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Failure of steady state thermodynamics in lattice gases under nonuniform drive

To be useful, steady state thermodynamics (SST) must be self-consistent and have predictive value. Although consistency of SST was recently verified for driven lattice gases under global weak exchange, I show here that it does not predict the coexisting densities in athermal stochastic lattice gases under a nonuniform drive. I consider the lattice gas with nearest-neighbor exclusion on the square lattice, with nearest-neighbor hopping (NNE dynamics), and with hopping to both nearest and next-nearest neighbors (NNE2 dynamics). Part of the system is subject to a drive $D$ that favors hopping along one direction, while the other part is free of the drive. Thus the steady state represents coexistence between two subsystems, one far from equilibrium and the other in equilibrium, which exchange particles along the interfaces. The dimensionless chemical potential $μ^*(ρ,D) = μ/k_B T$ for the lattice gas with density $ρ$ is readily determined in studies of a uniform system. Under the nonuniform drive, however, equating the chemical potentials of the coexisting subsystems does not yield the coexisting densities. The steady state chemical potential is, moreover, different in the coexisting bulk regions, contrary to the basic principles of thermodynamics. These results cast serious doubt on the predictive value of SST.