Diffusion in periodic, correlated random forcing landscapes

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 $D_L^{\mathrm{typ}}\sim \exp (-\beta L^H)$, we show via an exact analytical approach that the positive moments ($k>0$) scale like $⟨D_L^k⟩\sim \exp [-c^{\prime }(k\beta L^H{)}^{1/(1+H)}]$, and the negative ones as $⟨D_L^{-k}⟩\sim \exp (a^{\prime }(k\beta L^H{)}^2)$, $c^{\prime }$ and $a^{\prime }$ 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.

5 pages (main text) + 2 pages (supplemental material); v2: 9 pages, 3 figures, published version

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