A measure of the violation of the 'detailed balance' criterion: a possible definition of a 'distance' from equilibrium

TL;DR

This paper introduces a new measure based on the p-norm of probability currents to quantify how far a non-equilibrium steady state deviates from detailed balance, linking it to entropy production and particle currents.

Contribution

It proposes a novel p-norm based metric for detailed balance violation and explores its relation to entropy production and particle currents in non-equilibrium models.

Findings

The p-norm of probability currents quantifies detailed balance violation.

The measure relates to entropy production distribution.

Particle current serves as an exact violation indicator in specific models.

Abstract

Motivated by the classification of non-equilibrium steady states suggested by R.K.P. Zia and B. Schmittmann in J. Stat. Mech. P07012 (2007), we propose to measure the violation of the 'detailed balance' criterion by the $p$-norm ($||K||_p$) of the matrix formed by the probability currents. Its asymptotic analysis, for the totally asymmetric exclusion process, motivates the definition of a 'distance' from equilibrium $K^*$ obtained for $p=1$. In addition, we show that the latter quantity and the average activity $A^*$ are both related to the probability distribution of the entropy production. Finally, considering the open-ASEP and open-ZRP, we show that the current of particles gives an exact measure of the violation of 'detailed balance'.