Universal anomalous diffusion of weakly damped particles

Abstract

We show that anomalous diffusion arises in two different models for the motion of randomly forced and weakly damped particles: one is a generalisation of the Ornstein-Uhlenbeck process with a random force which depends on position as well as time, the other is a generalisation of the Chandrasekhar-Rosenbluth model of stellar dynamics, encompassing non-Coulombic potentials. We show that both models exhibit anomalous diffusion of position $x$ and momentum $p$ with the same exponents: $<x^2> \sim C_x t^2$ and $<p^2> \sim C_p t^{2/5}$. We are able to determine the prefactors $C_x$, $C_p$ analytically.