Abstract
We investigate a stochastic version of cellular automata used for simulating hydrodynamical flows, e.g. the HPP and FHP models. The extra stochasticity consists of “random exchanges” between neighboring cells which conserve momentum. We prove that, in suitable limits, these models satisfy the appropriate continuous Boltzmann and hydrodynamic equations, the same as those conjectured for the original models (except that there is no negative viscosity contribution). The results are obtained by proving a very strong form of propagation of chaos and by using Hilbert-Chapman-Enskog type expansions. Explicit proofs are presented for the stochastic HPP model.
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References
Frish, U., Hasslacher, B., Pomeau, Y.: Lattice gas automata for Navier Stokes equation. Phys. Rev. Lett.56, 1505 (1986)
Frisch, U., d'Humières, D., Hasslacher, B., Lallemand, P., Pomeau, Y., Rivet, J.-P.: Lattice gas hydrodynamics in two and three dimensions. Complex Syst.1, 648 (1987)
Wolfram, S.: Cellular Automaton fluids. 1. Basic theory. J. Stat. Phys.45, 471 (1986)
Hardy, J., de Pazzis, O., Pomeau, Y.: Molecular dynamics of classical lattice gas: Transport properties and time correlation functions. Phys. Rev. A13, 1949 (1976)
DeMasi, A., Esposito, R. Lebowitz, J.L., Presutti, E.: Rigorous results on some stochastic cellular automata, Proceedings of the Conference: Discrete kinetic theory, lattice gas dynamics and foundations of hydrodynamics held in Torino, Sept. 20–23, 1988. World Scientific (1989)
DeMasi, A., Esposito, R., Lebowitz, J.L.: Incompressible Navier-Stokes and Euler limits of the Boltzmann equation. Commun. Pure Appl. Math. (to appear)
Boghosian, B., Levermore, D.: A cellular automaton for Burgers equation. Complex Syst.1, 17–30 (1987)
Higuera, F.J., Succi, S., Benzi, R.: Lattice gas dynamics with enhanced collisions. Preprint 1989
McNamara, G., Zannetti, G.: Phys. Rev. Lett. (1988)
Lebowitz, J.L., Orlandi, E., Presutti, E.: Convergence of stochastic cellular automaton to Burgers' equation: fluctuations and stability. Physica D33, 165–188 (1988)
Calderoni, P., Pellegrinotti, A., Presutti, E., Vares, M.E.: Transient bimodality in interacting particle systems. J. Stat. Phys. (to appear)
Lanford III, O.E.: Time evolution of large classical systems; Dynamical systems, theory and applications. Moser, J., (ed.) Lecture Notes in Physics, vol. 38, p. 1, Berlin, Heidelberg, New York: Springer 1975
Caflisch, R.: The fluid dynamic limit of the nonlinear Boltzmann equation. Commun. Pure Appl. Math.33, 651–666 (1980)
Bardos, C., Golse, F., Levermore, D.: Compte Rendu (to appear)
Ladyzhenskaja, O.A.: The mathematical theory of Viscous incompressible flows. New York: Gordon and Breach 1969
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Communicated by A. Jaffe
Dedicated to Roland Dobrushin
Research partially supported by CNR-PS-MMAIT and NSF grant no. 86–12369
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DeMasi, A., Esposito, R., Lebowitz, J.L. et al. Hydrodynamics of stochastic cellular automata. Commun.Math. Phys. 125, 127–145 (1989). https://doi.org/10.1007/BF01217773
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DOI: https://doi.org/10.1007/BF01217773