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Cambridge University Press eBooks

Introduction

J. W. Halleý

Published Nov 16, 2006• 0 citations
Publisher

Abstract

The problems of statistical mechanics are those which involve systems with a larger number of degrees of freedom than we can conveniently follow explicitly in experiment, theory or simulation. The number of degrees of freedom which can be followed explicitly in simulations has been changing very rapidly as computers and algorithms improve. However, it is important to note that, even if computers continue to improve at their present rate, characterized by Moore's “law,” scientists will not be able to use them for a very long time to predict many properties of nature by direct simulation of the fundamental microscopic laws of physics. This point is important enough to emphasize. Suppose that, T years from the present, a calculation requiring computation time t 0 at present will require computation time t ( T ) = t 0 2 − T /2 (Moore's “law,” see Figure 1). Currently, state of the art numerical solutions of the Schrödinger equation for a few hundred atoms can be carried out fast enough so that the motion of these atoms can be followed long enough to obtain thermodynamic properties. This is adequate if one wishes to predict properties of simple homogeneous gases, liquids or solids from first principles (as we will be discussing later). However, for many problems of current interest, one is interested in entities in which many more atoms need to be studied in order to obtain predictions of properties at the macroscopic level of a centimeter or more.

DOI
10.1017/cbo9780511815843.002

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The problems of statistical mechanics are those which involve systems with a larger number of degrees of freedom than we can conveniently follow explicitly in experiment, theory or simulation. The number of degrees of freedom which can be followed explicitly in simulations has been changing very rapidly as computers and algorithms improve. However, it is important to note that, even if computers continue to improve at their present rate, characterized by Moore's “law,” scientists will not be able to use them for a very long time to predict many properties of nature by direct simulation of the fundamental microscopic laws of physics. This point is important enough to emphasize. Suppose that, T years from the present, a calculation requiring computation time t 0 at present will require computation time t ( T ) = t 0 2 − T /2 (Moore's “law,” see Figure 1). Currently, state of the art numerical solutions of the Schrödinger equation for a few hundred atoms can be carried out fast enough so that the motion of these atoms can be followed long enough to obtain thermodynamic properties. This is adequate if one wishes to predict properties of simple homogeneous gases, liquids or solids from first principles (as we will be discussing later). However, for many problems of current interest, one is interested in entities in which many more atoms need to be studied in order to obtain predictions of properties at the macroscopic level of a centimeter or more.
Abstract Statistical mechanics is the art of predicting the behaviour of a system with a large number of degrees of freedom, given the laws governing its microscopic behaviour. It divides naturally into two parts. The first is the more fundamental. It tries to answer general questions concerning the behaviour of systems with many degrees of freedom, such as how order can arise from the apparent chaos of the microscopic behaviour, and how time-symmetric microscopic laws can give rise to the time asymmetry observed in the macroscopic world. The second part of statistical mechanics is concerned with using the results of this analysis to determine the behaviour of specific systems. This is what we will be doing in this book. Statistical mechanics will be used as a tool to study critical phenomena in particular systems.
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Abstract We have now seen how powerful the tools of statistical mechanics can be in the study of complex systems in thermal equilibrium and how they can be used to determine exactly the behaviour of some simplified, but nevertheless interesting, models of phase transitions. However, these exact solutions are sometimes lengthy and difficult and there is no guarantee that any particular model will yield exact results. For example, we were able to solve the two-dimensional, but not the three-dimensional, Ising model, and that only in zero magnetic field. So, it is natural to consider whether we can use a computer to simulate these systems. In this chapter we shall see that this is indeed possible; a great deal of our understanding of critical phenomena is based wholly or partly on the results of such numerical work. However, as we shall also see, such simulations are not entirely straightforward and a number of tricks are needed to obtain accurate results. This is partly a consequence of the very physics that makes the phase transitions interesting to start with. One should probably think of numerical simulation as a whole subject, almost an art, of its own; we shall have time only to scratch at the surface and discuss a few of the most important ideas.
In 2005 we celebrate the hundredth anniversary of the publication of five seminal papers by Albert Einstein establishing the basis of three fundamental fields of physics: the theory of relativity, quantum theory and the theory of Brownian motion. This happened at a time when the reality of atoms and molecules was still far from certain. The microscopic view Einstein took of molecular motion led to the calculation of Avogadro's number and the size of molecules by studying the motion of small particles in liquids. Combining kinetic theory and classical thermodynamics finally led Einstein to the conclusion that the displacement of a microparticle under Brownian motion varies as the square root of time. The experimental demonstration of this law three years later was considered as the striking proof that atoms and molecules are physically real. Today single atoms are probed directly in experiments, and we are able to observe the individual paths of colliding or dissociating particles. These new techniques are described in some of the papers of this issue in a very impressive way. An even more exciting development was initiated with the demonstration of Bose–Einstein condensation of dilute gases of ultracold atoms, a phenomenon first predicted by Einstein in papers published between 1916 and 1924. The first paper of Einstein's famous series of 1905, 'On an heuristic point of view about the creation and conversion of light', was an explanation of the photoelectric effect, demonstrated in several experiments over the previous few years. The explanation was based on the quantum hypothesis introduced by Max Planck five years earlier, and was considered as an additional and important proof of Planck's hypothesis. Today quantum theory and photons are the basis for much of modern science and technology. We have learned to experiment with single photons, and we have demonstrated the advantages of information transfer by single photons. Photons permit new and incredibly precise time and frequency standards. These are not only technically important, but may also lead to even more stringent tests of relativity and other fundamental laws and concepts of physics. Examples of these applications and many other uses of modern optics are discussed in this issue. Special relativity, the third of Einstein's 1905 topics, has always been closely connected with atomic, molecular and optical physics. Today atomic physics provides some of the most stringent tests of special relativity. Furthermore, fast electron beams are used in advanced light sources such as synchrotron radiation generators and free electron lasers. Those aspects and related applications are also discussed here. We hope that this special issue will be of great interest to the reader by highlighting recent advances in atomic, molecular and optical physics. This field continues to provide one of the most fertile areas for research one hundred years after it first emerged from the work of Einstein published in 1905, his 'miraculous year'.
The issue is that one may enter an upper level course on statistical and thermal physics with only an appreciation of the classical macroscopic view never having attached quantum mechanics to anything thermodynamic. Statistical mechanics makes accurate predictions about systems which may contain 1023 or more particles. A sense of disbelief may be partly responsible for the widely held, but, incorrect view that statistical mechanics has to be difficult. A true understanding of entropy only comes from the microscopic treatment of the quantum approach made by statistical mechanics. The more fundamental, issue is that classical statistical mechanics is internally inconsistent, and can only be made to work by the importation from quantum theory of non-classical ideas such as indistinguishability. Gibbs’ paradox, over which the classical books agonize at length, vanishes into thin air in the quantum approach.
In the present monograph, we study infinite classical statistical systems, i.e., systems of infinitely many particles in the infinite phase space. The states of these systems are completely defined by infinite sequences of correlation functions. The key problem of statistical mechanics is to determine the state of a many-particle system at any time if the initial state is known, i.e., to determine its evolution. The other important problem is to determine and investigate equilibrium (stationary) states of statistical systems.