PHYSICAL APPROACH TO THE ERGODIC BEHAVIOR OF STOCHASTIC CELLULAR-AUTOMATA WITH GENERALIZATION TO RANDOM-PROCESSES WITH INFINITE MEMORY

The large time behavior of stochastic cellular automata is investigate d by means of an analogy with Van Kampen's approach to the ergodic beh avior of Markov processes in continuous time and with a discrete state space. A stochastic cellular automaton with a finite number of cells may display an extremely large, but however finite number M of lattice configurations. Since the different configurations are evaluated acco rding to a stochastic local rule connecting the variables correspondin g to two successive time steps, the dynamics of the process can be des cribed in terms of an inhomogeneous Markovian random walk among the M configurations of the system. An infinite Lippman-Schwinger expansion for the generating function of the total times q(1),...,q(M) spent by the automaton in the different M configurations is used for the statis tical characterization of the system. Exact equations for the moments of all times q(1),...,q(M) are derived in terms of the propagator of t he random walk. It is shown that for large values of the total time q = Sigma(u)q(u) the average individual times [q(u)] attached to the dif ferent configurations u = 1,..., M are proportional to the correspondi ng stationary state probabilities P-u(st): [q(u)] similar to qP(u)(st) , u = 1,...,M. These asymptotic laws show that in the long run the cel lular automaton is ergodic, that is, for large times the ensemble aver age of a property depending on the configurations of the lattice is eq ual to the corresponding temporal average evaluated over a very large time interval. For large total times q the correlation functions of th e individual sojourn times q(1),...,q(M) increase linearly with the to tal number of time steps q: [Delta q(u) Delta q(u')) similar to q as q --> infinity which corresponds to non-intermittent fluctuations. An a lternative approach for investigating the ergodic behavior of Markov p rocesses in discrete space and time is suggested on the basis of a mul tiple averaging of a Kronecker symbol; this alternative approach can b e extended to non-Markovian random processes with infinite memory. The implications of the results for the numerical analysis of the large t ime behavior of stochastic cellular automata are also investigated.