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

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

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

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

Weak solutions to Allen-Cahn-like equations modelling consolidation of porous media

We study the weak solvability of a system of coupled Allen–Cahn–like equations resembling cross–diffusion which is arising as a model for the consolidation of saturated porous media. Besides using energy like estimates, we cast the special structure of the system in the framework of the Leray–Schauder fixed point principle and ensure this way the local existence of strong solutions to a regularised version of our system. Furthermore, weak convergence techniques ensure the existence of weak solutions to the original consolidation problem. The uniqueness of global-in-time solutions is guaranteed in a particular case. Moreover, we use a finite difference scheme to show the negativity of the vector of solutions. Keywords: Weak solutions; cross–diffusion system; energy method; Leray–Schauder fixed point theorem; finite differences; consolidation of porous media

Persistence exponent in a superantiferromagnetic quenching

We measure the persistence exponent in a phase separating two-dimensional spin system with non-conserved dynamics quenched in a region with four coexisting stripe phases. The system is an Ising model with nearest neighbor, next-to-the-nearest neighbor and plaquette interactions. Due the particular nature of the ground states, the order parameter is defined in terms of blocks of spins. Our estimate of the persistence exponent, $\theta=0.42$, differs from those of the two-dimensional Ising and four state Potts models. Our procedure allows the study of persistence properties also at finite temperature $T$: our results are compatible with the hypothesis that $\theta$ does not depend on $T$ below the critical point.Comment: LaTeX file with postscript figure

Phase ordering in chaotic map lattices with conserved dynamics

Dynamical scaling in a two-dimensional lattice model of chaotic maps, in contact with a thermal bath, is numerically studied. The model here proposed is equivalent to a conserved Ising model with coupligs which fluctuate over the same time scale as spin moves. When couplings fluctuations and thermal fluctuations are both important, this model does not belong to the class of universality of a Langevin equation known as model B; the scaling exponents are continuously varying with the temperature and depend on the map used. The universal behavior of model B is recovered when thermal fluctuations are dominant.Comment: 6 pages, 4 figures. Revised version accepted for publication on Physical Review E as a Rapid Communicatio

Phase transitions for the cavity approach to the clique problem on random graphs

We give a rigorous proof of two phase transitions for a disordered system designed to find large cliques inside Erdos random graphs. Such a system is associated with a conservative probabilistic cellular automaton inspired by the cavity method originally introduced in spin glass theory.Comment: 36 pages, 4 figure

Folding transition of the triangular lattice in a discrete three--dimensional space

A vertex model introduced by M. Bowick, P. Di Francesco, O. Golinelli, and E. Guitter (cond-mat/9502063) describing the folding of the triangular lattice onto the face centered cubic lattice has been studied in the hexagon approximation of the cluster variation method. The model describes the behaviour of a polymerized membrane in a discrete three--dimensional space. We have introduced a curvature energy and a symmetry breaking field and studied the phase diagram of the resulting model. By varying the curvature energy parameter, a first-order transition has been found between a flat and a folded phase for any value of the symmetry breaking field.Comment: 11 pages, latex file, 2 postscript figure