On two reversible cellular automata with two particle species

TL;DR

This paper introduces two reversible cellular automata with two particle species, exploring their symmetries, conserved charges, equilibrium states, and hydrodynamic behavior, revealing unexpected slow relaxation phenomena.

Contribution

It provides explicit solutions for conserved charges, matrix product states for equilibrium, and empirical evidence of algebraic integrability in one model, advancing understanding of reversible cellular automata.

Findings

Closed-form conserved charges identified

Explicit matrix product form of Gibbs states derived

Hydrodynamic predictions do not match observed sound velocity

Abstract

We introduce a pair of time-reversible models defined on the discrete space-time lattice with 3 states per site, specifically, a vacancy and a particle of two flavours (species). The local update rules reproduce the rule 54 reversible cellular automaton when only a single species of particles is present, and satisfy the requirements of flavour exchange (C), space-reversal (P), and time-reversal (T) symmetries. We find closed-form expressions for three local conserved charges and provide an explicit matrix product form of the grand canonical Gibbs states, which are identical for both models. For one of the models this family of Gibbs states seems to be a complete characterisation of equilibrium (i.e. space and time translation invariant) states, while for the other model we empirically find a sequence of local conserved charges, one for each support size larger than 2, hinting to its algebraic integrability. Finally, we numerically investigate the behaviour of spatio-temporal correlation functions of charge densities, and test the hydrodynamic prediction for the model with exactly three local charges. Surprisingly, the numerically observed 'sound velocity' does not match the hydrodynamic value. The deviations are either significant, or they decay extremely slowly with the simulation time, which leaves us with an open question for the mechanism of such a glassy behaviour in a deterministic locally interacting system.