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

Ivan Calvo, B. A. Carreras, R. Sanchez, B. Ph. van Milligen

Published 2007-08-23, updated 2007-10-23Version 2

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.