Nonequilibrium thermodynamics and phase transition of Ehrenfest urns with interactions

Ehrenfest urns with interaction that are connected in a ring is considered as paradigm model for non-equilibrium thermodynamics and shown to exhibit two distinct steady states (NESS) of uniform non-uniform particle distributions. As the inter-particle attraction varies, first order phase transition occurs between these NESSs characterized by coexistence regime. The boundaries, NESS distributions near saddle points associated fluxes, average urn population fractions, relaxational dynamics obtained analytically verified numerically. A generalized law also obtained, which explicitly identifies heat, work, energy entropy system.