WHAT IS A MONTE CARLO SIMULATION?

In a Monte Carlo simulation we attempt to follow the ‘time dependence’ of a model for which change, or growth, does not proceed in some rigorously predefined fashion (e.g. according to Newton's equations of motion) but rather in a stochastic manner which depends on a sequence of random numbers which is generated during the simulation. With a second, different sequence of random numbers the simulation will not give identical results but will yield values which agree with those obtained from the first sequence to within some ‘statistical error’. A very large number of different problems fall into this category: in percolation an empty lattice is gradually filled with particles by placing a particle on the lattice randomly with each ‘tick of the clock’. Lots of questions may then be asked about the resulting ‘clusters’ which are formed of neighboring occupied sites. Particular attention has been paid to the determination of the ‘percolation threshold’, i.e. the critical concentration of occupied sites for which an ‘infinite percolating cluster’ first appears. A percolating cluster is one which reaches from one boundary of a (macroscopic) system to the opposite one. The properties of such objects are of interest in the context of diverse physical problems such as conductivity of random mixtures, flow through porous rocks, behavior of dilute magnets, etc. Another example is diffusion limited aggregation (DLA) where a particle executes a random walk in space, taking one step at each time interval, until it encounters a ‘seed’ mass and sticks to it.

As we have seen on several occasions, a great many of the physical phenomena of interest concern collective behaviour or strongly correlated particles. While it is sometimes possible to obtain useful insight using approximation methods, such as perturbation theory, reliable controlled approximations do not exist. In such situations, numerical simulations have become an indispensable tool.

We shall not attempt in this chapter to discuss computer programming nor shall we discuss many of the powerful methods, both algorithmic and in data analysis, that have been developed since numerical simulations have become an important research tool. Instead, we shall assume that the reader already has enough familiarity with computer programming to apply the notions which we discuss here. Our goal is to explain enough of the basics of the Monte Carlo method (classical and quantum) to allow the reader to apply it readily to interesting equilibrium statistical physics problems of the type discussed in this book. The problems in this chapter have been tailored with this in mind. The programs needed are relatively simple and the physics illustrates many of the phenomena already seen: phase transitions with and without spontaneous symmetry breaking, critical exponents, diverging susceptibilities, scaling, vortices, superfluids etc. Consequently, some of the problems are somewhat long and may be better considered as mini-projects.

Markov chains, convergence and detailed balance

Even relatively innocent looking models are impossible to solve exactly. The exact solution of the two-dimensional Ising model is known but not in the presence of an external magnetic field. In three dimensions, even in the absence of a magnetic field, the exact solution is not known. In fact, relatively few models can be solved exactly; see Reference [14] for an excellent review of the subject.

In Chapter 3 we exhibited the limitations of a purely classical approach. For example, if the temperature is below a threshold value, some degrees of freedom become ‘frozen’ and the equipartition theorem is no longer valid for them. The translational degrees of freedom of an ideal gas appear to escape this limitation of the classical (or more precisely, semi-classical) approximation. We shall see in this chapter that, in fact, this is not so: if the temperature continues to decrease below some reference temperature, the classical approximation will deteriorate progressively. However, in this case, the failure of the classical approximation is not related to freezing degrees of freedom but rather to the symmetry properties of the wave function for identical particles imposed by quantum mechanics. A rather spectacular consequence is that the kinetic energy is no longer a measure of the temperature. In a classical gas, even in the presence of interactions, the average kinetic energy is equal to 3kT/2, but this result does not hold when the temperature is low enough, even for an ideal gas. For example, if we consider the conduction electrons in a metal as an ideal gas, we shall show that the average kinetic energy of an electron is not zero even at zero temperature. In addition, this kinetic energy is about 100 times kT at normal temperature. Let us consider another example. In a gaseous mixture of helium-3 and helium-4 at low temperature, the average kinetic energies of the two isotopes are different: the average kinetic energy of helium-3 is larger than 3/2kT while that of helium-4 is smaller.

In the preceding chapters, we have limited our analysis to equilibrium situations. This is rather restrictive since non-equilibrium phenomena, such as heat conduction or diffusion, are of great interest and cannot be ignored. To remedy this, we focus in this chapter on an introduction to non-equilibrium phenomena. Further developments of the subject will be found in Chapters 8 and 9.

We have seen that equilibrium statistical mechanics is built on a general and systematic approach, namely the Boltzmann–Gibbs distribution. No such general approach is available for non-equilibrium situations; instead, we find a large variety of methods suited to particular cases and situations. What we are able to control well are cases close to equilibrium where we can rely on rather general methods like linear response theory, which will be described in Sections 9.1 and 9.2. In the present chapter, we shall consider a macroscopic approach, that of transport coefficients, which is the non-equilibrium analogue of equilibrium thermodynamics. At this stage, we shall not attempt a calculation of these transport coefficients from a microscopic theory. We shall only show that these coefficients satisfy a number of general properties, their actual values being taken from experiments. This parallels equilibrium thermodynamics where we uncovered a number of general relations between thermodynamic quantities while we did not attempt to calculate, for example, the specific heat from a microscopic theory but took its value from experiments.

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.