Ergodic properties of fractional Brownian-Langevin motion.

We investigate the time average mean-square displacement delta;{2}[over ](x(t))=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 ](x(t))];{2}-delta;{2}[over ](x(t));{2}]/delta;{2}[over ](x(t));{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.