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.
Highlights
Solved models are a major cornerstone of statistical mechanics and physics in general
Space evolution can be formulated as a composition of nondeterministic three-site maps and projectors to a subspace of configurations, as we show in subsection 7.4, where we introduce a convenient tensor-network representation of dynamics that provides an algebraic interpretation of the matrix product ansatz (MPA) encoding multi-time correlation functions
This review paper provides a coherent overview of a series of explicit results, appearing over the last five years—most of which were co-contributed by the authors, on dynamics and nonequilibrium statistical mechanics of a particular interacting reversible cellular automaton (RCA54)
Summary
Solved models are a major cornerstone of statistical mechanics and physics in general. It is of utmost importance to have at our disposal another type of model—or a class of models—with generic physical behaviour and for which dynamical physical quantities are accessible by a rigorous analysis free from assumptions Within such a class of models we can achieve the ‘holy grail’ of nonequilibrium statistical physics, which is to derive macroscopic transport laws from reversible and deterministic microscopic equations of motion [17]. In the last several years it has been recognised that such a model exists and seemingly belongs to a class distinct from regular Bethe-ansatz solvable models It is the rule 54 of the family of reversible cellular automata (RCA54) proposed and classified in [18]. The purpose of this review is to provide a comprehensive overview of recent results on the rule 54 model and discuss their comparison with simple predictions of hydrodynamic theory
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