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Age-dependent Dynamics of Locomotion in Caenorhabditis elegans: A Lyapunov Exponent Analysis
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Exact large deviation statistics and trajectory phase transition of a deterministic boundary driven cellular
Berislav Buča1, Juan P Garrahan2, Tomaž Prosen3
1Clarendon Laboratory, University of Oxford, Parks Road, Oxford OX1 3PU, United Kingdom.
Physical Review. E
|October 3, 2019
Summary
We analyze the rule 54 reversible cellular automaton (CA), a model related to kinetically constrained models (KCMs). Our findings reveal phase coexistence in CA dynamics and provide methods for studying rare events.
Area of Science:
- Statistical Mechanics
- Complex Systems
- Theoretical Physics
Background:
- Cellular automata (CA) offer simplified models for complex systems.
- The rule 54 CA is a discrete, deterministic analog of the Fredrickson-Andersen kinetically constrained model (KCM).
- Understanding long-time dynamics and phase transitions in such systems is crucial.
Purpose of the Study:
- To investigate the statistical properties of the long-time dynamics of the rule 54 reversible CA under stochastic boundary conditions.
- To compute exact large deviation functions and analyze phase behavior.
- To determine the finite size scaling and identify dynamics for rare events.
Main Methods:
- Utilizing a matrix product ansatz to derive exact solutions.
- Calculating large deviation cumulant generating functions for various observables.
- Analyzing rate functions and conditioned long-time distributions.
- Investigating finite size scaling behavior.
Main Results:
- The CA dynamics exhibit phase coexistence between active and inactive phases, mirroring standard KCMs.
- Exact expressions for large deviation functions and rate functions were obtained.
- The finite size scaling of trajectory transitions was precisely determined.
- An explicit "Doob-transformed" dynamics for rare events was identified.
Conclusions:
- The rule 54 CA, under stochastic boundary driving, operates at a critical point of phase coexistence.
- The matrix product ansatz provides a powerful tool for exact analysis of CA dynamics.
- The study elucidates the behavior of rare dynamical events and offers a method for their optimal realization.
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