Towards deterministic equations for Lévy walks: the fractional material derivative

Lévy walks are random processes with an underlying spatiotemporal coupling. This coupling penalizes long jumps, and therefore Lévy walks give a proper stochastic description for a particle's motion with broad jump length distribution. We derive a generalized dynamical formulation for Lévy walks, in which the fractional equivalent of the material derivative occurs. Our approach is expected to be useful for the dynamical formulation of Lévy walks in an external force field or in phase space, for which the description in terms of the continuous time random walk or its corresponding generalized master equation are less well suited.