Weakly Non-Ergodic Statistical Physics

Abstract

For weakly non ergodic systems, the probability density function of a time average observable $\overline{{\mathcal{O}}}$ is $f_{\alpha}(\overline{{\mathcal{O}}})=-{1\over \pi}\lim_{\epsilon\to 0}\mbox{Im}{\sum_{j=1}^{L}p^{\mathrm{eq}}_{j}(\overline{{\mathcal{O}}}-{\mathcal{O}}_{j}+i\epsilon)^{\alpha -1}\over \sum_{j=1}^{L}p^{\mathrm{eq}}_{j}(\overline{{\mathcal{O}}}-{\mathcal{O}}_{j}+i\epsilon)^{\alpha}}$ where ${\mathcal{O}}_{j}$ is the value of the observable when the system is in state j=1,…L. p ^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 ^eq _j is Boltzmann’s canonical probability. Within the unbiased sub-diffusive continuous time random walk model, the exponent 0<α<1 is the anomalous diffusion exponent 〈x ²〉∼t ^α found for free boundary conditions. When α→1 ergodic statistical mechanics is recovered $\lim_{\alpha \to 1}f_{\alpha}(\overline{{\mathcal{O}}})=\delta (\overline{{\mathcal{O}}}-\langle {\mathcal{O}}\rangle )$ . We briefly discuss possible physical applications in single particle experiments.