In Chapter 3, we examined two examples of phase transitions, the paramagnetic–ferromagnetic transition (Section 3.1.4) and the liquid–vapour transition (Section 3.5.4). In the latter case, thermodynamic functions such as entropy or specific volume are discontinuous at the transition. When, for example, one varies the temperature T at constant pressure, the transition takes place at a temperature Tc where the Gibbs potentials of the two phases are equal, but the two phases coexist at Tc and are either stable or metastable in the vicinity of Tc. Each of the phases carries with it its own entropy, specific volume, etc., which are in general different at T = Tc, hence the discontinuities. Such a phase transition is called a first order phase transition. The picture is quite different in the paramagnetic–ferromagnetic phase transition: the thermodynamic quantities are continuous at the transition, the transition is not linked to the crossing of two thermodynamic potentials and one never observes metastability. Such a transition is called a second order, or continuous, phase transition. The transition temperature is called the critical temperature and is denoted by Tc.

The new and remarkable feature which we shall encounter in continuous phase transitions is the existence of cooperative phenomena. To be specific, think of a spin system, for example an Ising or a Heisenberg model, where the interactions between spins are limited to nearest neighbours. More generally we shall consider short range interactions, which decrease rapidly as a function of the distance between spins, and we exclude from our study all long range interactions.

The goal of this first chapter is to give a presentation of thermodynamics, due to H. Callen, which will allow us to make the most direct connection with the statistical approach of the following chapter. Instead of introducing entropy by starting with the second law, for example with the Kelvin statement ‘there exists no transformation whose sole effect is to extract a quantity of heat from a reservoir and convert it entirely to work’, Callen assumes, in principle, the existence of an entropy function and its fundamental property: the principle of maximum entropy. Such a presentation leads to a concise discussion of the foundations of thermodynamics (at the cost of some abstraction) and has the advantage of allowing direct comparison with the statistical entropy that we shall introduce in Chapter 2. Clearly, it is not possible in one chapter to give an exhaustive account of thermodynamics; the reader is, instead, referred to classic books on the subject for further details.

Thermodynamic equilibrium

Microscopic and macroscopic descriptions

The aim of statistical thermodynamics is to describe the behaviour of macroscopic systems containing of the order of N ≈ 1023 particles. An example of such a macroscopic system is a mole of gas in a container under standard conditions of temperature and pressure. This gas has 6 × 1023 molecules in incessant motion, continually colliding with each other and with the walls of the container. To a first approximation, which will be justified in Chapter 2, we may consider these molecules as classical objects.

This book attempts to give at a graduate level a self-contained, thorough and pedagogic exposition of the topics that, we believe, are most fundamental in modern statistical thermodynamics. It follows a balanced approach between the macroscopic (thermodynamic) and microscopic (statistical) points of view.

The first half of the book covers equilibrium phenomena. We start with a thermodynamic approach in the first chapter, in the spirit of Callen, and we introduce the concepts of equilibrium statistical mechanics in the second chapter, deriving the Boltzmann–Gibbs distribution in the canonical and grand canonical ensembles. Numerous applications are given in the third chapter, in cases where the effects of quantum statistics can be neglected: ideal and non-ideal classical gases, magnetism, equipartition theorem, diatomic molecules and first order phase transitions. The fourth chapter deals with continuous phase transitions. We give detailed accounts of symmetry breaking, discrete and continuous, of mean field theory and of the renormalization group and we illustrate the theoretical concepts with many concrete examples. Chapter 5 is devoted to quantum statistics and to the discussion of many physical examples: Fermi gas, black body radiation, phonons and Bose–Einstein condensation including gaseous atomic condensates.

Chapter 6 offers an introduction to macroscopic non-equilibrium phenomena. We carefully define the notion of local equilibrium and the transport coefficients together with their symmetry properties (Onsager). Hydrodynamics of simple fluids is used as an illustration. Chapter 7 is an introduction to numerical methods, in which we describe in some detail the main Monte Carlo algorithms.

