Conservative random walks in confining potentials

TL;DR

This paper extends Lévy walk models to include external force fields, resulting in conservative, energy-preserving motion that combines deterministic forces with stochastic velocity reversals, providing new insights into confined superdiffusive processes.

Contribution

It introduces a novel conservative Lévy walk framework influenced by external potentials, integrating deterministic energy-conserving dynamics with stochastic reversals.

Findings

Velocity and position distributions depend on potential steepness

Motion remains on a constant energy surface, differing from thermal motion

Results applicable to systems with external confinement

Abstract

L\'evy walks are continuous time random walks with spatio-temporal coupling of jump lengths and waiting times, often used to model superdiffusive spreading processes such as animals searching for food, tracer motion in weakly chaotic systems, or even the dynamics in quantum systems such as cold atoms. In the simplest version L\'evy walks move with a finite speed. Here, we present an extension of the L\'evy walk scenario for the case when external force fields influence the motion. The resulting motion is a combination of the response to the deterministic force acting on the particle, changing its velocity according to the principle of total energy conservation, and random velocity reversals governed by the distribution of waiting times. For the fact that the motion stays conservative, that is, on a constant energy surface, our scenario is fundamentally different from thermal motion in the same external potentials. In particular, we present results for the velocity and position distributions for single well potentials of different steepness. The observed dynamics with its continuous velocity changes enriches the theory of L\'evy walk processes and will be of use in a variety of systems, for which the particles are externally confined.