Ergodic Properties of Fractional Brownian-Langevin Motion

Abstract

We investigate the time average mean square displacement $\overline{{\delta }^2}(x(t))={\int }_0^{t-\Delta }[x(t^{\prime }+\Delta )-x(t^{\prime }){]}^{2}d{t}^{\prime }/(t-\Delta )$ for fractional Brownian and Langevin motion. Unlike the previously investigated continuous time random walk model $\overline{{\delta }^2}$ converges to the ensemble average $<x^2>\sim t^{2H}$ 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.

8 pages, 6 figures

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