Non-equilibrium phase transition in a two-species driven-diffusive model of classical particles

A two-species driven-diffusive model of classicalparticles is introduced on a lattice with periodicboundary condition. The model consists of a finite numberof first class particles in the presence of a second classparticle. While the first class particles can only hop forward,the second class particle is able to hop both forwardand backward with specific rates. We have shown that thepartition function of this model can be calculated exactly.The model undergoes a non-equilibrium phase transitionwhen a condensation of the first class particles occursbehind the second class particle. The phase transition pointand the spatial correlations between the first class particlesare calculated exactly. On the other hand, we have shownthat this model can be mapped onto a two-dimensionalwalk model. The random walker can only move on the firstquarter of a two-dimensional plane and that it takes thepaths which can start at any height and end at any heightupper than the height of the starting point. The initialvertex (starting point) and the final vertex (end point) ofeach lattice path are weighted. The weight of the outsetpoint depends on the height of that point while the weightof the end point depends on the height of both the outsetpoint and the end point of each path. The partition functionof this walk model is calculated using a transfer matrixmethod.