Two examples of memory-induced effects in Brownian motion and random walk

Abstract

Two examples of memory induced transitions are discussed in this talk. The first example deals with a continuous time random walk with a complete memory of its history. Depending on the value of correlation parameter the walker may exhibit localization around the origin, uncorrelated random walk, or net drift in the direction of the initial bias. In the latter case the mean square displacement may either scale linearly with the number of steps N, or as N ln N, or as a power law in N with an exponent that depends on the correlation parameter. Memory induced transitions leading to non-vanishing third and fourt order cumulants show that the Fokker-Planck approximations are inadequate in the superdiffusive regime. The second example deals with the Brownian motion of a charged particle in a uniform magnetic field driven internally by an exponentially correlated noise. A strong dissipation regime is described in which the ensemble-averaged fluctuations of the velocity exhibit transient oscillations that arise from memory effects. The results are extended to the general case of internal driving by correlated Gaussian stochastic forces with finite autocorrelation times.