We have given in two previous chapters a first introduction to non-equilibrium phenomena. The present chapter is devoted to a presentation of more general approaches, in which time dependence will be made explicit, whereas in practice we had to limit ourselves to stationary situations in Chapters 6 and 8. In the first part of the chapter, we examine the relaxation toward equilibrium of a system that has been brought out of equilibrium by an external perturbation. The main result is that, for small deviations from equilibrium, this relaxation is described by equilibrium time correlation functions, called Kubo (or relaxation) functions: this result is also known as ‘Onsager's regression law’. The Kubo functions turn out to be basic objects of non-equilibrium statistical mechanics. First they allow one to compute the dynamical susceptibilities, which describe the response of the system to an external time dependent perturbation: the dynamical susceptibilities are, within a multiplicative constant, the time derivatives of Kubo functions. A second crucial property is that transport coefficients can be expressed in terms of time integrals of Kubo functions. As we limit ourselves to small deviations from equilibrium, our theory is restricted to a linear approximation and is known as linear response theory. The classical version of linear reponse is somewhat simpler than the quantum one, and will be described first in Section 9.1. We shall turn to the quantum theory in Section 9.2, where one of our main results will be the proof of the fluctuationdissipation theorem.

This chapter is devoted to some important applications of the canonical and grand canonical formalisms. We shall concentrate on situations where the effects of quantum statistics, to be studied in Chapter 5, may be considered as negligible. The use of the formalism is quite straightforward when one may neglect the interactions between the elementary constituents of the system. Two very important examples are the ideal gas (non-interacting molecules) and paramagnetism (non-interacting spins), which are the main subjects of Section 3.1. In the following section, we show that at high temperatures one may often use a semi-classical approximation, which leads to simplifications in the formalism. Important examples are the derivation of the Maxwell velocity distribution for an interacting classical gas and of the equipartition theorem. These results are used in Section 3.3 to discuss the behaviour of the specific heat of diatomic molecules. In Section 3.4 we address the case where interactions between particles cannot be neglected, for example in a liquid, and we introduce the concept of pair correlation function. We show how pressure and energy may be related to this function, and we describe briefly how it can be measured experimentally. Section 3.5 shows the fundamental rôle played by the chemical potential in the description of phase transitions and of chemical reactions. Finally Section 3.6 is devoted to a detailed exposition of the grand canonical formalism, including a discussion of fluctuations and of the equivalence with the canonical ensemble.

We are almost ready to fully exploit the connection, established in earlier chapters, between the statistics of a self-avoiding random walk and the statistical mechanics of a magnet near the phase transition from its paramagnetic and ferromagnetic states. Because of the mathematical similarity between the two systems, we will be able to make use of an array of calculational strategies that, collectively, represent realizations of the renormalization group. This generic method for the study of systems with long-range correlations has fundamentally altered the way in which physicists view the world around them. The method is so powerful and so widespread in its application, that it seems worthwhile to do a little more than simply explain how to use it in the present context. This chapter consists of a discussion of the philosophy underlying the renormalization group and of a general description of the way in which it is applied. We will finish off by taking the reader through a simple calculation that is relevant to random walks and the associated magnetic system. Then, we will generalize the method to encompass a wide class of systems, the O(n) model being one of them. In the next chaper, the reader will be subjected to a full-blown introduction to the method, as it applies to the self-avoiding walk. Those already familiar with the renormalization group may wish to skip directly to Chapter 12.

Scale invariance in mathematics and nature

The notion of scale invariance is not exactly new. A famous poem by Jonathan Swift goes as follows:

The notion and quantification of shape

Shape is an intuitively accessible notion. We organize visual information in terms of shapes, and the shape of an object represents one of the first of its qualities referred to in an informal descriptive rendering of it. While our language presents us with a wide repertoire of verbal images for the approximate portrayal of the shape of a physical entity (“round,” “oblong,” “crescent,” “stellate” …) the precise characterization of a shape, in terms of a number, or set of numbers, has remained elusive. This is with good reason. It is well-known to mathematicians that the class consisting of the set of all curves is a higher order of infinity than the set of all real numbers. This means that there can be no one-to-one correspondence between curves and real numbers. As shapes, intuitively at least, bear a conceptual relationship to curves, it is plausible that the set of all shapes dwarfs in magnitude the set of real numbers, or of finite sets of real numbers.

