An exactly solvable evaporation-deposition PCA with long-distance interactions

TL;DR

This paper introduces an exactly solvable probabilistic cellular automaton modeling evaporation and deposition with long-distance interactions, providing explicit formulas for its stationary distribution, partition function, and conditions for reversibility.

Contribution

It presents a novel exactly solvable PCA with long-distance interactions, deriving explicit formulas for its stationary distribution and analyzing its reversibility conditions.

Findings

The Markov chain is ergodic.

Explicit formulas for the stationary distribution and partition function are provided.

Conditions for reversibility are characterized.

Abstract

We consider a probabilistic cellular automaton (PCA) of evaporation-deposition on the one-dimensional lattice having $n$ sites with periodic boundary conditions, in which each site, during each epoch, can be in one of two states: $0$ and $1$. Fix a positive integer $m\geqslant 2$. There are two types of transitions at each discrete time, which are as follows: (i) the first site in every contiguous block of $m$ $0$s becomes a $1$ with probability $p_1$, and (ii) the first site in every contiguous block of $(m-1)$ $0$s followed immediately by a $1$ also becomes a $1$ with probability $(1-p_2)$. As in a PCA, all of these transitions occur simultaneously. We show that the resulting discrete-time Markov chain is ergodic, and we give an explicit formula for its limiting distribution, the partition function and the density. We also propose necessary and sufficient conditions for this Markov chain to be reversible. For $m=2$, we provide a fully analytical expression for the free energy of this model.