  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.
