Linear response, fluctuation-dissipation, and finite-system-size effects in superdiffusion

Abstract

Lévy walks (LWs) are a popular stochastic tool to model anomalous diffusion and have recently been used to describe a variety of phenomena. We study the linear response behavior of this generic model of superdiffusive LWs in finite systems to an external force field under both stationary and nonstationary conditions. These finite-size LWs are based on power-law waiting time distributions with a finite-time regularization at τ(c), such that the physical requirements are met to apply linear response theory and derive the power spectrum with the correct short frequency limit, without the introduction of artificial cutoffs. We obtain the generalized Einstein relation for both ensemble and time averages over the entire process time and determine the turnover to normal Brownian motion when the full system is explored. In particular, we obtain an exact expression for the long time diffusion constant as a function of the scaling exponent of the waiting time density and the characteristic time scale τ(c).