Fractional Langevin equation far from equilibrium: Riemann-Liouville fractional Brownian motion, spurious nonergodicity and aging

Abstract

We consider the fractional Langevin equation far from equilibrium (FLEFE) to describe stochastic dynamics which do not obey the fluctuation-dissipation theorem, unlike the conventional fractional Langevin equation (FLE). The solution of this equation is Riemann-Liouville fractional Brownian motion (RL-FBM), also known in the literature as FBM II. Spurious nonergodicity, stationarity, and aging properties of the solution are explored for all admissible values $\alpha>1/2$ of the order $\alpha$ of the time-fractional Caputo derivative in the FLEFE. The increments of the process are asymptotically stationary. However when $1/2<\alpha<3/2$, the time-averaged mean-squared displacement (TAMSD) does not converge to the mean-squared displacement (MSD). Instead, it converges to the mean-squared increment (MSI) or structure function, leading to the phenomenon of spurious nonergodicity. When $\alpha\ge 3/2$, the increments of FLEFE motion are nonergodic, however the higher order increments are asymptotically ergodic. We also discuss the aging effect in the FLEFE by investigating the influence of an aging time $t_a$ on the mean-squared displacement, time-averaged mean-squared displacement and autocovariance function of the increments. We find that under strong aging conditions the process becomes ergodic, and the increments become stationary in the domain $1/2<\alpha<3/2$.