Irreversible models with Boltzmann-Gibbs probability distribution and entropy production

We analyze irreversible interacting spin models evolving according to a master equation with spin flip transition rates that do not obey detailed balance but obey global balance with a Boltzmann-Gibbs probability distribution. Spin flip transition rates with up-down symmetry are obtained for a linear chain, a square lattice, and a cubic lattice with a stationary state corresponding to the Ising model with nearest neighbor interactions. We show that these irreversible dynamics describes the contact of the system with particle reservoirs that cause a flux of particles through the system. Using a microscopic definition, we determine the entropy production rate of these irreversible models and show that it can be written as a macroscopic bilinear form in the forces and fluxes. Exact expressions for this property are obtained for the linear chain and the square lattice. In this last case the entropy production rate displays a singularity at the phase transition point of the same type as the entropy itself.