Continuous Time Random Walks in periodic systems: fluid limit and fractional differential equations on the circle

TL;DR

This paper investigates continuous time random walks on a circle, deriving fractional differential equations that describe anomalous transport phenomena influenced by topology and Levy flights, with explicit solutions in simple cases.

Contribution

It introduces a periodic fractional Riemann-Liouville operator framework for fluid limits of CTRWs on the circle, highlighting topology effects and providing explicit propagator expressions.

Findings

Derived generalized master equation for CTRWs on the circle

Formulated fluid limit equations with periodic fractional derivatives

Computed explicit propagators in simple cases

Abstract

In this article, the continuous time random walk on the circle is studied. We derive the corresponding generalized master equation and discuss the effects of topology, especially important when Levy flights are allowed. Then, we work out the fluid limit equation, formulated in terms of the periodic version of the fractional Riemann-Liouville operators, for which we provide explicit expressions. Finally, we compute the propagator in some simple cases. The analysis presented herein should be relevant when investigating anomalous transport phenomena in systems with periodic dimensions.