Exponential Convergence to Equilibrium in Cellular Automata Asymptotically Emulating Identity

Abstract

We consider the problem of finding the density of 1's in a configuration obtained by n iterations of a given cellular automaton (CA) rule, starting from disordered initial condition. While this problems is intractable in full generality for a general CA rule, we argue that for some sufficiently simple classes of rules it is possible to express the density in terms of elementary functions. Rules asymptotically emulating identity are one example of such a class, and density formulae have been previously obtained for several of them. We show how to obtain formulae for density for two further rules in this class, 160 and 168, and postulate likely expression for density for eight other rules. Our results are valid for arbitrary initial density. Finally, we conjecture that the density of 1's for CA rules asymptotically emulating identity always approaches the equilibrium point exponentially fast.

An Example of Computation of the Density of Ones in Probabilistic Cellular Automata by Direct Recursion

“…Although this method is obviously applicable only to PCA rules with rather simple dynamics, it may be possible to find other PCA rules with complete sets, thus making the method useful for them. Generalization of the rule used in this paper to larger neighbourhood sizes comes to mind as a first possibility, and sufficiently simple deterministic rules, such as asymptotic emulators of identity investigated in [8], are also possible candidates.…”

Orbits of Bernoulli Measures in Cellular Automata

“…Indeed, for many other elementary CA rules for which formulae for P n (0) are either known or conjectured, the exponential convergence to P ∞ (0) can be observed most frequently. This includes 15 elementary rules which are known as asymptotic emulators of identity (Rogers and Want, 1994;Fukś and Soto, 2014). Formulae for P n (1) for the initial measure µ 1/2 for these rules are shown below.…”

“…Formulae for P n (1) for the initial measure µ 1/2 for these rules are shown below. Starred rules are those for which a formal proof has been published in the literature (see Fukś and Soto, 2014, and references therein).…”

An Example of a Deterministic Cellular Automaton Exhibiting Linear-exponential Convergence to the Steady State

“…One can easily show (in a manner similar as done in [14]) that if one starts with a bi-infinite string of symbols drawn from a Bernoulli distribution where probabilities of 0, 1 and 2 are, respectively, p, q and r, then the expected proportion of sites in state k after n iterations of rule f is given by Ψ f −n (k) (p, q, r). This quantity will be called density of symbols k after n iterations of f .…”

“…Cellular automata are infinitely-dimensional dynamical systems, yet a behaviour similar to hyperbolicity in finite-dimensional systems has been observed in many of them. In particular, in some binary cellular automata in one dimension, known as asymptotic emulators of identity, if the initial configuration is drawn from a Bernoulli distribution, the expected proportion of ones (or zeros) usually tends to its stationary value exponentially fast [14]. This type of behaviour is quite common in many other dynamical systems.…”