Mean first passage time for anomalous diffusion

Gitterman, Moshe

doi:10.1103/physreve.62.6065

Cited by 99 publications

(101 citation statements)

References 12 publications

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“…Doing so leads to erroneous analytical results, in contrast e.g. with Monte Carlo simulations [41,42,43,44]. This also implies that standard techniques such as the method of images are not applicable [12,32].…”

Section: Boundary Conditions For the Eigenvalue Problemmentioning

confidence: 99%

Fractional Laplacian in bounded domains

Zoia

Rosso

Kardar³

2007

Phys. Rev. E

233

290

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The fractional Laplacian operator, −(−△) α 2 , appears in a wide class of physical systems, including Lévy flights and stochastic interfaces. In this paper, we provide a discretized version of this operator which is well suited to deal with boundary conditions on a finite interval. The implementation of boundary conditions is justified by appealing to two physical models, namely hopping particles and elastic springs. The eigenvalues and eigenfunctions in a bounded domain are then obtained numerically for different boundary conditions. Some analytical results concerning the structure of the eigenvalues spectrum are also obtained.

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Section: Boundary Conditions For the Eigenvalue Problemmentioning

confidence: 99%

Fractional Laplacian in bounded domains

Zoia

Rosso

Kardar³

2007

Phys. Rev. E

233

290

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“…In general, the first passage time for subdiffusion was recently a matter of considerable interest and dispute in the literature [16,17,18]. It is now understood [19,20,21] that the mean first passage time diverges for subdiffusion, because a subdiffusing walker tends to remain too long on the place that it once reached.…”

Section: /2 With Time T)mentioning

confidence: 99%

First passage times and asymmetry of DNA translocation

Lua

Grosberg

2005

Phys. Rev. E

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Motivated by experiments in which single-stranded DNA with a short hairpin loop at one end undergoes unforced diffusion through a narrow pore, we study the first passage times for a particle, executing one-dimensional brownian motion in an asymmetric sawtooth potential, to exit one of the boundaries. We consider the first passage times for the case of classical diffusion, characterized by a mean-square displacement of the form (∆x) 2 ∼ t, and for the case of anomalous diffusion or subdiffusion, characterized by a mean-square displacement of the form (∆x) 2 ∼ t γ with 0 < γ < 1. In the context of classical diffusion, we obtain an expression for the mean first passage time and show that this quantity changes when the direction of the sawtooth is reversed or, equivalently, when the reflecting and absorbing boundaries are exchanged. We discuss at which numbers of 'teeth' N (or number of DNA nucleotides) and at which heights of the sawtooth potential this difference becomes significant. For large N , it is well known that the mean first passage time scales as N 2 . In the context of subdiffusion, the mean first passage time does not exist. Therefore we obtain instead the distribution of first passage times in the limit of long times. We show that the prefactor in the power relation for this distribution is simply the expression for the mean first passage time in classical diffusion. We also describe a hypothetical experiment to calculate the average of the first passage times for a fraction of passage events that each end within some time t * . We show that this average first passage time scales as N 2/γ in subdiffusion.

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“…One can mention some erroneous results for Lévy flights obtained in Ref. [Gitterman, 2000], because author used the traditional conditions at two absorbing boundaries (see the related correspondence [Yuste & Lindenberg, 2004;Gitterman, 2004]). The numerical results for the first passage time of free Lévy flights confined in a finite interval were presented in Ref.…”

Section: Barrier Crossingmentioning

confidence: 99%

Lévy Flight Superdiffusion: An Introduction

Dubkov

Spagnolo

Учайкин

2008

Int. J. Bifurcation Chaos

300

295

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After a short excursion from discovery of Brownian motion to the Richardson "law of four thirds" in turbulent diffusion, the article introduces the Lévy flight superdiffusion as a self-similar Lévy process. The condition of self-similarity converts the infinitely divisible characteristic function of the Lévy process into a stable characteristic function of the Lévy motion. The Lévy motion generalizes the Brownian motion on the base of the α-stable distributions theory and fractional order derivatives. The further development of the idea lies on the generalization of the Langevin equation with a non-Gaussian white noise source and the use of functional approach. This leads to the Kolmogorov's equation for arbitrary Markovian processes. As particular case we obtain the fractional Fokker-Planck equation for Lévy flights. Some results concerning stationary probability distributions of Lévy motion in symmetric smooth monostable potentials, and a general expression to calculate the nonlinear relaxation time in barrier crossing problems are derived. Finally we discuss results on the same characteristics and barrier crossing problems with Lévy flights, recently obtained with different approaches.

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Mean first passage time for anomalous diffusion

Cited by 99 publications

References 12 publications

Fractional Laplacian in bounded domains

Fractional Laplacian in bounded domains

First passage times and asymmetry of DNA translocation

Lévy Flight Superdiffusion: An Introduction

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