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Exact solution of the “Rule 150” reversible cellular automaton

Joseph W. P. Wilkinson1,2,*, Tomaž Prosen3, and Juan P. Garrahan1,2

  • 1School of Physics and Astronomy, University of Nottingham, Nottingham, NG7 2RD, United Kingdom
  • 2Centre for the Mathematics and Theoretical Physics of Quantum Non-equilibrium Systems, University of Nottingham, Nottingham, NG7 2RD, United Kingdom
  • 3Department of Physics, Faculty of Mathematics and Physics, University of Ljubljana, SI-1000 Ljubljana, Slovenia

  • *Corresponding author: joseph.wilkinson@nottingham.ac.uk
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Phys. Rev. E 105, 034124 – Published 17 March, 2022

DOI: https://doi.org/10.1103/PhysRevE.105.034124

Abstract

We study the dynamics and statistics of the Rule 150 reversible cellular automaton (RCA). This is a one-dimensional lattice system of binary variables with synchronous (Floquet) dynamics that corresponds to a bulk deterministic and reversible discretized version of the kinetically constrained “exclusive one-spin facilitated” (XOR) Fredrickson-Andersen (FA) model, where the local dynamics is restricted: A site flips if and only if its adjacent sites are in different states from each other. Similar to other RCA that have been recently studied, such as Rule 54 and Rule 201, the Rule 150 RCA is integrable, however, in contrast is noninteracting: The emergent quasiparticles, which are identified by the domain walls, behave as free fermions. This property allows us to solve the model by means of matrix product ansatz. In particular, we find the exact equilibrium and nonequilibrium stationary states for systems with closed (periodic) and open (stochastic) boundaries, respectively, resolve the full spectrum of the time evolution operator and, therefore, gain access to the relaxation dynamics, and obtain the exact large deviation statistics of dynamical observables in the long-time limit.

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Phys. Rev. E 105, 034124– Published 17 March, 2022
  • Received 3 December 2021
  • Accepted 18 February 2022
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