Ergodic properties of fractional Brownian-Langevin motion

We investigate the time average mean-square displacement {^2} (x (t) ) =₀^t-x ({t^'+) -x (t^') }^2dt^'∕ (t-) for fractional Brownian-Langevin motion where x (t) is the stochastic trajectory and is the lag time. Unlike the previously investigated continuous-time random-walk model, {^2} converges to the ensemble average ⟨x^2⟩t^2H in the long measurement time limit. The convergence to ergodic behavior is slow, however, and surprisingly the Hurst exponent H=34 marks the critical point of the speed of convergence. When H<34, the ergodicity breaking parameter E₁=⟨[{{^2} (x (t) ) }^2⟩-⟨{{^2} (x (t) ) ⟩}^2]/⟨{{^2} (x (t) ) ⟩}^2 (H) t^-1, when H=34, E₁ (916) (ln0. 2em{0ex}t) t^-1, and when 34<H<1, E₁ (H) ^4-4Ht^4H-4. In the ballistic limit H1 ergodicity is broken and E₁2. The critical point H=34 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.