model considered in this paper. For the solutions of the derived PDEs, the boundary layer phenomena are observed, which inspires the motivation for developing the boundary layer theory for nonlocal PDEs.Introduction. -Anomalous diffusion phenomena are widely found in natural world; the subdiffusion includes, e.g., motion of lipids on membranes, solute transport in porous media, translocation of polymers; and the superdiffuion is observed in, e.g., turbulent flow, optical materials, motion of predators, human travel, etc [1]. The types of diffusion are usually distinguished by the exponent of the evolution of the second order moment of a stochastic process x(t) with respect to the time t, i.e., x T (t)x(t) ∼ t γ ; when γ = 1, it is normal diffusion; γ < 1 corresponds to subdiffusion and γ > 1 superdiffusion.

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“…The term Bessel derives from the fact that for v 1, Eq. (1) is mathematically related to the Bessel process which describes the radial component of Brownian motion in arbitrary dimensions [12,[30][31][32]. Clearly the statistics of χ and the zero crossing times, τ , determines the random position of the particle, x(t).…”

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confidence: 99%

Large Fluctuations for Spatial Diffusion of Cold Atoms

2017

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Large-deviations theory deals with tails of probability distributions and the rare events of random processes, for example spreading packets of particles. Mathematically, it concerns the exponential fall-of of the density of thin-tailed systems. Here we investigate the spatial density Pt(x) of laser cooled atoms, where at intermediate length scales the shape is fat-tailed. We focus on the rare events beyond this range, which dominate important statistical properties of the system. Through a novel friction mechanism induced by the laser fields, the density is explored with the recently proposed nonnormalized infinite-covariant density approach. The small and large fluctuations give rise to a bi-fractal nature of the spreading packet.PACS numbers: 05.40. Jc,46.65.+g In diffusion processes such as Brownian motion, the concentration of particles starting at the origin spreads out like a Gaussian, which is fully characterized by the mean squared-displacement. This is the result of the widely applicable Gaussian central limit theorem (CLT) [1]. Of no less importance is large-deviations theory [2], which deals with the rare fluctuations of processes such as simple coin tossing random walks (see e.g., [3]), extreme variations of the surface height in the KardarParisi-Zhang model [4] and the tails of the position distribution in single-file diffusion [5,6]. Mathematically, a prerequisite of the theory is that the cumulant generating function be "well behaved", i.e. smooth and differentiable. Large-deviations theory works when the decay of the probability of the observable of interest is exponential (see details in [2]). However many systems do not meet this requirement [2], for example Lévy fat-tailed processes [7][8][9], where the decay rate is a power-law. This is the case for a cloud of atoms undergoing Sisyphus laser-cooling [10], where both theoretically [11,12] and experimentally [13], it was shown that the central part of the spreading particle packet is described by the Lévy CLT [14]. The latter deals with the sum of independent identically-distributed random variables, but unlike the classical Gaussian CLT, here the summands' own distribution is heavy-tailed. As a result, Lévy's CLT yields an infinite mean squared-displacement for the sum [14], and consequently also the second cumulant. Largedeviations theory mainly deals with thin-tailed processes where extreme events are rare, but in Lévy processes these large fluctuations are dominant. To study the fluctuations in this system, we will show that the relevant tool is the asymptotic moment-generating function, which yields an infinite-covariant density (ICD) [12,15]. We will discuss the generality of this approach and its results below.In an experimental situation, diverging moments are unphysical. For example, although the experiment in [13] shows a nice fit of the particles' density to a symmetric Lévy distribution, clearly at finite times no particles traveling at finite velocities can ever be found infinitely far from their origin. The finiteness of all t...

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First-passage and first-exit times of a Bessel-like stochastic process

Cited by 47 publications

References 68 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

Mean exit time and escape probability for the anomalous processes with the tempered power-law waiting times

Large Fluctuations for Spatial Diffusion of Cold Atoms

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