Bar-Ilan University

About: Bar-Ilan University is a education organization based out in Ramat Gan, Israel. It is known for research contribution in the topics: Population & Poison control. The organization has 12835 authors who have published 34964 publications receiving 995648 citations. The organization is also known as: Bar Ilan University & BIU.

Ergodic properties of fractional Brownian-Langevin motion

Abstract: We investigate the time average mean-square displacement $\overline{{\ensuremath{\delta}}^{2}}\mathbf{(}x(t)\mathbf{)}={\ensuremath{\int}}_{0}^{t\ensuremath{-}\ensuremath{\Delta}}{[x({t}^{\ensuremath{'}}+\ensuremath{\Delta})\ensuremath{-}x({t}^{\ensuremath{'}})]}^{2}d{t}^{\ensuremath{'}}∕(t\ensuremath{-}\ensuremath{\Delta})$ for fractional Brownian-Langevin motion where $x(t)$ is the stochastic trajectory and $\ensuremath{\Delta}$ is the lag time. Unlike the previously investigated continuous-time random-walk model, $\overline{{\ensuremath{\delta}}^{2}}$ converges to the ensemble average $⟨{x}^{2}⟩\ensuremath{\sim}{t}^{2H}$ in the long measurement time limit. The convergence to ergodic behavior is slow, however, and surprisingly the Hurst exponent $H=\frac{3}{4}$ marks the critical point of the speed of convergence. When $Hl\frac{3}{4}$, the ergodicity breaking parameter ${E}_{B}={\mathbf{[}⟨[\overline{{\ensuremath{\delta}}^{2}}\mathbf{(}x(t)\mathbf{)}\mathbf{]}}^{2}⟩\ensuremath{-}{⟨\overline{{\ensuremath{\delta}}^{2}}\mathbf{(}x(t)\mathbf{)}⟩}^{2}\mathbf{]}/{⟨\overline{{\ensuremath{\delta}}^{2}}\mathbf{(}x(t)\mathbf{)}⟩}^{2}\ensuremath{\sim}k(H)\ensuremath{\Delta}{t}^{\ensuremath{-}1}$, when $H=\frac{3}{4}$, ${E}_{B}\ensuremath{\sim}(\frac{9}{16})(\mathrm{ln}\phantom{\rule{0.2em}{0ex}}t)\ensuremath{\Delta}{t}^{\ensuremath{-}1}$, and when $\frac{3}{4}lHl1$, ${E}_{B}\ensuremath{\sim}k(H){\ensuremath{\Delta}}^{4\ensuremath{-}4H}{t}^{4H\ensuremath{-}4}$. In the ballistic limit $H\ensuremath{\rightarrow}1$ ergodicity is broken and ${E}_{B}\ensuremath{\sim}2$. The critical point $H=\frac{3}{4}$ is marked by the divergence of the coefficient $k(H)$. Fractional Brownian motion as a model for recent experiments of subdiffusion of mRNA in the cell is briefly discussed, and a comparison with the continuous-time random-walk model is made.