Introduction

In this chapter we discuss metastability and nucleation for several short range lattice spin systems at low temperature. We already analysed the case of the Curie–Weiss model in Chapter 4. Due to the fact that the intermolecular interaction does not decay at infinity, the Curie–Weiss model exhibits a mean field behaviour without any spatial structure; the configurations are well described, especially for large volumes, by the values of a unique macroscopic order parameter, i.e. the magnetization, so that the configuration space becomes one-dimensional. In contrast, for short range stochastic Ising models the geometrical aspects are particularly relevant. In this last case the configuration space can be viewed as a space of families of contours and it tends to be infinite dimensional in the thermodynamic limit. Indeed it is well known that the description of pure coexisting equilibrium phases at low temperature is naturally given in terms of a gas of contours (see Chapter 3, proof of Proposition 3.30, and [280]). It is clear that in the analysis of dynamical phenomena taking place in large finite systems, the geometrical description in terms of contours will also play a relevant role.

The central question in the description of the decay from a metastable to a stable phase for short range stochastic Ising models is, in addition to determination of the typical escape time, the characterization of the typical ‘nucleation pattern’ namely the typical sequences, in shape and size, of droplets along which nucleation of the stable phase takes place.

Introduction

In the analysis of a system with a large number of interacting components (at a microscopic level) it is of clear importance to find out about its collective, or macroscopic, behaviour. This is quite an old problem, going back to the origins of statistical mechanics, in the search for a mathematical characterization of ‘equilibrium states’ in thermodynamical systems. Though the problem is old, and the foundations of equilibrium statistical mechanics have been settled, the general question remains of interest, especially in the set-up of non-equilibrium systems. We could then take as the object of study a (non-stationary) time evolution with a large number (n) of components, where the initial condition and/or the dynamics present some randomness. One example of such a collective description is the so-called hydrodynamic limit. Passing by a space-time scale change (micro → macro) it allows, through a limiting procedure, the derivation of a reduced description in terms of macroscopic variables, such as density and temperature. Other limits, besides the hydrodynamic, may also appear in different situations, giving rise to macroscopic equations.

In all such cases the macroscopic equation indicates the typical behaviour in a limiting situation (n → +∞, and proper rescaling). Thus, it is essential to know something about:

1. rates of convergence, i.e. how are the fluctuations of the macroscopic random fields (for example, the empirical density) around the prescribed value given by the macroscopic equation?

2. how to estimate the chance of observing something quite different than what is prescribed by the macroscopic equation.

1. …

This book has germinated from the lecture notes of a course ‘Large deviations and metastability’ given by one of us at the ‘CIMPA First School on Dynamical and Disordered Systems’, at Universidad de la Frontera, Temuco, during the summer of 1992 [293].

Since then a large amount of new material on metastability has been accumulated, and our goal was to combine a basic introduction to the theory of large deviations with a wide overview of the metastable behaviour of stochastic dynamics.

Typical examples of metastable states are supersaturated vapours and magnetic systems with magnetization opposite to the external field. Metastable behaviour is characterized by a long period of apparent equilibrium of a pure thermodynamic phase followed by an unexpected fast decay towards the stable equilibrium of a different pure phase or of a mixture, e.g. homogeneous nucleation of the liquid phase inside a highly supersaturated vapour, due to spontaneous density fluctuations. The point of view of metastability as a genuinely dynamical phenomenon is now widely accepted. Approaches which aim to describe static aspects of metastability (such as determination of the metastable branch of the equation of state of a fluid) in the Gibbs equilibrium set-up are, in their ‘naïve form’, applicable only in a mean field context. In this case, the physically unacceptable assumption that the range of the interaction equals the linear dimension of the container gives rise to pathological behaviour of non-convex free energy that implies negative compressibility, namely, thermodynamic instability.

The van der Waals–Maxwell theory

Metastability is a relevant phenomenon for thermodynamic systems close to a first order phase transition. Examples are supercooled vapours and liquids, super-saturated vapours and solutions, as well as ferromagnets in the part of the hysteresis loop where the magnetization is opposite to the external magnetic field. A metastable state occurs when some thermodynamic parameter such as the temperature, pressure or magnetic field is changed from a value giving rise to a stable state with a unique phase, say X, to one for which at least part of the system should be in some new equilibrium phase Y. Then, in particular experimental situations, instead of undergoing the phase transition, the system goes over continuously into a ‘false’ equilibrium state with a unique phase X′, far from Y but actually close to the initial equilibrium phase X. It is this apparent equilibrium situation that is called a ‘metastable state’. Its properties are very similar to those of the stable equilibrium state; for example for a supersaturated vapour one can determine the pressure experimentally as a function of the temperature and the specific volume. We speak of the ‘metastable branch’ of the isothermal curve.

