## Abstract

We investigate the time average mean-square displacement δ2 ̄ (x (t)) = ∫0 t-Δ [x (t′ +Δ) -x (t′)] 2 d 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, δ2 ̄ converges to the ensemble average x2 ∼ t2H 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 EB = [[δ2 ̄ (x(t))]2 - δ2 ̄ (x(t))2] / δ2 ̄ (x(t))2 ∼k (H) Δ t-1, when H= 3 4, EB ∼ (9 16) (ln t) Δ t-1, and when 3 4 <H<1, EB ∼k (H) Δ4-4H t4H-4. In the ballistic limit H→1 ergodicity is broken and EB ∼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.

Original language | English |
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Article number | 011112 |

Journal | Physical Review E |

Volume | 79 |

Issue number | 1 |

DOIs | |

State | Published - 5 Jan 2009 |