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Exact solution of the "Rule 150" reversible cellular automaton
Joseph W P Wilkinson1,2, Tomaž Prosen3, Juan P Garrahan1,2
1School of Physics and Astronomy, University of Nottingham, Nottingham, NG7 2RD, United Kingdom.
Physical Review. E
|April 16, 2022
Summary
The Rule 150 reversible cellular automaton (RCA) exhibits noninteracting dynamics, allowing for exact solutions. Its emergent quasiparticles behave as free fermions, simplifying complex system analysis.
Area of Science:
- Statistical Mechanics
- Complex Systems
- Theoretical Physics
Background:
- Reversible cellular automata (RCAs) are discrete dynamical systems with applications in physics.
- Kinetically constrained models, like the Fredrickson-Andersen (FA) model, exhibit complex emergent behavior.
- Integrable systems offer analytical tractability for understanding dynamics.
Purpose of the Study:
- To investigate the dynamics and statistics of the Rule 150 reversible cellular automaton (RCA).
- To analyze the behavior of emergent quasiparticles in this specific RCA.
- To determine equilibrium and nonequilibrium stationary states and relaxation dynamics.
Main Methods:
- Utilizing the integrability of Rule 150 RCA.
- Employing the matrix product ansatz for analytical solutions.
- Analyzing domain walls as emergent quasiparticles behaving as free fermions.
Main Results:
- The Rule 150 RCA is shown to be noninteracting, with domain walls acting as free fermions.
- Exact solutions for equilibrium and nonequilibrium stationary states are derived for both closed and open boundary conditions.
- The full spectrum of the time evolution operator is resolved, enabling analysis of relaxation dynamics.
Conclusions:
- The noninteracting nature of Rule 150 RCA simplifies its analysis significantly.
- The model provides a tractable framework for studying statistical mechanics in systems with kinetic constraints.
- Exact solutions for dynamics and statistics are achievable for this class of integrable, noninteracting RCAs.
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