Breakdown of the Onsager principle as a sign of aging

TL;DR

This paper investigates the relationship between CTRW and GME under the Onsager principle, revealing that their equivalence holds only for exponential waiting times and exploring implications for aging and non-Markovian dynamics.

Contribution

It establishes conditions under which CTRW and GME are equivalent, highlighting the role of stationarity and aging, and introduces a non-Markovian GME consistent with the Onsager principle.

Findings

Equivalence between CTRW and GME holds only for exponential waiting times.

The Onsager principle is valid in fully aged systems regardless of waiting time distribution.

Non-Markovian GME can be consistent with the Onsager principle under stationarity.

Abstract

We discuss the problem of the equivalence between Continuous Time Random Walk (CTRW) and Generalized Master Equation (GME). The walker, making instantaneous jumps from one site of the lattice to another, resides in each site for extended times. The sojourn times have a distribution psi(t) that is assumed to be an inverse power law. We assume that the Onsager principle is fulfilled, and we use this assumption to establish a complete equivalence between GME and the Montroll-Weiss CTRW. We prove that this equivalence is confined to the case when psi(t) is an exponential. We argue that is so because the Montroll-Weiss CTRW, as recently proved by Barkai [E. Barkai, Phys. Rev. Lett. 90, 104101 (2003)], is non-stationary, thereby implying aging, while the Onsager principle, is valid only in the case of fully aged systems. We consider the case of a dichotomous fluctuation, and we prove that the Onsager principle is fulfilled for any form of regression to equilibrium provided that the stationary condition holds true. We set the stationary condition on both the CTRW and the GME, thereby creating a condition of total equivalence, regardless the nature of the waiting time distribution. As a consequence of this procedure we create a GME that it is a "bona fide" master equation, in spite of being non-Markovian. We note that the memory kernel of the GME affords information on the interaction between system of interest and its bath. The Poisson case yields a bath with infinitely fast fluctuations. We argue that departing from the Poisson form has the effect of creating a condition of infinite memory and that these results might be useful to shed light into the problem of how to unravel non-Markovian master equations.