Exponential convergence to equilibrium in cellular automata asymptotically emulating identity

TL;DR

This paper investigates the asymptotic behavior of the density of 1's in cellular automata rules that emulate the identity, providing explicit formulas for some rules and conjecturing exponential convergence to equilibrium.

Contribution

It derives density formulas for two new CA rules in the class of identity-emulating rules and proposes a general conjecture on exponential convergence for all such rules.

Findings

Density formulas for rules 160 and 168 derived.

Conjecture that density approaches equilibrium exponentially fast.

Likely density expressions proposed for eight additional rules.

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.