Classical motion in force fields with short range correlations

TL;DR

This paper investigates the universal long-term behavior of fast particles in time-dependent random force fields with short-range spatial correlations, revealing distinct superballistic and ballistic regimes depending on the force being a gradient field.

Contribution

It demonstrates universal power-law behaviors for particle motion in such fields, distinguishing between gradient and non-gradient force fields across dimensions.

Findings

Superballistic motion in 1D with q^2(t) ~ t^{12/5}

Ballistic motion in higher dimensions with q^2(t) ~ t^{2}

Different power laws for non-gradient force fields, p^2(t) ~ t^{2/3} and q^2(t) ~ t^{8/3}

Abstract

We study the long time motion of fast particles moving through time-dependent random force fields with correlations that decay rapidly in space, but not necessarily in time. The time dependence of the averaged kinetic energy and mean-squared displacement is shown to exhibit a large degree of universality; it depends only on whether the force is, or is not, a gradient vector field. When it is, p^{2}(t) ~ t^{2/5} independently of the details of the potential and of the space dimension. Motion is then superballistic in one dimension, with q^{2}(t) ~ t^{12/5}, and ballistic in higher dimensions, with q^{2}(t) ~ t^{2}. These predictions are supported by numerical results in one and two dimensions. For force fields not obtained from a potential field, the power laws are different: p^{2}(t) ~ t^{2/3} and q^{2}(t) ~ t^{8/3} in all dimensions d\geq 1.