Abstract
We study integrability properties of a reversible deterministic cellular automaton (Rule 54 of (Bobenko et al 1993 Commun. Math. Phys. 158 127)) and present a bulk algebraic relation and its inhomogeneous extension which allow for an explicit construction of Liouvillian decay modes for two distinct families of stochastic boundary driving. The spectrum of the many-body stochastic matrix defining the time propagation is found to separate into sets, which we call orbitals, and the eigenvalues in each orbital are found to obey a distinct set of Bethe-like equations. We construct the decay modes in the first orbital (containing the leading decay mode) in terms of an exact inhomogeneous matrix product ansatz, study the thermodynamic properties of the spectrum and the scaling of its gap, and provide a conjecture for the Bethe-like equations for all the orbitals and their degeneracy.
Highlights
Understanding the emergence of laws governing macroscopic physical phenomena, such as transport and relaxation, from deterministic and reversible microscopic dynamics is one of the most prominent fundamental problems of statistical mechanics
We found that the spectrum of the Markov matrix of a deterministic boundary driven cellular automaton (Rule 54) organizes into orbitals
We found explicit matrix product forms of the eigenvectors in two main orbitals – the nonequilibrium steady states (NESS)-orbital
Summary
Understanding the emergence of laws governing macroscopic physical phenomena, such as transport and relaxation, from deterministic and reversible microscopic dynamics is one of the most prominent fundamental problems of statistical mechanics. Are given in terms of 4 × 4 stochastic matrices PL and PR (to be specified later), which in turn imply that the full 2n × 2n propagator U itself is a stochastic matrix and conserves total probability during the time evolution This dynamics which is bulk-deterministic and boundary-stochastic should be contrasted with related, though distinct, discrete time asymmetric exclusion process models [4,5,6], which feature both stochastic bulk dynamics as well as stochastic driving. According to Perron-Frobenius theorem, the non-equilibrium steady state (NESS) eigenvector, corresponding to Λ0 = 1, is unique and all other eigenvalues Λj1 are bounded by |Λj | < 1 and the corresponding components of the state vector decay during the time evolution. Bernoulli driving has been introduced and studied for the steady state in [2]
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