Measure of the violation of the detailed balance criterion: a possible definition of a "distance" from equilibrium.

Motivated by the classification of nonequilibrium steady states suggested by R. K. P. Zia and B. Schmittmann, in J. Stat. Mech. (2007) P07012, 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 simple 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 asymmetric simple exclusion process and open zero-range process, we show that the current of particles gives an exact measure of the violation of detailed balance.