Phase Diagram in Stored-Energy-Driven L\'evy Flight

Abstract

Phase diagram based on the mean square displacement (MSD) and the distribution of diffusion coefficients of the time-averaged MSD for the stored-energy-driven L\'evy flight (SEDLF) is presented. In the SEDLF, a random walker cannot move while storing energy, and it jumps by the stored energy. The SEDLF shows a whole spectrum of anomalous diffusions including subdiffusion and superdiffusion, depending on the coupling parameter between storing time (trapping time) and stored energy. This stochastic process can be investigated analytically with the aid of renewal theory. Here, we consider two different renewal processes, i.e., ordinary renewal process and equilibrium renewal process, when the mean trapping time does not diverge. We analytically show the phase diagram according to the coupling parameter and the power exponent in the trapping-time distribution. In particular, we find that distributional behavior of time-averaged MSD intrinsically appears in superdiffusive as well as normal diffusive regime even when the mean trapping time does not diverge.