129 research outputs found
Slow dynamics for the dilute Ising model in the phase coexistence region
In this paper we consider the Glauber dynamics for a disordered ferromagnetic
Ising model, in the region of phase coexistence. It was conjectured several
decades ago that the spin autocorrelation decays as a negative power of time
[Huse and Fisher, Phys. Rev. B, 1987]. We confirm this behavior by establishing
a corresponding lower bound in any dimensions , together with an
upper bound when . Our approach is deeply connected to the Wulff
construction for the dilute Ising model. We consider initial phase profiles
with a reduced surface tension on their boundary and prove that, under mild
conditions, those profiles are separated from the (equilibrium) pure plus phase
by an energy barrier.Comment: 44 pages, 6 figure
Colligative properties of solutions: II. Vanishing concentrations
We continue our study of colligative properties of solutions initiated in
math-ph/0407034. We focus on the situations where, in a system of linear size
, the concentration and the chemical potential scale like and
, respectively. We find that there exists a critical value \xit such
that no phase separation occurs for \xi\le\xit while, for \xi>\xit, the two
phases of the solvent coexist for an interval of values of . Moreover, phase
separation begins abruptly in the sense that a macroscopic fraction of the
system suddenly freezes (or melts) forming a crystal (or droplet) of the
complementary phase when reaches a critical value. For certain values of
system parameters, under ``frozen'' boundary conditions, phase separation also
ends abruptly in the sense that the equilibrium droplet grows continuously with
increasing and then suddenly jumps in size to subsume the entire system.
Our findings indicate that the onset of freezing-point depression is in fact a
surface phenomenon.Comment: 27 pages, 1 fig; see also math-ph/0407034 (both to appear in JSP
Metastability and Nucleation for the Blume-Capel Model. Different mechanisms of transition
We study metastability and nucleation for the Blume-Capel model: a
ferromagnetic nearest neighbour two-dimensional lattice system with spin
variables taking values in -1,0,+1. We consider large but finite volume, small
fixed magnetic field h and chemical potential "lambda" in the limit of zero
temperature; we analyze the first excursion from the metastable -1
configuration to the stable +1 configuration. We compute the asymptotic
behaviour of the transition time and describe the typical tube of trajectories
during the transition. We show that, unexpectedly, the mechanism of transition
changes abruptly when the line h=2*lambda is crossed.Comment: 96 pages, 44 tex-figures, 7 postscript figure
Critical droplets in Metastable States of Probabilistic Cellular Automata
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 and
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 is large enough and is
zero, and a unique phase when is nonzero. When the system evolves, at small
positive values of , from an initial state with all spins down, the PCA
dynamics give rise to a transition from a metastable to a stable phase when a
droplet of the favored phase inside the metastable phase reaches a
critical size. We give heuristic arguments to estimate the critical size in the
limit of zero ``temperature'' (), as well as estimates of the
time required for the formation of such a droplet in a finite system. Monte
Carlo simulations give results in good agreement with the theoretical
predictions.Comment: 5 LaTeX picture
Metastability in the dilute Ising model
Consider Glauber dynamics for the Ising model on the hypercubic lattice with
a positive magnetic field. Starting from the minus configuration, the system
initially settles into a metastable state with negative magnetization. Slowly
the system relaxes to a stable state with positive magnetization. Schonmann and
Shlosman showed that in the two dimensional case the relaxation time is a
simple function of the energy required to create a critical Wulff droplet.
The dilute Ising model is obtained from the regular Ising model by deleting a
fraction of the edges of the underlying graph. In this paper we show that even
an arbitrarily small dilution can dramatically reduce the relaxation time. This
is because of a catalyst effect---rare regions of high dilution speed up the
transition from minus phase to plus phase.Comment: 49 page
Finite-size effects for anisotropic bootstrap percolation: logarithmic corrections
In this note we analyze an anisotropic, two-dimensional bootstrap percolation
model introduced by Gravner and Griffeath. We present upper and lower bounds on
the finite-size effects. We discuss the similarities with the semi-oriented
model introduced by Duarte.Comment: Key words: Bootstrap percolation, anisotropy, finite-size effect
Rigorous Probabilistic Analysis of Equilibrium Crystal Shapes
The rigorous microscopic theory of equilibrium crystal shapes has made
enormous progress during the last decade. We review here the main results which
have been obtained, both in two and higher dimensions. In particular, we
describe how the phenomenological Wulff and Winterbottom constructions can be
derived from the microscopic description provided by the equilibrium
statistical mechanics of lattice gases. We focus on the main conceptual issues
and describe the central ideas of the existing approaches.Comment: To appear in the March 2000 special issue of Journal of Mathematical
Physics on Probabilistic Methods in Statistical Physic
Metastable behavior for bootstrap percolation on regular trees
We examine bootstrap percolation on a regular (b+1)-ary tree with initial law
given by Bernoulli(p). The sites are updated according to the usual rule: a
vacant site becomes occupied if it has at least theta occupied neighbors,
occupied sites remain occupied forever. It is known that, when b>theta>1, the
limiting density q=q(p) of occupied sites exhibits a jump at some
p_t=p_t(b,theta) in (0,1) from q_t:=q(p_t)p_t. We
investigate the metastable behavior associated with this transition.
Explicitly, we pick p=p_t+h with h>0 and show that, as h decreases to 0, the
system lingers around the "critical" state for time order h^{-1/2} and then
passes to fully occupied state in time O(1). The law of the entire
configuration observed when the occupation density is q in (q_t,1) converges,
as h tends to 0, to a well-defined measure.Comment: 10 pages, version to appear in J. Statist. Phy
Scaling Limit and Critical Exponents for Two-Dimensional Bootstrap Percolation
Consider a cellular automaton with state space
where the initial configuration is chosen according to a Bernoulli
product measure, 1's are stable, and 0's become 1's if they are surrounded by
at least three neighboring 1's. In this paper we show that the configuration
at time n converges exponentially fast to a final configuration
, and that the limiting measure corresponding to is in
the universality class of Bernoulli (independent) percolation.
More precisely, assuming the existence of the critical exponents ,
, and , and of the continuum scaling limit of crossing
probabilities for independent site percolation on the close-packed version of
(i.e., for independent -percolation on ), we
prove that the bootstrapped percolation model has the same scaling limit and
critical exponents.
This type of bootstrap percolation can be seen as a paradigm for a class of
cellular automata whose evolution is given, at each time step, by a monotonic
and nonessential enhancement.Comment: 15 page
Relaxation times of kinetically constrained spin models with glassy dynamics
We analyze the density and size dependence of the relaxation time for
kinetically constrained spin systems. These have been proposed as models for
strong or fragile glasses and for systems undergoing jamming transitions. For
the one (FA1f) or two (FA2f) spin facilitated Fredrickson-Andersen model at any
density and for the Knight model below the critical density at which
the glass transition occurs, we show that the persistence and the spin-spin
time auto-correlation functions decay exponentially. This excludes the
stretched exponential relaxation which was derived by numerical simulations.
For FA2f in , we also prove a super-Arrhenius scaling of the form
. For FA1f in = we
rigorously prove the power law scalings recently derived in \cite{JMS} while in
we obtain upper and lower bounds consistent with findings therein.
Our results are based on a novel multi-scale approach which allows to analyze
in presence of kinetic constraints and to connect time-scales and
dynamical heterogeneities. The techniques are flexible enough to allow a
variety of constraints and can also be applied to conservative stochastic
lattice gases in presence of kinetic constraints.Comment: 4 page
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