Ergodic properties of fractional Brownian-Langevin motion

We investigate the time average mean-square displacement ${{{δ}}²}(x(t))={{∫}}₀^{t{-}{Δ}}{[x({t}^{{'}}+{Δ}){-}x({t}^{{'}})]}²d{t}^{{'}}∕(t{-}{Δ})$ 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, ${{{δ}}²}$ 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 $H=3/4$ marks the critical point of the speed of convergence. When $H<3/4$, the ergodicity breaking parameter ${E}B={[⟨[{{{δ}}²}(x(t))]}²⟩{-}{⟨{{{δ}}²}(x(t))⟩}²]/{⟨{{{δ}}²}(x(t))⟩}²{~}k(H){Δ}{t}^{{-}1}$, when $H=3/4$, ${E}B{~}(9/16)(ln{{0.2em}{0ex}}t){Δ}{t}^{{-}1}$, and when $3/4<H<1$, ${E}B{~}k(H){{Δ}}^{4{-}4H}{t}^{4H{-}4}$. 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.