Anomalous Diffusion of particles with inertia in external potentials

TL;DR

This paper investigates the anomalous diffusion behavior of particles with inertia in external potentials, providing explicit solutions and analyzing how harmonic potentials and damping influence long-term particle localization and subdiffusive behavior.

Contribution

It introduces explicit cases of a harmonic potential and velocity-dependent damping into a new Kramers-Fokker-Planck framework, deriving exact moments and asymptotic behaviors.

Findings

Bounding potential and damping induce subdiffusive behavior.

Together, potential and damping cause long-term particle localization.

Explicit relations for moments are derived for specific cases.

Abstract

Recently a new type of Kramers-Fokker-Planck Equation has been proposed [R. Friedrich et al. Phys. Rev. Lett. {\bf 96}, 230601 (2006)] describing anomalous diffusion in external potentials. In the present paper the explicit cases of a harmonic potential and a velocity-dependend damping are incorporated. Exact relations for moments for these cases are presented and the asymptotic behaviour for long times is discussed. Interestingly the bounding potential and the additional damping by itself lead to a subdiffussive behaviour, while acting together the particle becomes localized for long times.