Generalized Khinchin Theorem for a Class of Aging Processes

TL;DR

This paper generalizes Khinchin's theorem to classify ergodic behavior in aging, non-stationary processes, providing analytical tools to quantify deviations from ergodicity and revealing universal features in anomalous dynamics.

Contribution

It extends Khinchin's theorem to aging processes, offering a universal analytical expression for correlation functions in non-stationary systems.

Findings

Derived a simple analytical expression for two-time correlation functions.

Revealed universality where the binding potential influences only the first two moments.

Applied results to experimental data on anomalous dynamics.

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 non-stationary processes in which correlation functions exhibit aging. We classify the ergodic behavior of such systems and provide a generalization of Khinchin's theorem. Our work 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.