Distribution of TimeAveraged Observables for Weak Ergodicity Breaking
Abstract
We find a general formula for the distribution of timeaveraged 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 X̄, the time average of the position of the particle, for unbiased and uniformly biased particles, and show that X̄ 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
 EPrint:
 5 pages, 2 figures