Distribution of Time-Averaged Observables for Weak Ergodicity Breaking
Abstract
We find a general formula for the distribution of time-averaged observables for systems modeled according to the subdiffusive 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 nonergodic statistical mechanical framework is constructed, which is based on Lévy’s generalized central limit theorem. As an example we calculate the distribution of X¯, the time average of the position of the particle, for unbiased and uniformly biased particles, and show that X¯ exhibits large fluctuations compared with the ensemble average ⟨X⟩.
- Publication:
-
Physical Review Letters
- Pub Date:
- November 2007
- DOI:
- 10.1103/PhysRevLett.99.210601
- arXiv:
- arXiv:0707.3865
- Bibcode:
- 2007PhRvL..99u0601R
- Keywords:
-
- 05.70.Ln;
- 05.20.Gg;
- 05.40.Fb;
- Nonequilibrium and irreversible thermodynamics;
- Classical ensemble theory;
- Random walks and Levy flights;
- Condensed Matter - Statistical Mechanics
- E-Print:
- 5 pages, 2 figures