Correlations in the Diffusive Maps With Quenched Disorder

Abstract

It was shown by G. Radons1 that, for a large class of one-dimensional maps, diffusion is suppressed by the presence of quenched disorder. Focusing on simple diffusive maps with discrete disorder, we investigate the behavior of the correlation functions χ1 (τ; t) and χ01 (τ; t), which arise naturally from the random walks induced by disorder of the system. Our numerical simulations show that both χ1 (τ; t) and χ01 (τ; t) decay with τ more slowly than the exponential decay, and both scale linearly with t; i.e. χ1 (τ; t) = tϕ1 (τ) and χ01 (τ; t) = tϕ01 (τ). Interestingly, we have also found that both [Formula: see text] and [Formula: see text] are mainly independent of the disorder configurations of the system.

Suppression and enhancement of diffusion in disordered dynamical systems

“…[2], and as further analyzed in Refs. [14], quenched disorder can change the deterministic diffusive dynamics in these maps profoundly: adding static randomness in form of a local bias with globally vanishing drift leads to dynamical localization of trajectories in a complicated potential landscape. On disordered lattices, this effect is well-known as the Golosov phenomenon [4,19] thus reappearing in the framework of deterministic dynamics.…”

“…To our knowledge, only very few cases of such models have been studied so far. Examples include random Lorentz gases for which Lyapunov exponents have been calculated by means of kinetic theory and by computer simulations [11], numerical studies of diffusion on disordered rough surfaces [12] and in disordered deterministic ratchets [13], as well as numerical and analytical studies of chaotic maps on the line with quenched disorder [2,14].…”