Diffusion in periodic, correlated random forcing landscapes

TL;DR

This paper investigates the complex diffusion behavior of a Brownian particle in a correlated random potential derived from fractional Brownian motion, revealing non-self-averaging properties and distinct statistical tails in the diffusion coefficient.

Contribution

It provides an exact analytical characterization of the moments and distribution of the diffusion coefficient in a novel correlated random landscape, highlighting non-trivial scaling and tail behaviors.

Findings

Positive moments of D_L scale as exp[-c' (k β L^H)^{1/(1+H)}]

Negative moments of D_L scale as exp[a' (k β L^H)^2]

Distribution of D_L exhibits a log-normal left tail and a singular, log-stable right tail

Abstract

We study the dynamics of a Brownian particle in a strongly correlated quenched random potential defined as a periodically-extended (with period $L$) finite trajectory of a fractional Brownian motion with arbitrary Hurst exponent $H \in (0,1)$. While the periodicity ensures that the ultimate long-time behavior is diffusive, the generalised Sinai potential considered here leads to a strong logarithmic confinement of particle trajectories at intermediate times. These two competing trends lead to dynamical frustration and result in a rich statistical behavior of the diffusion coefficient $D_L$: Although one has the typical value $D^{\rm typ}_L \sim \exp(-\beta L^H)$, we show via an exact analytical approach that the positive moments ($k>0$) scale like $\langle D^k_L \rangle \sim \exp{[-c' (k \beta L^{H})^{1/(1+H)}]}$, and the negative ones as $\langle D^{-k}_L \rangle \sim \exp(a' (k \beta L^{H})^2)$, $c'$ and $a'$ being numerical constants and $\beta$ the inverse temperature. These results demonstrate that $D_L$ is strongly non-self-averaging. We further show that the probability distribution of $D_L$ has a log-normal left tail and a highly singular, one-sided log-stable right tail reminiscent of a Lifshitz singularity.