Khinchin theorem and anomalous diffusion

TL;DR

This paper demonstrates that the Khinchin theorem remains valid for both normal and anomalous diffusion processes modeled by a Generalized Langevin Equation, clarifying misconceptions about irreversibility and ergodicity.

Contribution

It proves the validity of the Khinchin theorem across all types of diffusion described by a Generalized Langevin Equation, including anomalous diffusion.

Findings

Khinchin theorem holds for normal diffusion

Khinchin theorem holds for anomalous diffusion

Irreversibility is broader than ergodicity in these systems

Abstract

A recent paper [M. H. Lee, Phys. Rev. Lett. 98, 190601 (2007)] has called attention to the fact that irreversibility is a broader concept than ergodicity, and that therefore the Khinchin theorem [A. I. Khinchin, Mathematical Foundations of Statistical Mechanics (Dover, New York) 1949] may fail in some systems. In this Letter we show that for all ranges of normal and anomalous diffusion described by a Generalized Langevin Equation the Khinchin theorem holds.