Basic Ideas to Approach Metastability in Probabilistic Cellular Automata

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, are discrete time Markov chains on lattice with finite single--cell states whose distinguishing feature is the \textit{parallel} character of the updating rule. We review some of the results obtained about the metastable behavior of Probabilistic Cellular Automata and we try to point out difficulties and peculiarities with respect to standard Statistical Mechanics Lattice models.Comment: arXiv admin note: text overlap with arXiv:1307.823

Competitive nucleation in metastable systems

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

A comparison between different cycle decompositions for Metropolis dynamics

In the last decades the problem of metastability has been attacked on rigorous grounds via many different approaches and techniques which are briefly reviewed in this paper. It is then useful to understand connections between different point of views. In view of this we consider irreducible, aperiodic and reversible Markov chains with exponentially small transition probabilities in the framework of Metropolis dynamics. We compare two different cycle decompositions and prove their equivalence

Metastability for reversible probabilistic cellular automata with self--interaction

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 powerful approaches, not needing a complete description of the fixed points of the dynamics, but relying on few model dependent results. We compute the exit time, in the sense of logarithmic equivalence, and characterize the critical droplet that is necessarily visited by the system during its excursion from the metastable to the stable state. We need to supply two model dependent inputs: (1) the communication energy, that is the minimal energy barrier that the system must overcome to reach the stable state starting from the metastable one; (2) a recurrence property stating that for any configuration different from the metastable state there exists a path, starting from such a configuration and reaching a lower energy state, such that its maximal energy is lower than the communication energy

Controller Design for the Acquisition Phase of the LISA Mission using a Kalman Filter

LISA, the Laser Interferometer Space Antenna, and its technology-demonstrating precursor LISA Pathfinder, form an ESA/NASA collaborative project, selected as an ESA Cornerstone and included in NASA’s ‘Beyond Einstein’ initiative. The primary objective of the LISA mission is to detect and observe gravitational waves emitted from Massive Black Holes and galactic binaries in the low-frequency band which ranges from 0.1 mHz up to 1 Hz with a goal of extending the measurements down to 30 µHz. The underlying measurement principle is a laser interferometry system built up with three satellites that are flying in a triangular constellation with an edge length of 5 million km. The interferometric measurements of LISA are only possible once the three laser links between the three spacecraft of the LISA constellation are established. This phase is addressed as the constellation acquisition for LISA. LISA constellation acquisition is challenging, given the weak signal received by the spacecraft 5 million km away, inherent limits of the attitude sensors accuracy, orbit determination accuracy issues and the time required to phase-lock the incoming and outgoing laser signals. In order to counteract all these adverse constraints and make the LISA constellation acquisition possible, the laser pointing must satisfy challenging performance requirements during the whole duration of the acquisition phase. This thesis proposes a strategy for the acquisition control of the LISA formation based on the use of a Kalman filter: it pre-processes the measurement data providing enhanced signals for the controller, which has the very same structure used in the science mode. The Kalman filter is designed such that it realizes a continuous blend of the sensors data, providing a massive disturbance rejection. Simulations and sensitivity analysis are performed in order to demonstrate the feasibility of the proposed approach

Competitive nucleation in reversible Probabilistic Cellular Automata

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

Relaxation Height in Energy Landscapes: an Application to Multiple Metastable States

The study of systems with multiple (not necessarily degenerate) metastable states presents subtle difficulties from the mathematical point of view related to the variational problem that has to be solved in these cases. We introduce the notion of relaxation height in a general energy landscape and we prove sufficient conditions which are valid even in presence of multiple metastable states. We show how these results can be used to approach the problem of multiple metastable states via the use of the modern theories of metastability. We finally apply these general results to the Blume--Capel model for a particular choice of the parameters ensuring the existence of two multiple, and not degenerate in energy, metastable states