  Ehrenfest urns with interaction that are connected in a ring is considered as
a paradigm model for non-equilibrium thermodynamics and is shown to exhibit two
distinct non-equilibrium steady states (NESS) of uniform and non-uniform
particle distributions. As the inter-particle attraction varies, a first order
non-equilibrium 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 non-equilibrium thermodynamics law is
also obtained, which explicitly identifies the heat, work, energy and entropy
of the system.