On the other hand, if one is willing to content oneself with a general paradigm for the measurement of shape, there are ways of quantifying it in terms of numbers that have a certain descriptive and predictive utility. In fact, the numerical specification of shapes has acquired a certain urgency of late, in light of the widespread use of computer imaging and the concomitant focus on the development of codes for the creation and manipulation of pictorial quantities.

In this chapter, we will look at different ways of characterizing and measuring the shape of a random walk.

The concept of a field dates back to Euler, who introduced the notion to describe fluid flows in his study of hydrodynamics. Methods and concepts based on field theory now pervade the physical sciences and engineering. Field-theoretical ideas exert a strong influence on physical intuition and shape modern nomenclature. In addition, some of the most powerful analytical tools available to the modern scientist are those developed to study the behavior of fields.

In the context of models designed to describe the physical world, a field is a quantity that varies continuously in space and time. Examples are the electric and magnetic fields, the velocity and density distributions of a liquid or vapor and the quantum-mechanical wavefunction of a microscopic particle. In some cases, such as the velocity and density fields introduced by Euler, the notion of continuity must be taken advisedly. Because of the atomic structure of matter, one cannot carry the notion of a smooth density distribution down to the length scales on which molecules can be distinguished. There, the classical description is necessarily in terms of particles. Quantum mechanically, wavefunctions replace the classical density and velocity fields as the appropriate mode of description. This proviso notwithstanding, in the regimes in which density and velocity fields accurately describe the state of a liquid or vapor, they form the basis of an extremely useful theoretical model that yields important physical properties of these systems.

It turns out that the random walk also lends itself to description in terms of a field. As in the case of a liquid or vapor, the field-based description maintains its validity in a restricted range of length scales.

This entire book is, in one way or another, devoted to a single process: the random walk. As we will see, the rules that control the random walk are simple, even when we add elaborations that turn out to have considerable significance. However, as often occurs in mathematics and the physical sciences, the consequences of simple rules are far from elementary. We will also discover that random walks, as interesting as they are in themselves, provide a basis for the understanding of a wide range of phenomena. This is true in part because random walk processes are relevant to so many processes in such a wide range of contexts. It also follows from the fact that the solution of the random walk problem requires the use of so many of the mathematical techniques that have been developed and applied in contemporary twentieth-century physics. We'll start out simply, but it won't be long before we enounter aspects to the problem that invite – indeed require – intense scrutiny.

We begin our investigations by looking at the random walk in its most elementary manifestation. The reader may find that most of what follows in this chapter is familiar material. It is, nevertheless, useful to read through it. For one thing, review is always helpful. More importantly, connections that are hinted at in the early portions of this book will play an important role in later discussion.

The simplest walk

In the simplest example of a random walk the walker is confined to a straight line. This kind of walk is called, appropriately enough, a one-dimensional walk. In this case, steps take the walker in one direction or the other.

General introduction to generating functions

This book makes extensive use of generating functions. In that respect the discussions here are consistent with the approach that condensed matter physicists generally take when calculating properties of the random walk as it relates to problems of contemporary interest. This chapter is devoted to a discussion of the generating function and to an exploration of some of the ways in which the generating function method can be put to use in the study of the random walk. Many of the arguments in later chapters will call upon techniques and results that will be developed in the pages to follow. Thus, the reader is strongly urged to pay close attention to the discussion that follows, as topics and techniques that are introduced here will crop up repeatedly later on.

What is a generating function?

The generating function is a mathematical stratagem that simplifies a number of problems. Its range of applicability extends far beyond the mathematics of the random walk. Readers who have had an introduction to ordinary differential equations will have already seen examples of the use of the method of the generating function in the study of special functions. The generating function also plays a central role in graph theory and in the study of combinatorics, percolation theory, classical and quantum field theory and a myriad of other applications in physics and mathematics. Briefly, a generating function is a mathematical expression, depending on one or more variables, that admits a power series expansion. The coefficients of the expansion are the members of a family, or sequence, of numbers or functions.