Ergodic Properties of Fractional Brownian-Langevin Motion

TL;DR

This paper studies the ergodic properties of fractional Brownian and Langevin motion, revealing how the convergence to ergodicity depends on the Hurst exponent and identifying a critical point at H=3/4.

Contribution

It provides a detailed analysis of the ergodicity breaking parameter for fractional Brownian motion, highlighting the critical Hurst exponent and comparing with continuous time random walk models.

Findings

Ergodic convergence is slow and depends on the Hurst exponent.

H=3/4 is a critical point where convergence behavior changes.

In the ballistic limit, ergodicity is broken with EB approaching 2.

Abstract

We investigate the time average mean square displacement $\overline{\delta^2}(x(t))=\int_0^{t-\Delta}[x(t^\prime+\Delta)-x(t^\prime)]^2 dt^\prime/(t-\Delta)$ for fractional Brownian and Langevin motion. Unlike the previously investigated continuous time random walk model $\overline{\delta^2}$ converges to the ensemble average $<x^2 > \sim t^{2 H}$ in the long measurement time limit. The convergence to ergodic behavior is however slow, and surprisingly the Hurst exponent $H=3/4$ marks the critical point of the speed of convergence. When $H<3/4$, the ergodicity breaking parameter ${EB} = {Var} (\overline{\delta^2}) / < \overline{\delta^2} >^2\sim k(H) \cdot\Delta\cdot t^{-1}$, when $H=3/4$, ${EB} \sim (9/16)(\ln t) \cdot\Delta \cdot t^{-1}$, and when $3/4<H <1, {EB} \sim k(H)\Delta^{4-4H} t^{4H-4}$. In the ballistic limit $H \to 1$ ergodicity is broken and ${EB} \sim 2$. The critical point $H=3/4$ is marked by the divergence of the coefficient $k(H)$. Fractional Brownian motion as a model for recent experiments of sub-diffusion of mRNA in the cell is briefly discussed and comparison with the continuous time random walk model is made.