The distinguishing feature of metastability is that, eventually, either via an external perturbation or via a spontaneous fluctuation, a nucleus of the new phase appears, starting an irreversible process which leads to the stable equilibrium state Y, where the phase transition has taken place.

Summary Markov processes and the important subclass of Feller processes are introduced and shown to be determined by the associated semigroups. We take an analytic diversion into semigroup theory and investigate the important concepts of generator and resolvent. Returning to Lévy processes, we obtain two key representations for the generator: first, as a pseudo-differential operator; second, in ‘Lévy–Khintchine form’, which is the sum of a second-order elliptic differential operator and a (compensated) integral of difference operators. We also study the subordination of such semigroups and their action in Lp-spaces.

The structure of Lévy generators, but with variable coefficients, extends to a general class of Feller processes, via Courrège's theorems, and also to Hunt processes associated with symmetric Dirichlet forms, where the Lévy–Khintchine-type structure is apparent within the Beurling–Deny formula.

Markov processes, evolutions and semigroups

Markov processes and transition functions

Intuitively, a stochastic process is Markovian (or, a Markov process) if using the whole past history of the process to predict its future behaviour is no more effective than a prediction based only on a knowledge of the present. This translates into precise mathematics as follows.

Let (Ω, F, P) be a probability space equipped with a filtration (Ft, t ≥ 0). Let X = (X(t), t ≥ 0) be an adapted process.

Summary Section 1.1 is a review of basic measure and probability theory. In Section 1.2 we meet the key concepts of the infinite divisibility of random variables and of probability distributions, which underly the whole subject. Important examples are the Gaussian, Poisson and stable distributions. The celebrated Lévy–Khintchine formula classifies the set of all infinitely divisible probability distributions by means of a canonical form for the characteristic function. Lévy processes are introduced in Section 1.3. These are essentially stochastic processes with stationary and independent increments. Each random variable within the process is infinitely divisible, and hence its distribution is determined by the Lévy–Khintchine formula. Important examples are Brownian motion, Poisson and compound Poisson processes, stable processes and subordinators. Section 1.4 clarifies the relationship between Lévy processes, infinite divisibility and weakly continuous convolution semigroups of probability measures. Finally, in Section 1.5, we briefly survey recurrence and transience, Wiener–Hopf factorisation and local times for Lévy processes.

Review of measure and probability

The aim of this section is to give a brief resumé of key notions of measure theory and probability that will be used extensively throughout the book and to fix some notation and terminology once and for all. I emphasise that reading this section is no substitute for a systematic study of the fundamentals from books such as Billingsley, Itô, Ash and Doléans-Dade, Rosenthal, Dudley or, for measure theory without probability, Cohn.

Summary After a review of first-order differential equations and their associated flows, we investigate stochastic differential equations (SDEs) driven by Brownian motion and an independent Poisson random measure. We establish the existence and uniqueness of solutions under the standard Lipschitz and growth conditions, using the Picard iteration technique. We then turn our attention to investigating properties of the solution. These are exhibited as stochastic flows and as multiplicative cocycles. The interlacing structure is established, and we prove the continuity of solutions as a function of their initial conditions. We then show that solutions of SDEs are Feller processes and compute their generators. Perturbations are studied via the Feynman–Kac formula. We briefly survey weak solutions and associated martingale problems. Finally, we study solutions of Marcus canonical equations and discuss the respective conditions under which these yield stochastic flows of homeomorphisms and diffeomorphisms.

One of the most important applications of Itô's stochastic integral is in the construction of stochastic differential equations (SDEs). These are important for a number of reasons.

1. Their solutions form an important class of Markov processes where the infinitesimal generator of the corresponding semigroup can be constructed explicitly. Important subclasses that can be studied in this way include diffusion and jump-diffusion processes.

2. Their solutions give rise to stochastic flows, and hence to interesting examples of random dynamical systems.

3. They have many important applications to, for example, filtering, control, finance and physics.