In this paper we study metastability in large volumes at low temperatures. We consider both Ising spins subject to Glauber spin-flip dynamics and lattice gas particles subject to Kawasaki hopping dynamics. Let $\b$ denote the inverse temperature and let $Ł_\b \subset \Z^2$ be a square box with periodic boundary conditions such that $\lim_{\b\to\infty}|Ł_\b|=\infty$. We run the dynamics on $Ł_\b$ starting from a random initial configuration where all the droplets (= clusters of plus-spins, respectively, clusters of particles)are small. For large $\b$, and for interaction parameters that correspond to the metastable regime, we investigate how the transition from the metastable state (with only small droplets) to the stable state (with one or more large droplets) takes place under the dynamics. This transition is triggered by the appearance of a single \emph{critical droplet} somewhere in $Ł_\b$. Using potential-theoretic methods, we compute the \emph{average nucleation time} (= the first time a critical droplet appears and starts growing) up to a multiplicative factor that tends to one as $\b\to\infty$. It turns out that this time grows as $Ke^{Γ\b}/|Ł_\b|$ for Glauber dynamics and $K\b e^{Γ\b}/|Ł_\b|$ for Kawasaki dynamics, where $Γ$ is the local canonical, respectively, grand-canonical energy to create a critical droplet and $K$ is a constant reflecting the geometry of the critical droplet, provided these times tend to infinity (which puts a growth restriction on $|Ł_\b|$). The fact that the average nucleation time is inversely proportional to $|Ł_\b|$ is referred to as \emph{homogeneous nucleation}, because it says that the critical droplet for the transition appears essentially independently in small boxes that partition $Ł_\b$.
We study the hydrodynamic scaling limit for the Glauber-Kawasaki dynamics. It is known that, if the Kawasaki part is speeded up in a diffusive space-time scaling, one can derive the Allen-Cahn equation which is a kind of the reaction-diffusion equation in the limit. This paper concerns the scaling that the Glauber part, which governs the creation and annihilation of particles, is also speeded up but slower than the Kawasaki part. Under such scaling, we derive directly from the particle system the motion by mean curvature for the interfaces separating sparse and dense regions of particles as a …
We construct two types of equilibrium dynamics of infinite particle systems in a locally compact Polish space $X$, for which certain fermion point processes are invariant. The Glauber dynamics is a birth-and-death process in $X$, while in the case of the Kawasaki dynamics interacting particles randomly hop over $X$. We establish conditions on generators of both dynamics under which corresponding conservative Markov processes exist.
We construct two types of equilibrium dynamics of an infinite particle system in a locally compact metric space $X$ for which a permanental point process is a symmetrizing, and hence invariant measure. The Glauber dynamics is a birth-and-death process in $X$, while in the Kawasaki dynamics interacting particles randomly hop over $X$. In the case $X=\mathbb R^d$, we consider a diffusion approximation for the Kawasaki dynamics at the level of Dirichlet forms. This leads us to an equilibrium dynamics of interacting Brownian particles for which a permanental point process is a symmetrizing measure…
We derive a continuum mean-curvature flow as a certain hydrodynamic scaling limit of Glauber-Kawasaki dynamics with speed change. The Kawasaki part describes the movement of particles through particle interactions. It is speeded up in a diffusive space-time scaling. The Glauber part governs the creation and annihilation of particles. The Glauber part is set to favor two levels of particle density. It is also speeded up in time, but at a lesser rate than the Kawasaki part. Under this scaling, a mean-curvature interface flow emerges, with a homogenized `surface tension-mobility' parameter reflec…
We derive the hydrodynamic limit of Glauber-Kawasaki dynamics. The Kawasaki part is simple and describes independent movement of the particles with hard core exclusive interactions. It is speeded up in a diffusive space-time scaling. The Glauber part describes the birth and death of particles. It is set to favor two levels of particle density with a preference for one of the two. It is also speeded up in time, but at a lesser rate than the Kawasaki part. Under this scaling, the limiting particle density instantly takes either of the two favored density values. The interface which separates the…
We consider the Kawasaki dynamics of two types of particles under a killing effect on a $d$-dimensional square lattice. Particles move with possibly different jump rates depending on their types. The killing effect acts when particles of different types meet at the same site. We show the existence of a limit under the diffusive space-time scaling and suitably growing killing rate: segregation of distinct types of particles does occur, and the evolution of the interface between the two distinct species is governed by the two-phase Stefan problem. We apply the relative entropy method and combine…
A Kawasaki dynamics in continuum is a dynamics of an infinite system of interacting particles in $\mathbb{R}^d$ which randomly hop over the space. In this paper, we deal with an equilibrium Kawasaki dynamics which has a Gibbs measure $mu$ as invariant measure. We study a scaling limit of such a dynamics, derived through a scaling of the jump rate. Informally, we expect that, in the limit, only jumps of ``infinite length'' will survive, i.e., we expect to arrive at a Glauber dynamics in continuum (a birth-and-death process in $\mathbb{R}^d$). We prove that, in the low activity-high temperature …
