Diffusion in periodic, correlated random forcing landscapes

David S Dean 1 Shamik Gupta 2, * Gleb Oshanin 3 Alberto Rosso 2 Grégory Schehr 2

Journal of Physics A: Mathematical and Theoretical, IOP Publishing, 2014, 47 (37), pp.372001. <10.1088/1751-8113/47/37/372001>

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.

  • 1. LOMA - Laboratoire Ondes et Matière d'Aquitaine
  • 2. LPTMS - Laboratoire de Physique Théorique et Modèles Statistiques
  • 3. LPTMC - Laboratoire de Physique Théorique de la Matière Condensée