Inconsistencies in steady state thermodynamics

We address the issue of extending thermodynamics to nonequilibrium steady states. Using driven stochastic lattice gases, we ask whether consistent definitions of an effective chemical potential $\mu$, and an effective temperature $T_e$, are possible. These quantities are determined via zero-flux conditions of particles and energy between the driven system and a reservoir. For the models considered here, the fluxes are given in terms of certain stationary average densities, eliminating the need to perturb the system by actually exchanging particles; $\mu$ and $T_e$ are thereby obtained via open-circuit measurements, using a virtual reservoir. In the lattice gas with nearest-neighbor exclusion, temperature is not relevant, and we find that the effective chemical potential, a function of density and drive strength, satisfies the zeroth law, and correctly predicts the densities of coexisting systems. In the Katz-Lebowitz-Spohn driven lattice gas, both $\mu$ and $T_e$ need to be defined. We show analytically that the zeroth law is violated, and determine the size of the violations numerically. Our results highlight a fundamental inconsistency in the extension of thermodynamics to nonequilibrium steady states.