Generalized Boltzmann equation for lattice gas automata

In this paper a theory is formulated that predicts velocity and spatial correlations between occupation numbers that occur in lattice gas automata violating semi-detailed balance. Starting from a coupled BBGKY hierarchy for then-particle distribution functions, cluster expansion techniques are used to derive approximate kinetic equations. In zeroth approximation the standard nonlnear Boltzmann equation is obtained; the next approximation yields the ring kinetic equation, similar to that for hard-sphere systems, describing the time evolution of pair correlations. The ring equation is solved to determine the (nonvanishing) pair correlation functions in equilibrium for two models that violate semidetailed balance. One is a model of interacting random walkers on a line, the other one is a two-dimensional fluid-type model on a triangular lattice. The numerical predictions agree very well with computer simulations.

[1]  N. Balazs,et al.  Fundamental Problems in Statistical Mechanics , 1962 .

[2]  R. Piccirelli,et al.  Basis of the Functional Assumption in the Theory of the Boltzmann Equation , 1963 .

[3]  M. Ernst,et al.  Nonequilibrium fluctuations in μ space , 1981 .

[4]  Cohen,et al.  Dynamical processes in hard-sphere fluids. , 1987, Physical review. A, General physics.

[5]  Zanetti,et al.  Hydrodynamics of lattice-gas automata. , 1989, Physical review. A, General physics.

[6]  Ernst,et al.  Simulation of diffusion in a two-dimensional lattice-gas cellular automaton: A test of mode-coupling theory. , 1989, Physical review letters.

[7]  P. C. Rem,et al.  The Construction of Efficient Collision Tables for Fluid Flow Computations with Cellular Automata , 1989 .

[8]  M. Ernst,et al.  Lattice Lorentz gas , 1989 .

[9]  Ernst,et al.  Long-time tails in two-dimensional cellular automata fluids. , 1990, Physical review. A, Atomic, molecular, and optical physics.

[10]  U. Frisch,et al.  Low-viscosity lattice gases , 1990 .

[11]  Ernst,et al.  Kinetic theory for lattice-gas cellular automata. , 1991, Physical review. A, Atomic, molecular, and optical physics.

[12]  Anna T. Lawniczak,et al.  Reactive dynamics in a multispecies lattice‐gas automaton , 1992 .

[13]  Middleton,et al.  Self-organization and a dynamical transition in traffic-flow models. , 1992, Physical review. A, Atomic, molecular, and optical physics.

[14]  M. Hénon.  Implementation of the FCHC lattice gas model on the Connection Machine , 1992 .

[15]  D. Rothman,et al.  Lattice-gas and lattice-Boltzmann models of miscible fluids , 1992 .

[16]  Biased lattice gases with correlated equilibrium states , 1992 .

[17]  Ring kinetic theory for tagged-particle problems in lattice gases. , 1992, Physical review. A, Atomic, molecular, and optical physics.

[18]  M. Ernst,et al.  Lattice gases with static disorder: Renormalization of mean field theory , 1993 .

[19]  M. Ernst,et al.  Stress-stress correlation functions in lattice gases beyond Boltzmann's approximation , 1993 .

[20]  Ernst,et al.  Generalized hydrodynamics and dispersion relations in lattice gases. , 1993, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[21]  M. Ernst,et al.  Collective diffusion in a lattice gas automaton , 1993 .

[22]  Ernst,et al.  Lattice-gas automata with attractive and repulsive interactions. , 1993, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.