Sample-to-sample fluctuations of power spectrum of a random motion in a periodic Sinai model

TL;DR

This paper investigates the power spectrum of a diffusive process in a periodic Sinai model, revealing that most realizations exhibit Brownian-like low-frequency behavior with a disorder-dependent amplitude, characterized by multifractal moments and specific distribution tails.

Contribution

It provides a detailed analysis of the sample-to-sample fluctuations of the power spectrum in a periodic Sinai model, including the distribution and moments of the amplitude, supported by both analytical and numerical methods.

Findings

Most realizations show $S(f) \\sim {\\cal A}/f^2$ at low frequencies.

The amplitude ${\f A}$ has a multifractal moment structure.

Distribution of ${\f A}$ has a log-normal tail and an essential singularity.

Abstract

The Sinai model of a tracer diffusing in a quenched Brownian potential is a much studied problem exhibiting a logarithmically slow anomalous diffusion due to the growth of energy barriers with the system size. However, if the potential is random but periodic, the regime of anomalous diffusion crosses over to one of normal diffusion once a tracer has diffused over a few periods of the system. Here we consider a system in which the potential is given by a Brownian Bridge on a finite interval $(0,L)$ and then periodically repeated over the whole real line, and study the power spectrum $S(f)$ of the diffusive process $x(t)$ in such a potential. We show that for most of realizations of $x(t)$ in a given realization of the potential, the low-frequency behavior is $S(f) \sim {\cal A}/f^2$, i.e., the same as for standard Brownian motion, and the amplitude ${\cal A}$ is a disorder-dependent random variable with a finite support. Focusing on the statistical properties of this random variable, we determine the moments of ${\cal A}$ of arbitrary, negative or positive order $k$, and demonstrate that they exhibit a multi-fractal dependence on $k$, and a rather unusual dependence on the temperature and on the periodicity $L$, which are supported by atypical realizations of the periodic disorder. We finally show that the distribution of ${\cal A}$ has a log-normal left tail, and exhibits an essential singularity close to the right edge of the support, which is related to the Lifshitz singularity. Our findings are based both on analytic results and on extensive numerical simulations of the process $x(t)$.