Gibbsian nonequilibrium stationary states for two or three species of interacting particles

We construct explicit examples of one-dimensional driven diffusive systems for two and three species of interacting particles, defined by asymmetric dynamical rules which do not obey detailed balance, but whose nonequilibrium stationary states coincide with a prescribed equilibrium Gibbsian state. For simplicity, the states considered only involve nearest-neighbor interactions. For two species, the generic Gibbsian dynamics has five free parameters, and does not obey pairwise balance in general. The latter property is satisfied only by the totally asymmetric dynamics and the partially asymmetric dynamics with uniform bias, i.e., the cases originally considered by Katz, Lebowitz, and Spohn. For three species of interacting particles, with nearest-neighbor interactions between particles of the same species, the totally asymmetric Gibbsian dynamics has two free parameters, and obeys pairwise balance. These examples confirm that asymmetric stochastic dynamics producing a given nonequilibrium stationary state are far more constrained (in terms of numbers of free parameters) than the corresponding symmetric (equilibrium) dynamics.