Failure of Steady State Thermodynamics

Steady state thermodynamics (SST) is an attempt to extend
thermodynamics to nonequilibrium steady states arbitrarily far
from equilibrium, attributing intensive variables such as
temperature and chemical potential to these systems. To be
useful, SST must be self-consistent and have predictive value.
I examine this issue in the context of driven stochastic
lattice gases. Consistency of SST is verified for driven
lattice gases under global weak exchange, but only for a
particular class transition rates, those defined by Sasa and
Tasaki. Under local (pointwise) exchange, consistency only
holds in the limit of a vanishing exchange rate. SST fails to
predict the coexisting densities under a nonuniform drive, or
in the presence of a nonuniform density provoked by a wall or
nonuniform transition rates. The steady state chemical
potential profile is, moreover, nonuniform at coexistence,
contrary to the basic principles of thermodynamics. As a
further example, I discuss examples of a pair of systems
possessing identical steady states, but which do not coexist
when placed in contact. I then turn to the question of phase
coexistence in nonequilibrium steady states. Results on the
driven lattice gas with attractive interactions show that the
bulk densities of coexisting phases are different for phases
that (1) separate spontaneously in a single inhomogeneous
system, and (2) phases that coexist under particle exchange
between two homogeneous systems. These results cast serious
doubt on the consistency and predictive value of SST, and on
the notion of thermodynamic phases far from equilibrium.