Nonequilibrium thermodynamics and phase transition of Ehrenfest urns with interactions

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