Inspired by the recent results of C. Landim, G. Panizo and H.-T. Yau [LPY] on spectral gap and logarithmic Sobolev inequalities for unbounded conservative spin systems, we study uniform bounds in these inequalities for Glauber dynamics of Hamiltonian of the form V(x_1) + ... + V(x_n) + V(M-x_1 -...-x_n), (x_1,...,x_n) in R^n Specifically, we examine the case V is strictly convex (or small perturbation of strictly convex) and, following [LPY], the case V is a bounded perturbation of a quadratic potential. By a simple path counting argument for the standard random walk, uniform bounds for the Gl…
We deal with two following classes of equilibrium stochastic dynamics of infinite particle systems in continuum: hopping particles (also called Kawasaki dynamics), i.e., a dynamics where each particle randomly hops over the space, and birth-and-death process in continuum (or Glauber dynamics), i.e., a dynamics where there is no motion of particles, but rather particles die, or are born at random. We prove that a wide class of Glauber dynamics can be derived as a scaling limit of Kawasaki dynamics. More precisely, we prove the convergence of respective generators on a set of cylinder functions,…
We construct a new equilibrium dynamics of infinite particle systems in a Riemannian manifold $X$. This dynamics is an analog of the Kawasaki dynamics of lattice spin systems. The Kawasaki dynamics now is a process where interacting particles randomly hop over $X$. We establish conditions on the {\it a priori} explicitly given symmetrizing measure and the generator of this dynamics, under which a corresponding conservative Markov processes exists. We also outline two types of scaling limit of the equilibrium Kawasaki dynamics: one leading to an equilibrium Glauber dynamics in continuum (a birt…
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 $Λ\subset\mathbb{Z}^2$ be a finite box. Particles perform simple exclusion on $Λ$, but when they occupy neighboring sites they feel a binding energy $-U_1<0$ in the horizontal direction and $-U_2<0$ in the vertical one. Thus the Kawasaki dynamics is conservative inside the volume $Λ$. Along each bond touching the boundary of $Λ$ from the outside to the inside, particles are created with rate …
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 $Λ\subset\mathbb{Z}^2$ be a finite box. Particles perform simple exclusion on $Λ$, but when they occupy neighboring sites they feel a binding energy $-U_1<0$ in the horizontal direction and $-U_2<0$ in the vertical one. Thus the Kawasaki dynamics is conservative inside the volume $Λ$. Along each bond touching the boundary of $Λ$ from the outside to the inside, particles are created with rate $ρ=e^{-Δβ}$, while along each bond from …
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 $β>0$ be the inverse temperature and let $Λ\subsetΛ^β\subset\mathbb{Z}^2$ be two boxes. We consider the asymptotic regime corresponding to the limit as $β\rightarrow\infty$ for finite volume $Λ$ and $\lim_{β\rightarrow\infty}\frac{1}β\log|Λ^β|=\infty$. We study the simplified model, in which particles perform independent random walks on $Λ^β\setminusΛ$ and inside $Λ$ particles perform simple exclus…
This is the third in a series of three papers in which we study a two-dimensional lattice gas consisting of two types of particles subject to Kawasaki dynamics at low temperature in a large finite box with an open boundary. Each pair of particles occupying neighboring sites has a negative binding energy provided their types are different, while each particle has a positive activation energy that depends on its type. There is no binding energy between particles of the same type. At the boundary of the box particles are created and annihilated in a way that represents the presence of an infinite…
We show that some classes of birth-and-death processes in continuum (Glauber dynamics) may be derived as a scaling limit of a dynamics of interacting hopping particles (Kawasaki dynamics)
We develop a general technique, based on a Bochner-type identity, to estimate spectral gaps of a class of Markov operator. We apply this technique to various interacting particle systems. In particular, we give a simple and short proof of the diffusive scaling of the spectral gap of the Kawasaki model at high temperature. Similar results are derived for Kawasaki-type dynamics in the lattice without exclusion, and in the continuum. New estimates for Glauber-type dynamics are also obtained.
This is the second in a series of three papers in which we study a two-dimensional lattice gas consisting of two types of particles subject to Kawasaki dynamics at low temperature in a large finite box with an open boundary. Each pair of particles occupying neighboring sites has a negative binding energy provided their types are different, while each particle has a positive activation energy that depends on its type. There is no binding energy between particles of the same type. At the boundary of the box particles are created and annihilated in a way that represents the presence of an infinit…
We study the most probable way an interface moves on a macroscopic scale from an initial to a final position within a fixed time in the context of large deviations for a stochastic microscopic lattice system of Ising spins with Kac interaction evolving in time according to Glauber (non-conservative) dynamics. Such interfaces separate two stable phases of a ferromagnetic system and in the macroscopic scale are represented by sharp transitions. We derive quantitative estimates for the upper and the lower bound of the cost functional that penalizes all possible deviations and obtain explicit erro…
