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In nature the noisy signal representing a physical observable is in many cases unpredictable, though the long time average of the signal is identical to ensemble average (ergodicity). For this reason BoltzmannGibbs ergodic statistical mechanics is a powerful tool which gives predictions on time averages. Certain processes characterized by power law sojourn times, in such a way that the average waiting time in a state of the system is infinite are known to exhibit weak ergodicity breaking. For such scale free (in time) processes, time averages remain random even in the limit of long measurement time. The fundamental question is what statistical theory replaces standard ergodic statistical mechanics?
Recent examples of systems and models which exhibit weak ergodicity breaking include blinking quantum dots, anomalous diffusion of mRNA in the cell, the subdiffusive continuous time random walk model, and the quenched trap model. In these diverse systems the distribution of waiting times in a microstate of the system decays like a power law, with a diverging average sojourn time. For example blinking quantum dots have an on state in which many photons are emitted and an off state where the dot is in a dark state. The probability density function $\psi(t)$ of on and off times follows power law statistics with an average on and off time which is infinite $\psi(t)\sim t^{ (1 + \alpha)}$ and $0<\alpha<1$. Hence if we perform a time average it can never be made for long enough time to obtain ergodicity since the average on and off times diverge. A general theory of weak ergodicity is discussed and examples analyzed.
