The problem of competitive nucleation in the framework of Probabilistic Cellular Automata is studied from the dynamical point of view. The dependence of the metastability scenario on the self--interaction is discussed. An intermediate metastable phase, made of two flip--flopping chessboard configurations, shows up depending on the ratio between the magnetic field and the self--interaction. A behavior similar to the one of the stochastic Blume--Capel model with Glauber dynamics is found.
Molecular dynamics simulations for aqueous sodium chloride solutions were carried out at various concentrations. Supplementary to the Debye-Hückel theory, reversible transient nucleation of ions was observed even in dilute solutions. The average size of formed ion clusters and the lifetime of ion pairs increase with concentration until the saturation point, when ion clusters become stable and individual ions adjust their positions to form ordered lattice structures, leading to irreversible ion crystallization, which is beyond the description of the classical nucleation theory.
Cellular Automata are discrete-time dynamical systems on a spatially extended discrete space which provide paradigmatic examples of nonlinear phenomena. Their stochastic generalizations, i.e., Probabilistic Cellular Automata (PCA), are discrete time Markov chains on lattice with finite single-cell states whose distinguishing feature is the \emph{parallel} character of the updating rule. We study the ground states of the Hamiltonian and the low-temperature phase diagram of the related Gibbs measure naturally associated with a class of reversible PCA, called the \textit{cross PCA}. In such a mod…
A family of reversible deterministic cellular automata, including the rules 54 and 201 of [Bobenko et al., Commun. Math. Phys. 158, 127 (1993)] as well as their kinetically constrained quantum (unitary) or stochastic deformations, is shown to correspond to integrable Floquet circuit models with local interactions round-a-face. Using inhomogeneous solutions of the star-triangle relation with a one or two dimensional spectral parameter, changing their functional form depending on the orientation, we provide an explicit construction of the transfer matrix and establish its conservation law and in…
The problem of metastability for a stochastic dynamics with a parallel updating rule is addressed in the Freidlin--Wentzel regime, namely, finite volume, small magnetic field, and small temperature. The model is characterized by the existence of many fixed points and cyclic pairs of the zero temperature dynamics, in which the system can be trapped in its way to the stable phase. %The characterization of the metastable behavior %of a system in the context of parallel dynamics is a very difficult task, %since all the jumps in the configuration space are allowed. Our strategy is based on recent p…
Metastability is observed when a physical system is close to a first order phase transition. In this paper the metastable behavior of a two state reversible probabilistic cellular automaton with self-interaction is discussed. Depending on the self-interaction, competing metastable states arise and a behavior very similar to that of the three state Blume-Capel spin model is found.
We introduce a pair of time-reversible models defined on the discrete space-time lattice with 3 states per site, specifically, a vacancy and a particle of two flavours (species). The local update rules reproduce the rule 54 reversible cellular automaton when only a single species of particles is present, and satisfy the requirements of flavour exchange (C), space-reversal (P), and time-reversal (T) symmetries. We find closed-form expressions for three local conserved charges and provide an explicit matrix product form of the grand canonical Gibbs states, which are identical for both models. Fo…
A master equation approach is applied to a reversible and conservative cellular automata model (Q2R). The Q2R model is a dynamical variation of the Ising model for ferromagnetism that possesses quite a rich and complex dynamics. The configurational space is composed by a huge number of cycles with exponentially long periods. Following Nicolis and Nicolis [Phys. Rev. A 38, 427-433 (1988)], a coarse-graining approach is applied to the time series of the total magnetization leading to a master equation that governs the macroscopic irreversible dynamics of the Q2R automata. The methodology is repl…
We introduce a novel method of quantum emulation of a classical reversible cellular automaton. By applying this method to a chaotic cellular automaton, the obtained quantum many-body system thermalizes while all the energy eigenstates and eigenvalues are solvable. These explicit solutions allow us to verify the validity of some scenarios of thermalization to this system. We find that two leading scenarios, the eigenstate thermalization hypothesis scenario and the large effective dimension scenario, do not explain thermalization in this model.
Reversible cellular automata are seen as microscopic physical models, and their states of macroscopic equilibrium are described using invariant probability measures. We establish a connection between the invariance of Gibbs measures and the conservation of additive quantities in surjective cellular automata. Namely, we show that the simplex of shift-invariant Gibbs measures associated to a Hamiltonian is invariant under a surjective cellular automaton if and only if the cellular automaton conserves the Hamiltonian. A special case is the (well-known) invariance of the uniform Bernoulli measure …
We consider the problem of metastability for stochastic reversible dynamics with exponentially small transition probabilities. We generalize previous results in several directions. We give an estimate of the spectral gap of the transition matrix and of the mixing time of the associated dynamics in terms of the maximal stability level. These model-independent results hold in particular for a large class of Probabilistic Cellular Automata (PCA), which we then focus on. We consider the PCA in a finite volume, at small and fixed magnetic field, and in the limit of vanishing temperature. This model…
Classical reversible cellular automata (CAs), which describe the discrete-time dynamics of classical degrees of freedom in a finite state-space, can exhibit exact, nonthermal quantum eigenstates despite being classically chaotic. We show that every classical CA defines a family of generically non-integrable, periodically-driven (Floquet) quantum dynamics with exact, nonthermal eigenstates. These Floquet dynamics are nonergodic in the sense that certain product states on a periodic classical orbit fail to thermalize, while generic initial states thermalize as expected in a quantum chaotic syste…
Conway's cellular automaton Game of LIFE has been conjectured to be a critical (or quasicritical) dynamical system. This criticality is generally seen as a continuous order-disorder transition in cellular automata (CA) rule space. LIFE's mean-field return map predicts an absorbing vacuum phase ($ρ=0$) and an active phase density, with $ρ=0.37$, which contrasts with LIFE's absorbing states in a square lattice, which have a stationary density $ρ_{2D} \approx 0.03$. Here, we study and classify mean-field maps for $6144$ outer-totalistic CA and compare them with the corresponding behavior found in…
We consider the problem of metastability in a probabilistic cellular automaton (PCA) with a parallel updating rule which is reversible with respect to a Gibbs measure. The dynamical rules contain two parameters $β$ and $h$ which resemble, but are not identical to, the inverse temperature and external magnetic field in a ferromagnetic Ising model; in particular, the phase diagram of the system has two stable phases when $β$ is large enough and $h$ is zero, and a unique phase when $h$ is nonzero. When the system evolves, at small positive values of $h$, from an initial state with all spins down,…
