Fractional Klein-Kramers equation for superdiffusive transport: normal versus anomalous time evolution in a differential L{\'e}vy walk model

TL;DR

This paper introduces a fractional Klein-Kramers equation modeling superdiffusive transport with space-dependent forces, revealing complex relaxation behaviors and the coexistence of normal and anomalous diffusion in a differential Lévy walk framework.

Contribution

It presents a novel fractional Klein-Kramers equation for superdiffusion, demonstrating its non-negativity and detailed relaxation dynamics in phase space.

Findings

Velocity distribution relaxes via Mittag-Leffler functions

No stationary solution in space coordinate

Competition between Brownian and anomalous diffusion

Abstract

We introduce a fractional Klein-Kramers equation which describes sub-ballistic superdiffusion in phase space in the presence of a space-dependent external force field. This equation defines the differential L{\'e}vy walk model whose solution is shown to be non-negative. In the velocity coordinate, the probability density relaxes in Mittag-Leffler fashion towards the Maxwell distribution whereas in the space coordinate, no stationary solution exists and the temporal evolution of moments exhibits a competition between Brownian and anomalous contributions.