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## Über dieses Buch

This book has two main topics: large deviations and equilibrium statistical mechanics. I hope to convince the reader that these topics have many points of contact and that in being treated together, they enrich each other. Entropy, in its various guises, is their common core. The large deviation theory which is developed in this book focuses upon convergence properties of certain stochastic systems. An elementary example is the weak law of large numbers. For each positive e, P{ISn/nl 2: e} con­ verges to zero as n --+ 00, where Sn is the nth partial sum of indepen­ dent identically distributed random variables with zero mean. Large deviation theory shows that if the random variables are exponentially bounded, then the probabilities converge to zero exponentially fast as n --+ 00. The exponen­ tial decay allows one to prove the stronger property of almost sure conver­ gence (Sn/n --+ 0 a.s.). This example will be generalized extensively in the book. We will treat a large class of stochastic systems which involve both indepen­ dent and dependent random variables and which have the following features: probabilities converge to zero exponentially fast as the size of the system increases; the exponential decay leads to strong convergence properties of the system. The most fascinating aspect of the theory is that the exponential decay rates are computable in terms of entropy functions. This identification between entropy and decay rates of large deviation probabilities enhances the theory significantly.

## Inhaltsverzeichnis

### Chapter I. Introduction to Large Deviations

Abstract
One of the common themes of probability theory and statistical mechanics is the discovery of regularity in the midst of chaos. The laws of probability theory, which include laws of large numbers and central limit theorems, summarize the behavior of a stochastic system in terms of a few parameters (e.g., mean and variance). In statistical mechanics, one derives macroscopic properties of a substance from a probability distribution that describes the complicated interactions among the individual constituent particles. A central concept linking the two fields is entropy.1 The term was introduced into thermodynamics by Clausius in 1865 after many years of intensive work by him and others on the second law of thermodynamics. An early important step in its development and enrichment was the discovery by Boltzmann of a statistical interpretaton of entropy. Boltzmann’s discovery, which was published in 1877, has three parts. We have augmented part (c) to include the possibility of phase transitions.
Richard S. Ellis

### Chapter II. Large Deviation Property and Asymptotics of Integrals

Abstract
The previous chapter treated three levels of large deviations for independent, identically distributed (i.i.d.) random variables with a finite state space. The main results show the exponential decay of large deviation probabilities. A level-1 example is P ρ {|S n /nm ρ | ≥ ε}, where S n is the nth partial sum of the random variables and m ρ is their common mean. Levels-2 and 3 treat analogous probabilities for the empirical measures {L n } and the empirical processes {R n }, respectively. One purpose of the present chapter is to expand the scope of large deviations and to prepare the groundwork for applications to statistical mechanics later in the book.1 This chapter extends the ideas of Chapter I by considering random vectors taking values in ℝ d , where d≥1 is a fixed integer. ℝ d is a natural state space for stochastic models, including models in statistical mechanics. The theory, which is somewhat technical and detailed, is presented in a manner that closely parallels the development in Chapter I, where elementary proofs based upon combinatorics were possible because of the finite state space. The reader who followed that development should find the new theorems familiar looking and plausible. In order to reach illuminating applications of the theory to statistical mechanical models, the proofs of the theorems will be postponed until Chapters VI–IX.
Richard S. Ellis

### Chapter III. Large Deviations and the Discrete Ideal Gas

Abstract
In the next three chapters we apply the theory of large deviations to analyze some basic models in equilibrium statistical mechanics.1 This branch of physics applies probability theory to study equilibrium properties of systems consisting of a large number of particles. The systems fall into two groups: continuous systems, which include the solids, liquids, and gases common to everyday experience; and lattice systems, of which ferromagnets are the main example. This chapter introduces the continuous theory by treating a simple model called a discrete ideal gas. This model, which has no interactions, is a physical analog of i.i.d. random variables.
Richard S. Ellis

### Chapter IV. Ferromagnetic Models on ℤ

Abstract
Phase transitions are a familiar aspect of nature. Water boils, becoming water vapor, or water vapor, under compression, liquefies. These are examples of a liquid-gas phase transition. The liquid and the gas are said to be two phases of the same substance. One of the most interesting problems in equilibrium statistical mechanics is to explain phase transitions in terms of the probability distributions on configuration space which describe the microscopic behavior of physical systems. The simplest systems for which this is possible are ferromagnetic models on a lattice. The present chapter introduces these models.
Richard S. Ellis

