Weakly non-ergodic Statistical Physics

We find a general formula for the distribution of time averaged observables for weakly non-ergodic systems. Such type of ergodicity breaking is known to describe certain systems which exhibit anomalous fluctuations, e.g. blinking quantum dots and the sub-diffusive continuous time random walk model. When the fluctuations become normal we recover usual ergodic statistical mechanics. Examples of a particle undergoing fractional dynamics in a binding force field are worked out in detail. We briefly discuss possible physical applications in single particle experiments.