Failure of steady-state thermodynamics in nonuniform driven lattice gases

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.

Figure 1

Chemical potential ${\mu }^{*}$ versus particle density $\rho$ in the lattice gas with NN exclusion for (upper to lower) equilibrium, maximum drive ($D=1$) under nearest- and next-nearest neighbor hopping (NNE2 dynamics), and maximum drive under nearest-neighbor hopping (NNE dynamics). Data are simulation results for the square lattice, system size $L=80$. Uncertainties are smaller than the line width.

Figure 2

Chemical potential ${\mu }^{*}$ versus exchange rate $p_r$ under pointwise exchange (upper: driven system; lower: nondriven system) in the lattice gas with NNE2 dynamics. System size $L=20$, mean density 0.3.

Figure 3

Stationary particle density versus distance $r$ from exchange site under pointwise exchange (upper: driven system; lower: nondriven system) in the lattice gas with NNE2 dynamics. Open symbols: $p_r=0.05$; filled symbols: $p_r=0.005$. Parameters as in Fig. 2.

Figure 4

Density (upper) and chemical potential (lower) profiles in the half-driven NNE model. Drive $D=0$ for $x\le 200$; $D=1$ for $x>200$. Global densities $\overline{\rho }=0.2$ (left) and 0.25 (right). In the upper panels the dashed lines show the coexisting densities predicted by equating chemical potentials. In the lower panels the dashed lines show the expected uniform value of the chemical potential, and the dotted lines show the chemical potentials for the isolated systems, each at density $\overline{\rho }$.

Figure 5

Coexisting bulk particle densities ${\rho }_D$ and ${\rho }_0$, in driven and undriven regions, respectively, in the NNE model, as predicted by equating the chemical potentials in isolated systems (upper curve) and observed in simulations of the half-driven system (lower curve). Error are bars smaller than symbols. For purposes of comparison, the dashed line represents ${\rho }_D={\rho }_0$.

Figure 6

Density profiles in the half-driven NNE model with global density 0.22; drive $D=0$ for $x\le 100$; $D=1$ for $x>100$. Black: $p_r=1$; blue: $p_r=0.005$; red: $p_r=0.002$.

Figure 7

Density (upper) and chemical potential (lower) profiles in the half-driven NNE2 model with global density $\overline{\rho }=0.32$. Drive $D=0$ for $x\le 200$; $D=1$ for $x>200$. In the upper panel the dashed lines show the coexisting densities predicted by equating chemical potentials. In the lower panel the dashed line shows the expected uniform value of the chemical potential.

Figure 8

Coexisting bulk particle densities ${\rho }_D$ and ${\rho }_0$, in driven and undriven regions, respectively, of the NNE2 model, as predicted by equating the chemical potentials in isolated systems (lower curve at left) and observed in simulation (upper curve at left). For purposes of comparison, the dashed line represents ${\rho }_D={\rho }_0$.

Figure 9

Points: Bulk chemical potential versus bulk density in driven (lower) and undriven (upper) regions of half-driven NNE2 model. The curves show the corresponding values for isolated systems. Error bars are smaller than symbols.

Figure 10

Clusters on the square lattice. A uniform chemical potential requires that the ratio of the probabilities of clusters $a$ and $b$ be the same everywhere. A time-independent density profile perpendicular to the drive D requires that at each row parallel to D, the probabilities of clusters $c$ and $d$ be equal.

Figure 11

NNE2 lattice gas confined between hard walls parallel to the drive direction. Density profiles (upper) and chemical potential profiles (lower) in equilibrium (broken line) and under maximum drive (solid lines), for overall density $\rho =0.26$.

Figure 12

NNE lattice gas with periodic boundaries, with slow bonds (hopping probability $p_r=0.01$) between rows $y=L/2$ and $y=L/2+1$. Density profiles (upper) and chemical potential profiles (lower) in equilibrium (flat profiles) and under maximum drive (nonuniform profiles) for overall density $\rho =0.25$.

Figure 13

NNE lattice gas with all hopping rates reduced by $q=0.5$ in the central region (see text). Density profiles (upper) and chemical potential profiles (lower) in equilibrium (dashed lines) and under maximum drive (solid lines) for overall density $\rho =0.15$.