3,535 research outputs found
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
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
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
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