Anomalous Diffusion of Inertial, Weakly Damped Particles

TL;DR

This paper investigates the non-Gaussian, anomalous diffusion behavior of inertial particles under deterministic acceleration and random kicks, deriving a new fractional Kramers-Fokker-Planck equation with nonlocal couplings.

Contribution

It introduces a novel fractional equation in position-velocity space extending continuous time random walks, incorporating fractional derivatives to model anomalous diffusion of inertial particles.

Findings

Derived a new fractional Kramers-Fokker-Planck equation.

Found a closed-form solution for the force-free case.

Highlighted the role of fractional derivatives in nonlocal couplings.

Abstract

The anomalous (i.e. non-Gaussian) dynamics of particles subject to a deterministic acceleration and a series of 'random kicks' is studied. Based on an extension of the concept of continuous time random walks to position-velocity space, a new fractional equation of the Kramers-Fokker-Planck type is derived. The associated collision operator necessarily involves a fractional substantial derivative, representing important nonlocal couplings in time and space. For the force-free case, a closed solution is found and discussed.