A remark on the choice of stochastic transition rates in driven nonequilibrium systems

We study nonequilibrium steady states of the driven lattice gas with two particles, using the most general stochastic transition rules that satisfy the local detailed balance condition. We observe that i) the universal 1/rd long range correlation may be found already in the two-particle models, but ii) the magnitude (or even the existence/absence) of the long range correlation depends crucially on the rule for transition rates. The latter is in stark contrast with equilibrium dynamics, where all rules give essentially the same results provided that the detailed balance condition is satisfied. Stochastic processes with discrete state space are often studied as convenient idealized models of various physical systems. As for time evolution at or close to equilibrium, it has been established [1, 2] that transition rates must satisfy the detailed balance condition in order for the system to obey a macroscopic symmetry known as reciprocity. But no such criteria are known for systems far from equilibrium. A standard convention (see, for example, [3, 4, 5]) is to take transition rates satisfying the local detailed balance condition, which is a direct generalization of the detailed balance condition. It seems that an implicit assumption has been that the specific choice of transition rates does not affect physics in a serious manner. We here show that this is far from the case in driven nonequilibrium systems . We here study the driven lattice gas [3, 4, 5], which is one of the standard models of nonequilibrium systems driven by an external force. We use the most general transition rates satisfying the local detailed balance condition, and obtain the exact steady states of the models when there are only two particles. We first drive a simple condition under which the steady state is the same as the equilibrium state. The condition is satisfied in a large class of rules, but not in some of the standard ones including the Metropolis and the heat bath rules. Then we turn to transition rates which do not satisfy the condition, and investigate nonequilibrium corrections to the steady states. We find that, in the dimensions d ≥ 2, the two-particle models may exhibit the universal 1/r long range correlation, which is often regarded as an essential feature of nonequilibrium steady states. Moreover the magnitude of the long range correlation depends crucially on the choice of transition rates: it is extremely large in the Metropolis rule while it is absent in the exponential rule. Although there were some remarks [6] about rule dependence in the driven lattice gas, they were mainly quantitative and did not show the sharp implication on the long range correlation. As for the one-dimensional lattice gas driven by boundary conditions, it was shown recently [7] that the model with the exponential rule has a long-range correlation while that with the zero-range rule has no correlation. Department of Physics, Gakushuin University, Mejiro, Toshima-ku, Tokyo 171-8588, Japan