Deng Weihua, Barkai Eli
Department of Physics, Bar Ilan University, Ramat-Gan 52900, Israel.
Phys Rev E Stat Nonlin Soft Matter Phys. 2009 Jan;79(1 Pt 1):011112. doi: 10.1103/PhysRevE.79.011112. Epub 2009 Jan 13.
We investigate the time average mean-square displacement delta;{2}over =integral_{0};{t-Delta}[x(t;{'}+Delta)-x(t;{'})];{2}dt;{'}(t-Delta) for fractional Brownian-Langevin motion where x(t) is the stochastic trajectory and Delta is the lag time. Unlike the previously investigated continuous-time random-walk model, delta;{2}[over ] converges to the ensemble average x;{2} approximately t;{2H} in the long measurement time limit. The convergence to ergodic behavior is slow, however, 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 E_{B}=[[delta;{2}over ];{2}-delta;{2}over ;{2}]/delta;{2}over ;{2} approximately k(H)Deltat;{-1} , when H=3/4 , E_{B} approximately (9/16)(lnt)Deltat;{-1} , and when 3/4<H<1 , E_{B} approximately k(H)Delta;{4-4H}t;{4H-4} . In the ballistic limit H-->1 ergodicity is broken and E_{B} approximately 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 subdiffusion of mRNA in the cell is briefly discussed, and a comparison with the continuous-time random-walk model is made.
我们研究了分数布朗 - 朗之万运动的时间平均均方位移$\overline{\Delta^{2}}(x(t))=\int_{0}^{t - \Delta}\frac{[x(t' + \Delta)-x(t')]^{2}}{(t - \Delta)}dt'$，其中$x(t)$是随机轨迹，$\Delta$是滞后时间。与先前研究的连续时间随机游走模型不同，在长时间测量极限下，$\overline{\Delta^{2}}$近似于$t^{2H}$收敛到系综平均$\overline{x^{2}}$。然而，向遍历行为的收敛很慢，令人惊讶的是，赫斯特指数$H = 3/4$标志着收敛速度的临界点。当$H < 3/4$时，遍历性破坏参数$E_{B}=[\overline{(\Delta^{2}(x(t))})^{2}-\overline{\Delta^{2}(x(t))}^{2}]/\overline{\Delta^{2}(x(t))}^{2}\approx k(H)\Delta t^{-1}$；当$H = 3/4$时，$E_{B}\approx(9/16)(\ln t)\Delta t^{-1}$；当$3/4 < H < 1$时，$E_{B}\approx k(H)\Delta^{4 - 4H}t^{4H - 4}$。在弹道极限$H \to 1$时，遍历性被破坏，$E_{B}\approx 2$。临界点$H = 3/4$的特征是系数$k(H)$的发散。简要讨论了分数布朗运动作为细胞中mRNA亚扩散近期实验模型的情况，并与连续时间随机游走模型进行了比较。
