Rare-Event-Induced Ergodicity Breaking in Logarithmic Aging Systems

Ergodicity breaking and aging effects are fundamental challenges in out-of-equilibrium systems. Various mechanisms have been proposed to understand the non-ergodic and aging phenomena, possibly related to observations in systems ranging from structural glass and Anderson glasses to biological systems and mechanical systems. While anomalous diffusion described by Levy statistics efficiently captures ergodicity breaking, the origin of aging and ergodicity breaking in systems with ultraslow dynamics remain unclear. Here, we report a novel mechanism of ergodicity breaking in systems exhibiting log-aging diffusion. This mechanism, characterized by increasingly infrequent rare events with aging, yields statistics deviating significantly from Levy distribution, breaking ergodicity as shown by unequal time- and ensemble-averaged mean squared displacements and two distinct asymptotic probability distribution functions. Notably, although these rare events contribute negligibly to statistical averages, they dramatically change the system's characteristic time. This work lays the groundwork for microscopic understanding of out-of-equilibrium systems and provides new perspectives on glasses and Griffiths-McCoy singularities.