Exact results on Sinai's diffusion

Comtet, Alain; Dean, David S.

doi:10.1088/0305-4470/31/43/004

Cited by 41 publications

(46 citation statements)

References 17 publications

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“…Using the relation between the hitting probability and the maximum location it is possible to prove that the survival probability of a particle in a correlated disorder potential V (x) behaves as where θ V is the persistence exponent of the process V (x). For instance, in the Sinai model, using θ V = 1/2, H V = 1/2 we get S(t) ∼ 1/ log(t), in agreement with the exact result [55,56]. A random acceleration process potential would be self-affine with H V = 3/2 and persistence exponent θ V = 1/4 [44].…”

Section: A Digression On the Hitting Probabilitysupporting

confidence: 82%

First Passage Problems in Anomalous Diffusion

Rosso¹,

Zoia²

2014

First-Passage Phenomena and Their Applications

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Many physical and biological systems can be described in terms of a particle undergoing random displacements. When the spread of these particles at long times asymptotically scales as x 2 (t) ∼ t 2H , with H = 1/2, the walker diffusion is said to be anomalous, as opposed to regular diffusion, which corresponds to plain Brownian motion. Examples of anomalous diffusion emerge for instance when considering the motion of polymers through nanopores, or the percolation of tracer particles in heterogeneous soils. Determining the first-passage properties of such walkers is particularly difficult, because anomalous particles often display long-range correlations and/or wild distributions of time and space increments. Here we will review some relevant examples of anomalous diffusion and show by which mathematical and numerical tools the corresponding first-passage properties can be assessed.

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Section: A Digression On the Hitting Probabilitysupporting

confidence: 82%

First Passage Problems in Anomalous Diffusion

Rosso¹,

Zoia²

2014

First-Passage Phenomena and Their Applications

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“…Comtet and Dean [296] first computed Q(x 0 , t) for the Sinai model using an exact probabilistic approach and found that for large t and fixed x 0…”

Section: Persistence In the One Dimensional Sinai Modelmentioning

confidence: 99%

Persistence and first-passage properties in nonequilibrium systems

Bray

Majumdar

Schehr

2013

Advances in Physics

542

669

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In this review we discuss the persistence and the related first-passage properties in extended many-body nonequilibrium systems. Starting with simple systems with one or few degrees of freedom, such as random walk and random acceleration problems, we progressively discuss the persistence properties in systems with many degrees of freedom. These systems include spins models undergoing phase ordering dynamics, diffusion equation, fluctuating interfaces etc. Persistence properties are nontrivial in these systems as the effective underlying stochastic process is non-Markovian. Several exact and approximate methods have been developed to compute the persistence of such non-Markov processes over the last two decades, as reviewed in this article. We also discuss various generalisations of the local site persistence probability. Persistence in systems with quenched disorder is discussed briefly. Although the main emphasis of this review is on the theoretical developments on persistence, we briefly touch upon various experimental systems as well.

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“…Interestingly, a similar property arises in the problem of the coarsening of the pure 1D Φ 4 model at zero temperature, which can be treated exactly by successive elimination of the smallest domains in the system [38], a method reminiscent of the RSRG studied here. Finally note that since [18] appeared, several new papers have been devoted to the Sinai model [39][40][41][42].…”

Section: Introductionmentioning

confidence: 99%

Random walkers in one-dimensional random environments: Exact renormalization group analysis

1999

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Sinai's model of diffusion in one-dimension with random local bias is studied by a real space renormalization group which yields exact results at long times. The effects of an additional small uniform bias force are also studied. We obtain analytically the scaling form of the distribution of the position x(t) of a particle, the probability of it not returning to the origin and the distributions of first passage times, in an infinite sample as well as in the presence of a boundary and in a finite but large sample. We compute the distribution of meeting time of two particles in the same environment. We also obtain a detailed analytic description of the thermally averaged trajectories by computing quantities such as the joint distribution of the number of returns and of the number of jumps forward. These quantities obey multifractal scaling, characterized by generalized persistence exponents θ(g) which we compute. In the presence of a small bias, the number of returns to the origin becomes finite, characterized by a universal scaling function which we obtain. The full statistics of the distribution of successive times of return of thermally averaged trajectories is obtained, as well as detailed analytical information about correlations between directions and times of successive jumps. The two time distribution of the positions of a particle, x(t) and x(t ′ ) with t > t ′ , is also computed exactly. It is found to exhibit "aging" with several time regimes characterized by different behaviors. In the unbiased case, for t − t ′ ∼ t ′α with α > 1, it exhibits a ln t ln t ′ scaling, with a singularity at coinciding rescaled positions x(t) = x(t ′ ). This singularity is a novel feature, and corresponds to particles which remain in a renormalized valley. For closer times α < 1, the two time diffusion front exhibits a quasi-equilibrium regime with a ln(t − t ′ )/ ln t ′ behavior which we compute. The crossover to 1 a t/t ′ aging form in the presence of a small bias is also obtained analytically. Rare events corresponding to intermittent splitting of the thermal packet between separated wells which dominate some averaged observables, are also characterized in detail. Connections with the Green's function of a one-dimensional Schrödinger problem and quantum spin chains are discussed.

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Exact results on Sinai's diffusion

Cited by 41 publications

References 17 publications

First Passage Problems in Anomalous Diffusion

First Passage Problems in Anomalous Diffusion

Persistence and first-passage properties in nonequilibrium systems

Random walkers in one-dimensional random environments: Exact renormalization group analysis

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