Universal anomalous diffusion of weakly damped particles

TL;DR

This paper demonstrates that weakly damped particles under random forces exhibit universal anomalous diffusion in position and momentum, with specific scaling laws and analytically derived prefactors.

Contribution

It introduces two generalized models showing universal anomalous diffusion and analytically computes the diffusion prefactors for both position and momentum.

Findings

Both models exhibit $<x^2> \\sim C_x t^2$ and $<p^2> \\sim C_p t^{2/5}$.

The diffusion exponents are identical in both models.

Prefactors $C_x$ and $C_p$ are derived analytically.

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.