Persistent memory for a Brownian walker in a random array of obstacles

We show that for particles performing Brownian motion in a frozen
array of scatterers long-time correlations emerge in the
mean-square displacement. Defining the velocity autocorrelation
function (VACF) via the second time-derivative of the mean-square
displacement, power-law tails govern the long-time dynamics similar
to the case of ballistic motion. The physical origin of the
persistent memory is due to repeated encounters with the same
obstacle which occurs naturally in Brownian dynamics without
involving other scattering centers. This observation suggests that
in this case the VACF exhibits these anomalies already at first
order in the scattering density. Here we provide an analytic
solution for the dynamics of a tracer for a dilute planar Lorentz
gas and compare our results to computer simulations. Our result
support the idea that quenched disorder provides a generic
mechanism for persistent correlations irrespective of the
microdynamics of the tracer particle