A Sorting Problem

The following problem is considered: each cell of an integer-valued torus contains a particle that belongs to one of two types. At each step one selects at random a pair of neighboring particles of different type which are then exchanged with a probability that depends on the choice of the particles that are neighbors of the pair. The necessary and sufficient conditions are obtained for the existence of a final Gibbs measure on an arbitrary torus with an arbitrary number of particles.