Physical approach to the ergodic behavior of stochastic cellular automata with generalization to random processes with infinite memory

Abstract

The large time behavior of stochastic cellular automata is investigated by means of an analogy with Van Kampen's approach to the ergodic behavior 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 according to a stochastic local rule connecting the variables corresponding to two successive time steps, the dynamics of the process can be described 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 statistical characterization of the system. Exact equations for the moments of all times q 1, …, q M are derived in terms of the propagator of the random walk. It is shown that for large values of the total time q = Σ u q u the average individual times 〈 q u 〉 attached to the different configurations u = 1, …, M are proportional to the corresponding stationary state probabilities P u st: 〈 q u 〉 ∼ q P u st, u = 1, …, M. These asymptotic laws show that in the long run the cellular automaton is ergodic, that is, for large times the ensemble average of a property depending on the configurations of the lattice is equal to the corresponding temporal average evaluated over a very large time interval. For large total times q the correlation functions of the individual sojourn times q 1, …, q M increase linearly with the total number of time steps q: 〈 Δq u Δq u′ 〉 ∼ q as q → ∞ which corresponds to non-intermittent fluctuations. An alternative approach for investigating the ergodic behavior of Markov processes in discrete space and time is suggested on the basis of a multiple averaging of a Kronecker symbol; this alternative approach can be extended to non-Markovian random processes with infinite memory. The implications of the results for the numerical analysis of the large time behavior of stochastic cellular automata are also investigated.