Lévy flights from a continuous-time process

Abstract

Lévy flight dynamics can stem from simple random walks in a system whose operational time (number of steps $n)$ typically grows superlinearly with physical time t. Thus this process is a kind of continuous-time random walk (CTRW), dual to the typical Scher-Montroll model, in which n grows sublinearly with t. Models in which Lévy flights emerge due to a temporal subordination allow one easily to discuss the response of a random walker to a weak outer force, which is shown to be nonlinear. On the other hand, the relaxation of an ensemble of such walkers in a harmonic potential follows a simple exponential pattern, and leads to a normal Boltzmann distribution. Mixed models, describing normal CTRW’s in superlinear operational time and Lévy flights under the operational time of subdiffusive CTRW’s lead to a paradoxical diffusive behavior, similar to the one found in transport on polymer chains. The relaxation to the Boltzmann distribution in such models is slow, and asymptotically follows a power law.