Statistical Mechanics of Surjective Cellular Automata

Reversible cellular automata are seen as microscopic physical models, and their states of macroscopic equilibrium are described using invariant probability measures. Characterizing all the invariant measures of a cellular automaton could be challenging. Nevertheless, we establish a connection between the invariance of Gibbs measures (used in statistical mechanics to describe thermodynamic equilibrium) and the conservation of additive quantities in surjective cellular automata. Namely, we show that the simplex of shift-invariant Gibbs measures associated to a Hamiltonian is invariant under a surjective cellular automaton if and only if the cellular automaton conserves the Hamiltonian. A special case is the uniform Bernoulli measure, which is well known to be invariant under every surjective cellular automaton, and which corresponds to the conservation of the trivial Hamiltonian. As an application, we obtain that strongly transitive cellular automata have no invariant Gibbs measures other than the uniform Bernoulli measure. Another result is that a one-dimensional reversible cellular automaton that has a positively expansive transpose cannot preserve any full-support Markov measure other than the uniform Bernoulli measure. A general description of approach to equilibrium in reversible cellular automata is missing, but we speculate on the relevance of the randomization property of algebraic cellular automata (as suggested by others), and pose several open questions. As an aside, it turns out that a shift-invariant pre-image of a Gibbs measure under a preinjective factor map between shifts of finite type is always a Gibbs measure. We provide a sufficient condition under which the image of a Gibbs measure under a pre-injective factor map is not a Gibbs measure. We hint on a possible application of pre-injective factor maps in the study of phase transitions in statistical mechanical models.