Diffusion in periodic, correlated random forcing landscapes

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 (0, 1). While the periodicity ensures that the ultimate long time behavior is diffusive, the generalized 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 DL: although one has the typical value β ∼ − D L exp ( ) L H typ , we show via an exact analytical approach that the positive moments ( > k 0) scale like β 〈 〉 ∼ − ′ + D c k L exp [ ( ) ] L H H 1 (1 ) , and the negative ones as β 〈 〉 ∼ ′ − D a k L exp ( ( ) ) L k H 2 , ′ c and ′ a being numerical constants and β the inverse temperature. These results demonstrate that DL is strongly non-self-averaging. We further show that the probability distribution of DL has a log normal left tail and a highly singular, one sided log stable right tail reminiscent of a Lifshitz singularity.