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 $M\ge 4$, 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.

中文翻译：

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环上 Ehrenfest 多瓮模型中不同非均匀非平衡稳态的共存

最近提议的埃伦节$\mathrm{中号}$-环上相互作用的瓮模型被认为是一种范式模型，可以表现出各种不同的非平衡稳态。与之前由均匀稳态和非均匀非平衡稳态组成的环上三瓮模型不同，我们发现即使$\mathrm{中号}\ge 4$，额外的非平衡稳态可以与原始稳态共存。详细分析表明，这种额外的非平衡稳态是通过干草叉分岔出现的，如果$\mathrm{中号}$很奇怪。这种非平衡稳态的特性，例如稳定性和稳态通量，是针对四瓮情况进行分析得出的。还明确导出了带有相边界的完整相图。还分析了相关的热力学稳定性，证实了其稳定性。这些理论结果也通过三瓮和四瓮环模型的直接蒙特卡罗模拟得到了明确验证。