Aging and nonergodicity beyond the Khinchin theorem

Abstract

The Khinchin theorem provides the condition that a stationary process is ergodic, in terms of the behavior of the corresponding correlation function. Many physical systems are governed by nonstationary processes in which correlation functions exhibit aging. We classify the ergodic behavior of such systems and suggest a possible generalization of Khinchin's theorem. Our work also quantifies deviations from ergodicity in terms of aging correlation functions. Using the framework of the fractional Fokker-Planck equation, we obtain a simple analytical expression for the two-time correlation function of the particle displacement in a general binding potential, revealing universality in the sense that the binding potential only enters into the prefactor through the first two moments of the corresponding Boltzmann distribution. We discuss applications to experimental data from systems exhibiting anomalous dynamics.anomalous diffusion | ergodicity breaking | single particle trajectories | continuous time random walk | irreversibility I mportant tools for analyzing the generally complicated dynamics of physical observables O are their two-time correlation functions C O ðt 2 ;t 1 Þ. These are pair products of dynamic observables that are averaged over an ensemble, as defined below. When such a correlation function describes a stationary process, C O ðt 2 ;t 1 Þ is a function of the time difference only; that is, C O ðt 2 ;t 1 Þ ¼Ĉ O ðjt 2 − t 1 jÞ. For such processes the correlation function contains the dynamical information on other physical observables via fundamental theorems (1). For instance, the Wiener-Khinchin theorem relates the power spectrum to the correlation function, or the fluctuation-dissipation theorem connects correlation functions to linear response functions. Another well-known example is Khinchin's theorem (2), which provides a criterion for ergodicity of a process in terms of the corresponding stationary correlation function. However, the stationarity property is found to be violated in numerous systems (see below). These systems exhibit aging properties that are intimately connected to the nonstationary behavior. Correlation functions in aging systems behave very differently from their stationary counterparts (3-6). For instance, systems may be characterized by correlation functions of the type C O ðt 2 ;t 1 Þ ¼ h O ðt 1 ∕t 2 Þ (if t 2 ≥ t 1 ); i.e., the two times enter as a quotient rather than their difference. This nonstationary behavior is also connected to a breaking of ergodicity in the sense that long-time averages differ from ensemble averages of the same quantities (3, 7-10). The relation between correlation functions and ergodicity breaking can be quantified by the Edwards-Anderson parameter (3); see below.We here present a possible generalization of the Khinchin theorem for aging systems, namely, provide the condition for ergodicity for systems exhibiting aging. Moreover, for systems with broken ergodicity we relate fluctuations of time averages to the corresponding correlation function. I...

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