TY - JOUR

T1 - Convergence of stochastic cellular automation to Burgers' equation

T2 - Fluctuations and stability

AU - Lebowitz, Joel L.

AU - Orlandi, Enza

AU - Presutti, Errico

N1 - Funding Information:
The use of cellular automata to model hydrodynamical flows of all varieties has grown explosively in the last two years from the spark started by Frisch, Hasslacher and Pomeau (FHP) in their seminal paper \[1\]: for earlier work see \[2\],f or recent work \[3, 4\]. While the numerical results so far appear promising, at least to the devotees, the difficulties involved in proving convergence of the FHP model to the solutions of the Euler or Navier-Stokes equations appear almost as formidable as proving this for a system composed of atoms obeying Newtonian dynamics. In both cases the microscopic dynamics conserve mass, momentum and energy. The macroscopic fields corresponding to these quantities, obtained as sums of many microscopic variables, are then expected to evolve, on macroscopic space-time scales, according to suitable deterministic hydrodynamic equations \[5\].M ore precisely, we expect the non-conserved rapidly varying variables to accommodate themselves to the "instantaneous" values of the slowly varying ones to produce states of "local equilibrium" with parameters specified by the values of the slow variables. The central mathematical problem is then to show that the correlations created by the dynamics do not take the system out from such a local equilibrium state. (This is of course true by definition for a global equilibrium state-but such a state is of no interest for time evolution.) In order to get some insight into the questions posed by cellular automaton evolutions Boghosian and Levermore (BL) \[6\] invented a cellular automaton model for solving the one-dimensional Burgers' *Supported by NSF Grant No. DMR-86-12369. Part of this work was done while the authors were at the Institute for Theoretical Physics, University of California, Santa Barbara, California, where it was supported by NSF Grant No. Phy-82-17853 supplemented by funds from NASA. 1Also at Department of Physics. 2permanent Address: Dipartimento di Matematica, Universith di Roma "La Sapienza", P.le A.Moro 2, 00187 Roma, Italy. Partially supported by CNR PS-AITM grant. 3present Address: Department of Mathematics, University of Colorado, Boulder, CO 80309, USA.

PY - 1988

Y1 - 1988

N2 - We prove that for almost all realizations of the Boghosian-Levermore stochastic cellular automaton model the density profile converges, in the scaling limit, to the solution of Burgers' equation. The proof goes via the propagation of chaos and yields tight bounds on the fluctuations. These estimates also yield stability properties of the (smooth) shock front: at long times it remains well defined on a microscopic scale-but its location fluctuates.

AB - We prove that for almost all realizations of the Boghosian-Levermore stochastic cellular automaton model the density profile converges, in the scaling limit, to the solution of Burgers' equation. The proof goes via the propagation of chaos and yields tight bounds on the fluctuations. These estimates also yield stability properties of the (smooth) shock front: at long times it remains well defined on a microscopic scale-but its location fluctuates.

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U2 - 10.1016/S0167-2789(98)90017-3

DO - 10.1016/S0167-2789(98)90017-3

M3 - Article

AN - SCOPUS:0039324945

VL - 33

SP - 165

EP - 188

JO - Physica D: Nonlinear Phenomena

JF - Physica D: Nonlinear Phenomena

SN - 0167-2789

IS - 1-3

ER -