Ergodic Properties And Simulations Of Fractional Langevin Motion In Harmonic And Periodic Field

In this paper, we investigate the fluctuation of time average of displacement, namely x(t) for a particle whose motion is governed by fractional Langevin equations satisfying fluctuation dissipation relation and the system has an equilibrium state. For overdamped FLE, time average x(t) converges to the ensemble average <x(t)> in the long measurement time limit and limt�'∞δ2/xx=0, so the process is ergodic and the er-godicity follows a generic δ2/xx∝1/tα/tα behavior (0<α<1,1<α<2,α=2-2H), here H is the Hurst exponent and when α=1, δ2/xx decays with an exponential be-havior. For underdamped case, we obtain the similar results. A comparison with the numerical solution is made. We also consider the FLE in the periodic potential, find a transition from nonergodic to ergodic:α=0.6, we find that U0≥7, the fractional Langevin motion is ergodic, and U0<7 is nonergodic; For α=1.6, we also find a transition from nonergodic to ergodic, U0≥3 is ergodic, and U0<3 is nonergod-ic. Based on this, we also consider a two-component stochastic motion governed by fractional Langevin equation and CTRW-style dynamics in which heavy tailed waiting time is discussed. In particular, we analyze its ergodic behavior.