19 research outputs found
Critical Droplets and sharp asymptotics for Kawasaki dynamics with strongly anisotropic interactions
In this paper we analyze metastability and nucleation in the context of the
Kawasaki dynamics for the two-dimensional Ising lattice gas at very low
temperature. Let be a finite box. Particles
perform simple exclusion on , but when they occupy neighboring sites
they feel a binding energy in the horizontal direction and in
the vertical one. Thus the Kawasaki dynamics is conservative inside the volume
. Along each bond touching the boundary of from the outside
to the inside, particles are created with rate , while
along each bond from the inside to the outside, particles are annihilated with
rate , where is the inverse temperature and is an
activity parameter. Thus, the boundary of plays the role of an
infinite gas reservoir with density . We consider the parameter regime
also known as the strongly anisotropic regime. We take
, so that the empty (respectively full) configuration
is a metastable (respectively stable) configuration. We investigate how the
transition from empty to full takes place with particular attention to the
critical configurations that asymptotically have to be crossed with probability
1. The derivation of some geometrical properties of the saddles allows us to
identify the full geometry of the minimal gates and their boundaries for the
nucleation in the strongly anisotropic case. We observe very different
behaviors for this case with respect to the isotropic () and weakly
anisotropic () ones. Moreover, we derive mixing time, spectral gap
and sharp estimates for the asymptotic transition time for the strongly
anisotropic case.Comment: 51 pages and 14 figures. In this paper we consider, with the same
authors, the same model as our previous work arXiv:2108.02017, but in a
complementary range of parameter values. Thus sections 1, 2, 3 and 4.1 in the
present paper are similar to sections 1, 2, 3.1 and 4.1 in the previous paper
arXiv:2108.0201
Critical configurations and tube of typical trajectories for the Potts and Ising models with zero external field
We consider the ferromagnetic q-state Potts model with zero external field in
a finite volume evolving according to Glauber-type dynamics described by the
Metropolis algorithm in the low temperature asymptotic limit. Our analysis
concerns the multi-spin system that has q stable equilibria. Focusing on grid
graphs with periodic boundary conditions, we study the tunneling between two
stable states and from one stable state to the set of all other stable states.
In both cases we identify the set of gates for the transition and prove that
this set has to be crossed with high probability during the transition.
Moreover, we identify the tube of typical paths and prove that the probability
to deviate from it during the transition is exponentially small.Comment: 44 pages, 15 figures. All comments are welcome
Critical configurations and tube of typical trajectories for the Potts and Ising models with zero external field
Metastability for the Ising model on the hexagonal lattice
We consider the Ising model on the hexagonal lattice evolving according to
Metropolis dynamics. We study its metastable behavior in the limit of vanishing
temperature when the system is immersed in a small external magnetic field. We
determine the asymptotic properties of the transition time from the metastable
to the stable state and study the relaxation time and the spectral gap of the
Markov process. We give a geometrical description of the critical
configurations and show how not only their size but their shape varies
depending on the thermodynamical parameters. Finally we provide some results
concerning polyiamonds of maximal area and minimal perimeter
Metastability for General Dynamics with Rare Transitions: Escape Time and Critical Configurations
Metastability is a physical phenomenon ubiquitous in first order
phase transitions.
A fruitful mathematical way to approach this phenomenon
is the study of rare transitions Markov chains.
For Metropolis chains associated with Statistical Mechanics systems, this phenomenon has been
described in an elegant
way in terms of the energy landscape associated
to the Hamiltonian of the system.
In this paper, we provide a similar description in the
general rare transitions setup.
Beside their theoretical content, we believe that our results are a
useful tool to approach metastability for non--Metropolis systems
such as Probabilistic Cellular Automata
Crossover times in bipartite networks with activity constraints and time-varying switching rates
In this paper we study the performance of a bipartite network in which
customers arrive at the nodes of the network, but not all nodes are able to
serve their customers at all times. Each node can be either active or inactive,
and two nodes connected by a bond cannot be active simultaneously. This
situation arises in wireless random-access networks where, due to destructive
interference, stations that are close to each other cannot use the same
frequency band.
We consider a model where the network is bipartite, the active nodes switch
themselves off at rate 1, and the inactive nodes switch themselves on at a rate
that depends on time and on which half of the bipartite network they are in. An
inactive node cannot become active when one of the nodes it is connected to by
a bond is active. The switching protocol allows the nodes to share activity
among each other. In the limit as the activation rate becomes large, we compute
the crossover time between the two states where one half of the network is
active and the other half is inactive. This allows us to assess the overall
activity of the network depending on the switching protocol. Our results make
use of the metastability analysis for hard-core interacting particle models on
finite bipartite graphs derived in an earlier paper. They are valid for a large
class of bipartite networks, subject to certain assumptions. Proofs rely on a
comparison with switching protocols that are not time-varying, through coupling
techniques.Comment: 32 pages, 2 figur
Critical Droplets and sharp asymptotics for Kawasaki dynamics with weakly anisotropic interactions
In this paper we analyze metastability and nucleation in the context of the
Kawasaki dynamics for the two-dimensional Ising lattice gas at very low
temperature with periodic boundary conditions. Let be the inverse
temperature and let be two
boxes. We consider the asymptotic regime corresponding to the limit as
for finite volume and
. We
study the simplified model, in which particles perform independent random walks
on and inside particles perform
simple exclusion, but when they occupy neighboring sites they feel a binding
energy in the horizontal direction and in the vertical one.
Thus the Kawasaki dynamics is conservative inside the volume .
The initial configuration is chosen such that is empty and
particles are distributed randomly over
. Our results will use a deep analysis of a
local model, i.e., particles perform Kawasaki dynamics inside and
along each bond touching the boundary of from the outside to the
inside, particles are created with rate , while along
each bond from the inside to the outside, particles are annihilated with rate
, where is an activity parameter. Thus, in the local model the
boundary of plays the role of an infinite gas reservoir with density
. We take , so that the empty (respectively
full) configuration is a metastable (respectively stable) configuration. We
investigate how the transition from empty to full takes place in the local
model with particular attention to the critical configurations that
asymptotically have to be crossed with probability 1.Comment: 74 pages and 26 figure
Metastability in a lattice gas with strong anisotropic interactions under Kawasaki dynamics
In this paper we analyze metastability and nucleation in the context of a
local version of the Kawasaki dynamics for the two-dimensional strongly
anisotropic Ising lattice gas at very low temperature. Let
be a finite box. Particles perform simple
exclusion on , but when they occupy neighboring sites they feel a
binding energy in the horizontal direction and in the
vertical one. Thus the Kawasaki dynamics is conservative inside the volume
. Along each bond touching the boundary of from the outside
to the inside, particles are created with rate , while
along each bond from the inside to the outside, particles are annihilated with
rate , where is the inverse temperature and is an
activity parameter. Thus, the boundary of plays the role of an
infinite gas reservoir with density . We consider the parameter regime
also known as the strongly anisotropic regime. We take
and we prove that the empty (respectively full)
configuration is a metastable (respectively stable) configuration. We consider
the asymptotic regime corresponding to finite volume in the limit of large
inverse temperature . We investigate how the transition from empty to
full takes place. In particular, we estimate in probability, expectation and
distribution the asymptotic transition time from the metastable configuration
to the stable configuration. Moreover, we identify the size of the
\emph{critical droplets}, as well as some of their properties. We observe very
different behavior in the weakly and strongly anisotropic regimes. We find that
the \emph{Wulff shape}, i.e., the shape minimizing the energy of a droplet at
fixed volume, is not relevant for the nucleation pattern.Comment: 73 pages, 22 figure