Integrability of a deterministic cellular automaton driven by stochastic boundaries

TL;DR

This paper introduces an exactly solvable model combining a deterministic reversible cellular automaton with stochastic boundary conditions, leading to explicit steady state solutions and insights into integrability of boundary-driven lattice systems.

Contribution

It presents a novel integrable model with stochastic boundaries, providing explicit steady states and conservation laws for a deterministic cellular automaton.

Findings

Proven ergodicity and mixing for generic bath parameters.

Explicit construction of the non-equilibrium steady state using a simple matrix product ansatz.

Identification of local conservation laws and potential for full spectral solution.

Abstract

We propose an interacting many-body space-time-discrete Markov chain model, which is composed of an integrable deterministic and reversible cellular automaton (the rule 54 of [Bobenko et al, CMP 158, 127 (1993)]) on a finite one-dimensional lattice Z_2^n, and local stochastic Markov chains at the two lattice boundaries which provide chemical baths for absorbing or emitting the solitons. Ergodicity and mixing of this many-body Markov chain is proven for generic values of bath parameters, implying existence of a unique non-equilibrium steady state. The latter is constructed exactly and explicitly in terms of a particularly simple form of matrix product ansatz which is termed a patch ansatz. This gives rise to an explicit computation of observables and $k$-point correlations in the steady state as well as the construction of a nontrivial set of local conservation laws. Feasibility of an exact solution for the full spectrum and eigenvectors (decay modes) of the Markov matrix is suggested as well. We conjecture that our ideas can pave the road towards a theory of integrability of boundary driven classical deterministic lattice systems.