Statistical Mechanics Made Simple

2nd Edition

Daniel C Mattis, Robert Swendsen

• Publisher: WSPC
• Year: 2008
• Print ISBN: 9789812779090
• eBook ISBN: 9789814365383
• Edition: 2
• Topic: Physical Sciences
• Subtopic: Physics
• Index: Physics

This second edition extends and improves on the first, already an acclaimed and original treatment of statistical concepts insofar as they impact theoretical physics and form the basis of modern thermodynamics. This book illustrates through myriad examples the principles and logic used in extending the simple laws of idealized Newtonian physics and quantum physics into the real world of noise and thermal fluctuations.

In response to the many helpful comments by users of the first edition, important features have been added in this second, new and revised edition. These additions allow a more coherent picture of thermal physics to emerge. Benefiting from the expertise of the new co-author, the present edition includes a detailed exposition — occupying two separate chapters — of the renormalization group and Monte-Carlo numerical techniques, and of their applications to the study of phase transitions. Additional figures have been included throughout, as have new problems. A new Appendix presents fully worked-out solutions to representative problems; these illustrate various methodologies that are peculiar to physics at finite temperatures, that is, to statistical physics.

This new edition incorporates important aspects of many-body theory and of phase transitions. It should better serve the contemporary student, while offering to the instructor a wider selection of topics from which to craft lectures on topics ranging from thermodynamics and random matrices to thermodynamic Green functions and critical exponents, from the propagation of sound in solids and fluids to the nature of quasiparticles in quantum liquids and in transfer matrices.

Contents:

• Elementary Concepts in Statistics and Probability
• The Ising Model and the Lattice Gas
• Elements of Thermodynamics
• Statistical Mechanics
• The World of Bosons
• All About Fermions: Theories of Metals, Superconductors, Semiconductors
• Kinetic Theory
• The Transfer Matrix
• Monte Carlo and Other Computer Simulation Methods
• Critical Phenomena and the Renormalization Group
• Some Uses of Quantum Field Theory in Statistical Physics

Readership: Upper-level undergraduates, graduate students, academics and researchers in statistical physics.

Chapter 1

Elementary Concepts in Statistics and Probability

1.0. Random Variables and Their Distributions

Like the winning numbers of lottery tickets, physical variables often have a random component, with quantum mechanics adding an extra layer of uncertainty to the results. Some variables, such as the spatial position of an object r = (x, y, z), are continuous (x → x + dx), allowing differential calculus to be brought to bear. Others are discrete (e.g., the color of a sock in a drawer that contains a number of brown, grey and black socks, the identity of a card in a playing deck of 52, the spin of an electron, etc.) and require a distinct approach.

In this chapter we turn our attention to the most basic of discrete cases, the binary example of a coin toss (e.g., heads or tails.) Although a trained prestidigitator or a specially devised machine might be able to produce coin tosses that are always heads, under normal circumstances experience and common sense tell us that a coin toss results in 50% heads and 50% tails. That is, an infinitesimal initial variability in the toss results in maximum variability of the results. We study this phenomenon in detail with the aid of a remarkably simple tool, the binomial distribution. A bias toward one or the other outcome is then introduced. At the end of the chapter we generalize to an arbitrary number of discrete possibilities, using the multinomial distribution. Taken together, these intuitive results suggests a rôle for, and definition of, temperature, as the control parameter for the generation of random events.

1.1. The Binomial Distribution

We can obtain all the binomial coefficients from a simple generating function G_N:

where the

symbol^a (equally written

stands for the ratio of factorials N!/n₁!n₂!. Recalling the definition of the factorial of a positive integer n! = 1 × 2 × · · · × n, it is also conventional to define 0! = 1 by extension. This definition is required in order to satisfy a logical identity, that the number of ways to choose N objects out of a set of N is

. Both here and subsequently, n₂ = N – n₁.

If the p's are positive, each term in the sum is positive. If restricted to p₁ + p₂ = 1 they add to G_N(p₁,1 – p₁) ≡ 1. Thus, each term in the expansion on the right-hand side of (1.1) can be viewed as a probability of sorts.

Generally there are only three requirements for a function to be a probability: it must be non-negative, sum to 1, and it has to express the relative frequency of some stochastic (i.e. random) phenomenon in a meaningful way. The binary distribution which ensues from the generating function above can serve to label a coin toss (let 1 be “heads” and 2 “tails”), or to label spins “up” in a magnetic spin system by 1 and spins “down” by 2, or to identify copper atoms by 1 and gold atoms by 2 in a copper-gold alloy, etc. Indeed all non-quantum mechanical binary processes with a statistical component are similar and can be studied in the same way.

It follows (by inspection of Eq. (1.1) that we can define the probability of n₁ heads and n₂ = N – n₁ tails, in N tries, as

subject to p₂ = 1 – p₁. This chapter concerns in part the manner in which one chooses p₁ and p₂ in physical processes. These are the parameters that predetermine the relative a priori probabilities of the two events. Of course, by just measuring the relative frequency of the two events one could determine their respective values a posteriori after a sufficiently large number of tries N, and on the way measure all other properties of the binary distribution including as its width (second moment), etc.

But this is not required nor is it even desirable. One might attribute p₁ = p₂ = 1/2 by symmetry to a perfectly milled coin, without performing the experiment. Tossing it N times should either confirm the hypothesis or show up a hidden flaw. Similarly one can predict the width of the binary distribution from theory alone, without performing the experiment.

Thus it becomes quite compelling to understand the consequences of a probability distribution at arbitrary values of the parameters. Experiment can then be used not just to determine the numerical values of the parameters but also to detect systematic deviations from the supposed randomness.

These are just some of the good reasons not to insist on p₁ + p₂ = 1 at first. By allowing the generating function to depend on two independent parameters p₁ and p₂ it becomes possible ...

Table of contents

• Cover Page
• Title Page
• Copyright Page
• Contents
• Preface to Second Edition
• Preface to First Edition
• Introduction: Theories of Thermodynamics, Kinetic Theory and Statistical Mechanics
• Chapter 1: Elementary Concepts in Statistics and Probability
• Chapter 2: The Ising Model and the Lattice Gas
• Chapter 3: Elements of Thermodynamics
• Chapter 4: Statistical Mechanics
• Chapter 5: The World of Bosons
• Chapter 6: All About Fermions: Theories of Metals, Superconductors, Semiconductors
• Chapter 7: Kinetic Theory
• Chapter 8: The Transfer Matrix
• Chapter 9: Monte Carlo and Other Computer Simulation Methods
• Chapter 10: Critical Phenomena and the Renormalization Group
• Chapter 11: Some Uses of Quantum Field Theory in Statistical Physics
• Appendix A: Solutions to Selected Problems
• Bibliography
• Index