Failure of steady-state thermodynamics in nonuniform driven lattice gases

Abstract

To be useful, steady-state thermodynamics (SST) must be self-consistent and have predictive value. Consistency of SST was recently verified for driven lattice gases under global weak exchange. Here I verify consistency of SST under local (pointwise) exchange, but only in the limit of a vanishing exchange rate; for a finite exchange rate the coexisting systems have different chemical potentials. I consider the lattice gas with nearest-neighbor exclusion on the square lattice, with nearest-neighbor hopping, and with hopping to both nearest and next-nearest neighbors. I show that SST does not predict the coexisting densities under a nonuniform drive or in the presence of a nonuniform density provoked by a hard wall or nonuniform transition rates. The steady-state chemical potential profile is, moreover, nonuniform at coexistence, contrary to the basic principles of thermodynamics. Finally, I discuss examples of a pair of systems possessing identical steady states but which do not coexist when placed in contact. The results of these studies confirm the validity of SST for coexistence between spatially uniform systems but cast serious doubt on its consistency and predictive value in systems with a finite rate of particle exchange between coexisting regions exhibiting a nonuniform particle density.

Zeroth law and nonequilibrium thermodynamics for steady states in contact

“…For example, if the contact site is located in the bulk, the chemical potential has to be calculated with respect to the density and probability of the open site in the bulk. That is, even in these cases of nonuniform systems, the existence of the above mentioned chemical potential would then apparently restore an equilibriumlike thermodynamic structure, as formulated in [12,21] where an ITV equalizes upon contact and zeroth law is obeyed.…”

“…Next we consider previously studied athermal hardcore lattice gases, in two dimensions, with nearest neighbor exclusion (NNE) [12,21]. We study the simplest case where particles can be exchanged through a single-site or pointwise contact (v = 1) in each system, which can be readily generalized to other cases, e.g, when particles are exchanged globally (v = V ) or in higher dimensions.…”

“…We study the simplest case where particles can be exchanged through a single-site or pointwise contact (v = 1) in each system, which can be readily generalized to other cases, e.g, when particles are exchanged globally (v = V ) or in higher dimensions. The transition rates for particles hopping inside the individual systems (irrespective of that they are isolated or in contact with each other) can be chosen to be some specific nearest neighbor or next-nearest neighbor (or mixture of both) hopping rates in the presence of a driving field D; details of these rates, which can be found in [12,21], are omitted here as they are not explicitly required in the following analysis as long as the systems exchange particles very slowly.…”

“…Let us keep two such lattice gases, systems α = 1 and 2, in contact with each other [12,21] where particles are exchanged as follows. A site is called open if the site as well as all its nearest neighbours are unoccupied.…”

“…These models have a product measure or factorized steady state and therefore do not have any spatial correlations. In other studies, a class of lattice gas models with nonzero spatial correlations were considered [10,12,18,21,22] and, for some particular choice of mass exchange rates, zeroth law was found to be obeyed. However, the mass exchange rates, even in the limit of slow exchange, alters the fluctuation properties of the individual systems, leading to the breakdown of equivalence between canonical and grandcanonical ensembles.…”