Thermodynamical Aspects of Classical Lattice Systems

Abstract

In these lectures I consider the mathematical formalism of thermodynamics and statistical mechanics for lattice systems and show its tight connection with large deviations theory. The models I have in mind are the lattice models with compact state spaces, although the theory is developed in greater generality. The central notion studied here is the notion of an asymptotically decoupled p r obability measure. The main theme of these lectures is the thermodynamical like aspects of the law of large numbers. The main feature of the asymptotically decoupled probability measures is that the existence of important q u a n tities, like the speci c information gain, which h a ve a thermodynamical interpretation in statistical mechanics, can be established using the standard and simple arguments developed for proving the existence of thermodynamical limits (see e.g. YL] 1 ). A large part of these lectures is original. However, the principal results and methods have b e e n d e v eloped much earlier in a less general setting 2 . My presentation is strongly in uenced by m y longstanding collaboration with John Lewis and Wayne Sullivan in Dublin. It is based principally on LP] and LPS3]. The notion of asymptotically decoupled measures 3 has been introduced in lectures delivered in Grenoble in Spring 1999 on the same subject.Section 2 serves as an introductory section. I consider a special model and present the main features of statistical mechanics of lattice systems. The interplay b e t ween mathematical physics and large deviations theory is displayed. No proof is given, but almost all results presented in that section are proved with complete proofs in much greater generality in sections 3 to 5. Section 2 is independent of the rest of the lectures.In section 3 asymptotically decoupled measures are de ned and examples are given. Large deviations properties of such measures are expressed through two functionals: the speci c information gain h( j ), which is de ned on the space of translation invariant probability measures M + 1 , a n d p( j ) (called the pressure), which is de ned on the dual space of M + 1 , the space of the quasilocal functions F qloc . These functionals are conjugate to each other. Convexity p l a ys an important role in the thermodynamical formalism. The solutions of the equation h( j ) = 0 and the related notion of asymptotically I-null sequences, extending the notion of asymptotically quasi-independence of Csisz ar, are then studied. In subsection 3.4 a large deviations principle for the empirical measures in the multidimensional case is established.In section 4 the space L of bounded total oscillation functions is studied. L is a dense subset of the quasilocal functions. Nice properties are true on L. In particular 1 The importance of the thermodynamical limit in statistical mechanics was recognized in the late thirties. See in particular the very interesting introduction of KU].2 I did not write a review paper. The bibliography i s v ery incomplete, and the historical remarks very rare. More inform...

The large deviation approach to statistical mechanics

“…The problem with short-range models is that the empirical process is, in general, the only macrostate admitting an energy representation function. This is the case, for example, for the nearest-neighbor Ising model, which has been studied extensively in one and two dimensions from the point of view of the empirical process (see, e.g., [84,85,229,230]). We summarize in the next example the equilibrium properties of the mean magnetization of the 2D version of this model, obtained by contracting the large deviations of the empirical process down to the mean energy and mean magnetization.…”

“…We summarize in the next example the equilibrium properties of the mean magnetization of the 2D version of this model, obtained by contracting the large deviations of the empirical process down to the mean energy and mean magnetization. The main sources for this example are Ellis [85], Pfister [230], and Kastner [148]. For a discussion of the Ising model in one dimension, see [84,182].…”

“…This "slower" large deviation principle describes a surface effect related to a change of phase, and depends on boundary conditions imposed on the model (see, e.g., [141,230]). …”

Large deviations and fluctuation theorem for selectively decoupled measures on shift spaces

“…This weaker condition covers some important classes of measures (see the examples below), which are not covered by any result in the literature, as far as we are aware. The Ruelle-Lanford estimates, which are done in the spirit of [Pfi02], are noticeably complicated by the fact that we allow |ξ| ≤ τ t in the (SLD) condition.…”

“…Proof. For the reader's convenience, we include a complete proof, although this is a classical result (see [Pfi02,Proposition 3.5] or [DZ00, Lemmas 4.1.11 and 4.1.21]). First, the special case x = x 1 = x 2 in (3.17) immediately implies that the two infima in (3.16) are equal, so that s is well defined.…”