Wilkinson, Joseph W P
(2022)
Non-equilibrium dynamics of bulk-deterministic cellular automata.
PhD thesis, University of Nottingham.
Abstract
In this thesis we study simple one-dimensional nonequilibirum many-body systems, namely, reversible cellular automata (RCA). These are discrete time lattice models exhibiting emergent collective excitations---solitons---that move with fixed velocities and that interact via pairwise scattering. In particular, we study the attractively interacting Rule 201 RCA and noninteracting Rule 150 RCA which, together with the extensively studied repulsively interacting Rule 54 RCA constitute arguably the simplest one-dimensional microscopic physical models of strongly interacting and asymptotically freely propagating particles, to investigate interacting nonequilibrium many-body dynamics.
After a brief literature review of the field, we present the first publication-style chapter which considers the Rule 201 RCA. Here, we study the stationary or steady state properties of systems with periodic, deterministic, and stochastic boundary conditions. We demonstrate that, despite the complexities of the model, specifically, a reducible state space and nontrivial topological vacuum, the model exhibits a simple and intuitive quasiparticle interpretation, reminiscent of the simpler Rule 54 RCA. This enables us to obtain exact expressions for the steady states in terms of a highly versatile matrix product state (MPS) representation that takes an instructive generalized Gibbs ensemble form.
In the second publication-style chapter, we study the Rule 150 RCA. Due to its simplicity, originating from the noninteracting dynamics, we are able to obtain many exact results relating to its dynamics. To start, we generalize the MPS ansatz used to study the Rule 201 RCA, and find its exact steady state distribution for identical boundary conditions. We proceed to extend the MPS ansatz further and obtain a class of eigenvectors that form the dominant decay modes of the Markov propagator. Following this, we postulate a conjecture for the complete spectrum, which is in perfect agreement with numerics obtained via exact diagonalization of computationally tractable system sizes, providing access to the full relaxation dynamics. From here, we further utilise the ansatz to investigate the large deviation statistics and obtain exact expressions for its scaled cumulant generating function and rate function, which demonstrate the existence of a dynamical first order phase transition.
The third and final publication-style chapter focuses on the exact dynamical large deviations statistics of the Rule 201 RCA. Specifically, we employ the methods introduced to study the large deviations of the Rule 54 RCA and show that they fail here to provide any insight into the atypical dynamical behaviour of the Rule 201 RCA. We suggest that this is due to the restrictions imposed by the local dynamical rules, which limits the support of the local observables. In spite of this, we explicitly derived an exact analytic expression for the dominant eigenvalue of the tilted Markov propagator, from which several large deviation statistics can be obtained.
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