Weakly non-ergodic statistical physics

For weakly non ergodic systems, the probability density function of a time average observable Script O sign is f_α(Script O sign) = (Equation Presented) where Script O sign_j is the value of the observable when the system is in state j=1,... L. p _j ^eq is the probability that a member of an ensemble of systems occupies state j in equilibrium. For a particle undergoing a fractional diffusion process in a binding force field, with thermal detailed balance conditions, p _j^eq is Boltzmann's cnonical probability. Within the unbiased sub-diffusive continuous time random walk model, the exponent 0 < α < 1 is the anomalous diffusion exponent 〈x²〉 ∼t^α found for free boundary conditions. When α → 1 ergodic statistical mechanics is recovered lim_α→1 f_α(Script O sign) = δ(Script O sign - 〈Script O sign〉). We briefly discuss possible physical applications in single particle experiments.