Ergodic properties of fractional Brownian-Langevin motion

We investigate the time average mean-square displacement (delta(2)) over bar (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 bar converges to the ensemble average < x(2)> similar to 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 bar (x(t))](2)> - <(delta(2)) over bar (x(t))>(2)] / <(delta(2)) over bar (x(t))>(2) similar to k(H)Delta t(-1), when H = 3/4, E-B similar to (9/16) (ln t)Delta t(-1), and when 3/4 < H < 1, E-B similar to k(H)Delta(4-4H)t(4H-4). In the ballistic limit H -> 1 ergodicity is broken and E-B similar to 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.