Distribution of time-averaged observables for weak ergodicity breaking

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 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.