Weakly Non-Ergodic Statistical Physics

For weakly non ergodic systems, the probability density function of a time average observable (O) over bar is f alpha((O) over bar) = 1/pi lim(epsilon -> 0) Im Sigma(L)(j=1) P-j(eq) ((O) over bar -O-j+i epsilon)(alpha-1)/Sigma(L)(j=1) P-j(eq)((O) over bar -O-j+i epsilon)(alpha) where O-j is the value of the observable when the system is in state j = 1,... L. p(j)(eq) j 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 canonical probability. Within the unbiased sub-diffusive continuous time random walk model, the exponent 0 < alpha < 1 is the anomalous diffusion exponent < x(2)> similar to t(alpha) found for free boundary conditions. When alpha -> 1 ergodic statistical mechanics is recovered lim(alpha -> 1) f(alpha)((O) over bar) = delta((O) over bar - < O >). We briefly discuss possible physical applications in single particle experiments.