Zeroth law and nonequilibrium thermodynamics for steady states in contact

We ask what happens when two nonequilibrium systems in steady state are kept in contact and allowed to exchange a quantity, say mass, which is conserved in the combined system. Will the systems eventually evolve to a new stationary state where a certain intensive thermodynamic variable, like equilibrium chemical potential, equalizes following the zeroth law of thermodynamics and, if so, under what conditions is it possible? We argue that an equilibriumlike thermodynamic structure can be extended to nonequilibrium steady states having short-ranged spatial correlations, provided that the systems interact weakly to exchange mass with rates satisfying a balance condition—reminiscent of a detailed balance condition in equilibrium. The short-ranged correlations would lead to subsystem factorization on a coarse-grained level and the balance condition ensures both equalization of an intensive thermodynamic variable as well as ensemble equivalence, which are crucial for construction of a well-defined nonequilibrium thermodynamics. This proposition is proved and demonstrated in various conserved-mass transport processes having nonzero spatial correlations.

Figure 1

Schematic representation of “equilibration” of two steady-state systems in contact. Intensive thermodynamic variables ${\mu }_1\left(t\right)$ and ${\mu }_2\left(t\right)$, chemical potentials of systems 1 and 2 at time $t$, eventually equalize in the steady state, ${\mu }_1(t=\infty )={\mu }_2(t=\infty )$. The size of the contact region $v^{1/d},v$ the volume of the contact region in the $d$ dimension, is much larger than the individual correlation length ${\xi }_{\alpha }$.

Figure 2

“Equilibration” of steady states in contact. In (a)–(e), chemical potentials ${\mu }_1\left(t\right)$ and ${\mu }_2\left(t\right)$ of systems $1$ (red solid lines) and $2$ (blue dotted lines) vs rescaled time $u_{0}t$. ${\mu }_1$ and ${\mu }_2$, initially chosen to be different, eventually equalize. Densities $({\rho }_1,{\rho }_2)$ in the final steady states are, respectively, $(3.60,5.40)$ in (a), $(3.57,5.43)$ in (b), $(5.31,2.69)$ in (c), $\left(5.32,2.68\right)$ in (d), and $(3.32,6.68)$ in (e). In all cases, $p\left(\varepsilon \right)=p_b\left(\varepsilon \right)=exp(-\varepsilon ),u_0=0.1$, and $v=10$ [except in (c) and (d) where $v=1$].