Ergodic properties of fractional Brownian-Langevin motion

We investigate the time average mean-square displacement δ²̅ (x(t))=∫₀^t−Δ[x(t^′+Δ)−x(t^′)]²dt^′∕(t−Δ) for fractional Brownian-Langevin motion where x(t) is the stochastic trajectory and Δ is the lag time. Unlike the previously investigated continuous-time random-walk model, δ²̅ converges to the ensemble average ⟨x²⟩∼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=[⟨[δ²̅ (x(t))]²⟩−⟨δ²̅ (x(t))⟩²]/⟨δ²̅ (x(t))⟩²∼k(H)Δt⁻¹, when H=3/4, E_B∼(9/16)(ln t)Δt⁻¹, and when 3/4<H<1, E_B∼k(H)Δ^4−4Ht^4H−4. In the ballistic limit H→1 ergodicity is broken and E_B∼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.