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

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 ($\mathrm{\mid }\mathrm{\mid }K\mathrm{\mid }{\mathrm{\mid }}_p$) 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 $K^{\ast }$ obtained for $p=1$. In addition, we show that the latter quantity and the average activity $A^{\ast }$ 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’.