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

TL;DR

This paper studies a fractional Langevin equation far from equilibrium, revealing spurious nonergodicity, aging effects, and the conditions under which the process becomes ergodic or stationary, using Riemann-Liouville fractional Brownian motion.

Contribution

It introduces and analyzes the FLEFE model, highlighting spurious nonergodicity, aging, and stationarity properties for different fractional orders, extending understanding of non-equilibrium stochastic dynamics.

Findings

For 1/2<α<3/2, TAMSD converges to MSI, indicating spurious nonergodicity.

When α≥3/2, the process is nonergodic, but higher order increments are ergodic.

Strong aging can restore ergodicity and stationarity in the process.

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 $1/2<\alpha<3/2$.