How a single number intertwines space and time in one dimensional diffusive systems
Pre-print : https://arxiv.org/abs/2206.08739 Title : Role of initial conditions in 1D diffusive systems : compressibility, hyperuniformity and long-term memory Authors : Tirthankar Banerjee, Robert L. Jack and Michael E. Cates At the starting point of equilibrium Statistical Mechanics lies the 'ergodic hypothesis' [1]. In simplified terms, a system is taken to be ergodic if the ensemble and time averages of relevant observables are mutually equal. The hypothesis holds as long as the observation time is sufficiently larger than any intrinsic timescale associated with the system's own dynamics. At such long timescales, ergodic systems end up in unique steady states, thus rendering the 'initial conditions' irrelevant. Ergodicity naturally breaks down when there are diverging intrinsic (or microscopic) timescales in the system (for example, near phase transitions) [2,3]. In other words, 'non-ergodicity' is a signature of strong memory effects [4,5]. This lends ...