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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. mu and T_e are determined via coexistence, i.e., zero flux of particles and energy between the driven system and a 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 in this case the zeroth law is violated for Metropolis exchange rates, and determine the size of the violations numerically. The zeroth law appears to be violated for generic exchange rates. Remarkably, the system-reservoir coupling proposed by Sasa and Tasaki [J. Stat. Phys. 125, 125 (2006)], is free of inconsistencies, and the zeroth law holds. This is because the rate depends only on the state of the donor system, and is independent of that of the acceptor.