Distribution of Time-Averaged Observables for Weak Ergodicity Breaking

Details Distribution of Time-Averaged Observables for Weak Ergodicity Breaking Distribution of Time-Averaged Observables for Weak Ergodicity Breaking

2007-07-26

arxiv.org/abs/0707.3865v1

Phys. Rev. Letters 99, 210601 (2007)

Physics

Condensed Matter

Statistical Mechanics

5 pages, 2 figures

Scientific paper

10.1103/PhysRevLett.99.210601

We find a general formula for the distribution of time-averaged observables for systems modeled according to the sub-diffusive continuous time random walk. For Gaussian random walks coupled to a thermal bath we recover ergodicity and Boltzmann's statistics, while for the anomalous subdiffusive case a weakly non-ergodic statistical mechanical framework is constructed, which is based on L\'evy's generalized central limit theorem. As an example we calculate the distribution of $\bar{X}$: the time average of the position of the particle, for unbiased and uniformly biased particles, and show that $\bar{X}$ exhibits large fluctuations compared with the ensemble average $$.

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