Abstract
In this paper we consider a lattice gas as a discrete Markov process, with a Markov operator\(\mathfrak{L}\) acting on the phase space of the lattice gas cellular automata. We are interested in the asymptotic properties of the sequences of densities in both Liouville and Boltzmann descriptions. We show that under appropriate hypotheses, in both descriptions, the sequence of densities are asymptotically periodic. It is then possible, by introducing a slight modification in the transition process, to avoid the existence of cycles and to ensure the stability of the stationary densities. We point out the particular part played by the regular global linear invariants that define the asymptotic Gibbs states in a one-to-one way for most models.
Similar content being viewed by others
Referencens
D. Bernardin, Global invariants and equalibrium states in lattice gases,J. Stat. Phys. 68:457–495 (1992).
U. Frisch, B. Hasslacher, and Y. Pomeau, Lattice gas automata for the Navier-Stokes euations,Pys. Rev. Lett.46:15050–1508 (1986).
M. H. Ernst, Linear response theory for celular automata fluids, inFundamental Problems in Statistical Mechanics VII (North-Holland, Amsterdam, 1990), pp. 321–355.
U. Frisch, D. d'Humières, B. Hasslacher, Y. Pomeau, and J. P. Rivet, Lattice gas hydrodynamics in two and three dimensions.Complex Systems 1:649–707 (1987).
A Demasi, R. Esposito, J. L. Lebowitz, and E. Presutti, Rigourous results on some stochastic cellular automata, inDiscrete Kinetic Theory, Lattice Gas Dynamics and Foundatios of Hydrodynamics (World Scientific, Singapore, 1989), pp 93–101.
A. Lasota and M. C. Mackey,Probabilistic Properties of Deterministic Systems (Cambridge University Press, Cambridge 1985).
R. Gatignol,Théorie cinétique des gaz à répartition discrète de vitesses (Springer-Verlag, Berlin, 1975).
E. B. Dynkin,Théorie des processus markoviens (Dunod, 1963).
J. Fritz, Stationary states and hydrodynamics of FHP cellular automata,J. Stat. Phys. 77:53–76 (1994).
B. Boghosian and D. Levermore, A cellular automaton for Burgers equations,Complex Systems 1:17–30 (1987).
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Bernardin, D., Sero-Guillaume, O.E. Exact stability results in stochastic lattice gas cellular automata. J Stat Phys 81, 409–443 (1995). https://doi.org/10.1007/BF02179987
Received:
Accepted:
Issue Date:
DOI: https://doi.org/10.1007/BF02179987