Approaching stationarity: competition between long jumps and long waiting times

Abstract

Within the continuous-time random walk (CTRW) scenarios, properties of the overallmotion are determined by the waiting time and the jump length distributions. In thedecoupled case, with power-law distributed waiting times and jump lengths, the CTRWscenario is asymptotically described by the double (space and time) fractionalFokker–Planck equation. Properties of a system described by such an equation aredetermined by the subdiffusion parameter and the jump length exponent. Nevertheless, thestationary state is determined solely by the jump length distribution and thepotential. The waiting time distribution determines only the rate of convergenceto the stationary state. Here, we inspect the competition between long waitingtimes and long jumps and how this competition is reflected in the way in which astationary state is reached. In particular, we show that the distance between atime-dependent and a stationary solution changes in time as a double power law.