Anomalous fluctuations of currents in Sinai-type random chains with strongly correlated disorder

TL;DR

This paper investigates a generalized Sinai model with strongly correlated disorder, revealing how the steady-state current and displacement behave under different Hurst parameters, showing a paradoxical relationship between disorder correlation and transport.

Contribution

It introduces a fractional Brownian motion-based Sinai model and analytically characterizes how strong correlations affect current decay and displacement growth, revealing novel paradoxical behaviors.

Findings

Mean-square displacement grows as log^{2/H}(n)

Current moments decay as L^{-(1-H)}

Displacement decreases while current increases with H

Abstract

We study properties of a random walk in a generalized Sinai model, in which a quenched random potential is a trajectory of a fractional Brownian motion with arbitrary Hurst parameter H, 0< H <1, so that the random force field displays strong spatial correlations. In this case, the disorder-average mean-square displacement grows in proportion to log^{2/H}(n), n being time. We prove that moments of arbitrary order k of the steady-state current J_L through a finite segment of length L of such a chain decay as L^{-(1-H)}, independently of k, which suggests that despite a logarithmic confinement the average current is much higher than its Fickian counterpart in homogeneous systems. Our results reveal a paradoxical behavior such that, for fixed n and L, the mean square displacement decreases when one varies H from 0 to 1, while the average current increases. This counter-intuitive behavior is explained via an analysis of representative realizations of disorder.