Rule 54: Exactly solvable model of nonequilibrium statistical mechanics

TL;DR

This paper reviews the Rule 54 cellular automaton as an exactly solvable model in nonequilibrium statistical mechanics, providing detailed microscopic, hydrodynamic, and quantum insights into its dynamics and stationary states.

Contribution

It introduces exact solutions for nonequilibrium steady states, decay modes, large deviations, and quantum extensions, advancing understanding of integrable models in statistical physics.

Findings

Exact matrix product descriptions of steady states

Demonstration of fluid-like behavior with ballistic and diffusive transport

Explicit results on quantum operator spreading and entanglement

Abstract

We review recent results on an exactly solvable model of nonequilibrium statistical mechanics, specifically the classical Rule 54 reversible cellular automaton and some of its quantum extensions. We discuss the exact microscopic description of nonequilibrium dynamics as well as the equilibrium and nonequilibrium stationary states. This allows us to obtain a rigorous handle on the corresponding emergent hydrodynamic description, which is treated as well. Specifically, we focus on two different paradigms of Rule 54 dynamics. Firstly, we consider a finite chain driven by stochastic boundaries, where we provide exact matrix product descriptions of the nonequilibrium steady state, most relevant decay modes, as well as the eigenvector of the tilted Markov chain yielding exact large deviations for a broad class of local and extensive observables. Secondly, we treat the explicit dynamics of macro-states on an infinite lattice and discuss exact closed form results for dynamical structure factor, multi-time-correlation functions and inhomogeneous quenches. Remarkably, these results prove that the model, despite its simplicity, behaves like a regular fluid with coexistence of ballistic (sound) and diffusive (heat) transport. Finally, we briefly discuss quantum interpretation of Rule 54 dynamics and explicit results on dynamical spreading of local operators and operator entanglement.