  The recently proposed Ehrenfest M-urn model with interactions on a ring is
considered as a paradigm model which can exhibit a variety of distinct
nonequilibrium steady states. Unlike the previous three-urn model on a ring
which consists of a uniform steady state and a nonuniform nonequilibrium steady
state, it is found that for even M>=4, an additional nonequilibrium steady
state can coexist with the original ones. Detailed analysis reveals that this
additional nonequilibrium steady state emerged via a pitchfork bifurcation
which cannot occur if M is odd. Properties of this nonequilibrium steady state,
such as stability, and steady-state flux are derived analytically for the
four-urn case. The full phase diagram with the phase boundaries is also derived
explicitly. The associated thermodynamic stability is also analyzed, confirming
its stability. These theoretical results are also explicitly verified by direct
Monte Carlo simulations for the three-urn and four-urn ring models.
