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 $\overline{X}$, the time average of the position of the particle, for unbiased and uniformly biased particles, and show that $\overline{X}$ exhibits large fluctuations compared with the ensemble average $⟨X⟩$.

• Received 26 July 2007

DOI:https://doi.org/10.1103/PhysRevLett.99.210601