### Chapter V. Magnetic Models on ℤ D and on the Circle

Abstract
This chapter consists of two parts. Part 1 extends the results of Chapter IV to ferromagnetic models on the integer lattices ℤ D , D∈{1, 2, ...}. These results included properties of the specific Gibbs free energy, the specific magnetization, and infinite-volume Gibbs states. The dimension of the lattice enters in a dramatic way in the proof of spontaneous magnetization. For D = 1, the critical inverse temperature β c is finite only for interactions of infinite range; e.g., J(k) = |k|−α, k ≠ 0, for some 1 < α ≤ 2. By contrast, for D ≥ 2, β c is finite for any nontrivial interaction J [Theorem V.5.1]. We give a complete proof of spontaneous magnetization for D ≥ 2. First, a set of powerful moment inequalities is proved which allow us to reduce from a general model to the Ising model on ℤ2. Then spontaneous magnetization is shown for the latter by means of a combinatorial argument due to Peierls. The moment inequalities also yield monotonicity and concavity properties of the specific magnetization which were stated in Chapter IV without proof.
Richard S. Ellis

### Chapter VI. Convex Functions and the Legendre-Fenchel Transform

Abstract
The first part of this book illustrated the use of entropy concepts in analyzing stochastic and statistical mechanical systems. Convexity was a recurring theme. Suppose that ρ is a probability measure on ℝ d such that
$${c_p}\left( t \right) = \log \int_{{\mathbb{R}^d}} {\exp \left\langle {t,x} \right\rangle \rho \left( {dx} \right)}$$
is finite for all t in ℝ d . The function c ρ (t), called the free energy function of ρ, is a convex function on ℝ d [Example VII.1.2]. The Legendre-Fenchel transform of c ρ (t) is given by
$$I_\rho ^{\left( 1 \right)}\left( z \right) = \mathop {\sup }\limits_{t \in {\mathbb{R}^d}} \left\{ {\left\langle {t,z} \right\rangle } \right. - {c_\rho }\left. {\left( t \right)} \right\},z \in {\mathbb{R}^d}.$$
Richard S. Ellis

### Chapter VII. Large Deviations for Random Vectors

Abstract
In Section II.4, we stated three levels of large deviation properties for i.i.d. random vectors taking values in ℝ d . In Section II.6 we also stated a large deviation theorem for random vectors [Theorem 11.6.1] which generalized the level-1 property. In this chapter, Theorem 11.6.1 will be proved [Sections VII.2–VII.4] and the level-1 large deviation property will be derived as a corollary [Section VII.5]. The results on exponential convergence of random vectors stated in Theorem 1I.6.3 will be proved in Section VII.6.
Richard S. Ellis

### Chapter VIII. Level-2 Large Deviations for I.I.D. Random Vectors

Abstract
Theorem II.4.3 stated the level-2 large deviation property for i.i.d. random vectors taking values in ℝ d . This theorem follows from the results contained in Donsker and Varadhan (1975a, 1976a), which prove level-2 large deviation properties for Markov processes taking values in a complete separable metric space.1 In Chapter VIII, we will give an elementary, self-contained proof of Theorem I1.4.3 in the special case of i.i.d. random variables with a finite state space. This version of the theorem was applied in Chapter III to study the exponential convergence of velocity observables for the discrete ideal gas with respect to the microcanonical ensemble [Theorem III.4.4].
Richard S. Ellis

### Chapter IX. Level-3 Large Deviations for I.I.D. Random Vectors

Abstract
Theorem II.4.4 stated the level-3 large deviation property for i.i.d. random vectors taking values in ℝ d . In this chapter, we prove Theorem II.4.4 in the special case of i.i.d. random variables with a finite state space. This version of the theorem covers the applications of level-3 large deviations which were made in Chapters III, IV, and V to the Gibbs variational principle. Theorem II.4.4 can also be proved via the methods of Donsker and Varadhan (1983a). The main result in that paper is a level-3 theorem for continuous parameter Markov processes taking values in a complete separable metric space.1
Richard S. Ellis

### Backmatter

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