## Abstract

For weakly non ergodic systems, the probability density function of a time average observable Script O sign is f_{α}(Script O sign) = (Equation Presented) where Script O sign_{j} is the value of the observable when the system is in state j=1,... L. p _{j} ^{eq} is the probability that a member of an ensemble of systems occupies state j in equilibrium. For a particle undergoing a fractional diffusion process in a binding force field, with thermal detailed balance conditions, p _{j}^{eq} is Boltzmann's cnonical probability. Within the unbiased sub-diffusive continuous time random walk model, the exponent 0 < α < 1 is the anomalous diffusion exponent 〈x^{2}〉 ∼t^{α} found for free boundary conditions. When α → 1 ergodic statistical mechanics is recovered lim_{α→1} f_{α}(Script O sign) = δ(Script O sign - 〈Script O sign〉). We briefly discuss possible physical applications in single particle experiments.

Original language | English |
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Pages (from-to) | 565-586 |

Number of pages | 22 |

Journal | Journal of Statistical Physics |

Volume | 133 |

Issue number | 3 |

State | Published - Nov 2008 |

### Bibliographical note

Funding Information:Acknowledgements This work was supported by the Israel Science Foundation. E.B. thanks G. Margolin, S. Burov, and G. Bel for discussions.

### Funding

Acknowledgements This work was supported by the Israel Science Foundation. E.B. thanks G. Margolin, S. Burov, and G. Bel for discussions.

Funders | Funder number |
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Israel Science Foundation |

## Keywords

- Continuous time random walk
- Fractional Fokker-Planck equation
- Weak ergodicity breaking