Ehrenfest urn model with interaction

Coexistence of distinct nonuniform nonequilibrium steady states in Ehrenfest multiurn model on a ring

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.

Theory of nonequilibrium asymptotic state thermodynamics: Interacting Ehrenfest urn ring as an example

A generalized class of the nonequilibrium state, called the nonequilibrium asymptotic state (NEAS), is proposed. The NEAS is constructed within the framework of Fokker-Planck equations in thermodynamic limit. Besides the usual equilibrium state and nonequilibrium steady state (NESS), the class of NEAS could also cover the nonequilibrium periodic state (NEPS) in which its dynamics shows periodicity, the nonequilibrium quasiperiodic state (NEQPS), and nonequilibrium chaotic state (NECS) in which its dynamics becomes chaotic. Based on the theory of NEAS thermodynamics, the corresponding thermodynamics of different NEASs could also be determined. Finally the interacting Ehrenfest urn ring model is used as an example to illustrate how different kinds of NEAS (equilibrium state, uniform NESS, nonuniform NESS, NEPS) in the three-urn case are identified in our framework. In particular, the thermodynamics of NEPS and its phase transitions to other types of NEAS are studied.

Nonlinear Non-Equilibrium Thermodynamics Based on the Ehrenfest-Klein Model

Nonlinear non-equilibrium thermodynamic relations have been constructed based on the generalized Ehrenfest-Klein model. Using these relations, the behavior of the entropy and its production in time at arbitrary deviations from equilibrium has been studied. It has been shown that the transient fluctuation theorem is valid for this model if a dissipation functional is treated as the thermodynamic entropy production.

Phase transitions in Ehrenfest urn model with interactions: Coexistence of uniform and nonuniform states

A model based on the classic noninteracting Ehrenfest urn model with two urns is generalized to M urns with the introduction of interactions for particles within the same urn. As the inter-particle interaction strength is varied, phases of different levels of nonuniformity emerge and their stabilities are calculated analytically. In particular, coexistence of locally stable uniform and nonuniform phases connected by first-order transition occurs. The phase transition threshold and energy barrier can be derived exactly together with the phase diagram obtained analytically. These analytic results are further confirmed by Monte Carlo simulations.