Thermodynamics and phase coexistence in nonequilibrium steady states
Ronald Dickman (UFMG)

I review recent work focussing on whether thermodynamics can be extended to nonequilibrium steady states (NESS). A central issue - possibility of consistent definitions of temperature $T$ and chemical potential $\mu$ for systems in NESS - is analyzed using simple, far-from-equilibrium lattice models with stochastic dynamics. Each model includes a drive that maintains the system far from equilibrium, provoking particle and/or energy flows; for zero drive the system approaches equilibrium. Analysis and numerical simulation show that for spatially uniform NESS, coexistence with an appropriate reservoir yields consistent definitions of $T$ and $\mu$, provided a particular kind of rate (that proposed by Sasa and Tasaki [1]) is used for exchanges of particles and energy between systems [2]. Remarkably, the associated entropy function is not the Shannon entropy of the stationary probability distribution on configuration space. Consistent definitions of intensive parameters are not possible for nonuniform NESS [3]. The functions $T$ and $\mu$ for isolated phases cannot be used to predict the properties of coexisting phases in a single, phase-separated system [4]. Investigation of simple far-from-equilibrium systems exhibiting phase separation leads to the conclusion that phase coexistence is not well defined in this context. This is because the properties of the coexisting nonequilibrium systems depend on how they make contact, as verified in the driven lattice gas with attractive interactions, and in the two-temperature lattice gas, under (a) weak global exchange between uniform systems, and (b) phase-separated (nonuniform) systems. Thus, far from equilibrium, the notions of universality of phase coexistence (i.e., independent of how systems exchange particles and/or energy), and of phases with intrinsic properties (independent of their environment) are lost.

[1] S. Sasa and H. Tasaki, J. Stat. Phys. 125, 125 (2006).
[2] R. Dickman and R. Motai, Phys Rev E 89, 032134 (2014).
[3] R. Dickman, Phys Rev E 90, 062123 (2014).
[4] R. Dickman, New J. Phys. 18, 043034 (2016).