Disentangling roots of ergodicity breakdown by spectral analyses

Abstract

The connection between ergodicity breaking for a forced system and its two-time velocity correlation function is demonstrated using the Khinchin theorem, where the particle dynamics is described by a generalized Langevin equation. Products of observables at a pair of time points are constructed that are correlated with their initial preparations. Three types of nonergodic behaviors are elucidated, specifically the velocity correlation function of the system either approaches a constant, oscillates with time around a non-vanishing constant or oscillates around zero-value at large times. The corresponding noise spectral densities behave as low-hindering and high-passing, band-passing as well as low-passing and high-hindering or simply band-hindering. When a harmonic potential is imposed on the system, the first situation can be transformed into ergodicity whereas the latter two types cannot. Furthermore, the application to the Debye Brownian oscillator as well as the verification of several famous models are discussed.