Transport in disordered systems: The single big jump approach

Wang, Wanli; Vezzani, A.; Burioni, Raffaella; Barkai, Eli

doi:10.1103/physrevresearch.1.033172

Cited by 30 publications

(19 citation statements)

References 56 publications

(122 reference statements)

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“… If the jump length PDF is sub-exponential, the far tail of

will deviate from what we found here. Most likely the principle of the single big jump [ 64 , 65 , 66 ] will hold in some form, but the details of the theory are left unknown. Recently Dechant et al showed how the CTRW picture emerges from an under-damped Langevin description of a particle in a periodic potential [ 29 ].…”

Section: Discussionmentioning

confidence: 99%

Large Deviations for Continuous Time Random Walks

Wang

Barkai

Burov

2020

Entropy

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Recently observation of random walks in complex environments like the cell and other glassy systems revealed that the spreading of particles, at its tails, follows a spatial exponential decay instead of the canonical Gaussian. We use the widely applicable continuous time random walk model and obtain the large deviation description of the propagator. Under mild conditions that the microscopic jump lengths distribution is decaying exponentially or faster i.e., Lévy like power law distributed jump lengths are excluded, and that the distribution of the waiting times is analytical for short waiting times, the spreading of particles follows an exponential decay at large distances, with a logarithmic correction. Here we show how anti-bunching of jump events reduces the effect, while bunching and intermittency enhances it. We employ exact solutions of the continuous time random walk model to test the large deviation theory.

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“… If the jump length PDF is sub-exponential, the far tail of

Section: Discussionmentioning

confidence: 99%

Large Deviations for Continuous Time Random Walks

Wang

Barkai

Burov

2020

Entropy

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“…Thus, so far, the Scher-Berkowitz theoretical framework is based on a random walk picture [30] and not on a governing fractional advection-diffusion equation. Both the CTRW and the BSMW frameworks and the experiments in the field agree on one thing: advection-diffusion is anomalous and non-symmetric [31,34], however otherwise these schools promote widely different philosophies.…”

mentioning

confidence: 88%

Fractional advection-diffusion-asymmetry equation

Wang,

Barkai

2020

Preprint

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Fractional kinetic equations employ non-integer calculus to model anomalous relaxation and diffusion in many systems. While this approach is well explored, it so far failed to describe an important class of transport in disordered systems. Motivated by work on contaminant spreading in geological formations we propose and investigate a fractional advection-diffusion equation describing the biased spreading. While usual transport is described by diffusion and drift, we find a third term describing symmetry breaking which is omnipresent for transport in disordered systems. Our work is based on continuous time random walks with a finite mean waiting time and a diverging variance, a case that on the one hand is very common and on the other was missing in the kaleidoscope literature of fractional equations. The fractional space derivative stems from long trapping times while previously it was interpreted as a consequence of spatial Lévy flights.

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“…This section draws on extreme value statistics. We refer the reader to [25][26][27] for an introduction to extreme value statistics and applications in statistical mechanics, and we mention [28] as an example of recent work which uses extreme value statistics as a tool to analyse stochastic models for dispersion. We have seen that when D is small the integrals defining the mean first passage time are approximated by sums over extrema of the potential, as described by equation (17).…”

Section: Statistics Of Extreme-weighted Sumsmentioning

confidence: 99%

Flooding dynamics of diffusive dispersion in a random potential

2021

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We discuss the combined effects of overdamped motion in a quenched random potential and diffusion, in one dimension, in the limit where the diffusion coefficient is small. Our analysis considers the statistics of the mean first-passage time T(x) to reach position x, arising from different realisations of the random potential. Specifically, we contrast the median $${\bar{T}}(x)$$ T ¯ ( x ) , which is an informative description of the typical course of the motion, with the expectation value $$\langle T(x)\rangle $$ ⟨ T ( x ) ⟩ , which is dominated by rare events where there is an exceptionally high barrier to diffusion. We show that at relatively short times the median $${\bar{T}}(x)$$ T ¯ ( x ) is explained by a ‘flooding’ model, where T(x) is predominantly determined by the highest barriers which are encountered before reaching position x. These highest barriers are quantified using methods of extreme value statistics.

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Transport in disordered systems: The single big jump approach

Cited by 30 publications

References 56 publications

Large Deviations for Continuous Time Random Walks

Large Deviations for Continuous Time Random Walks

Fractional advection-diffusion-asymmetry equation

Flooding dynamics of diffusive dispersion in a random potential

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