Basics of Statistical Physics

Harald J W Müller-Kirsten

• Publisher: WSPC
• Year: 2013
• Print ISBN: 9789814449533
• eBook ISBN: 9789814449557
• Edition: 2
• Topic: Physical Sciences
• Subtopic: Mathematical & Computational Physics

Statistics links microscopic and macroscopic phenomena, and requires for this reason a large number of microscopic elements like atoms. The results are values of maximum probability or of averaging.

This introduction to statistical physics concentrates on the basic principles, and attempts to explain these in simple terms supplemented by numerous examples. These basic principles include the difference between classical and quantum statistics, a priori probabilities as related to degeneracies, the vital aspect of indistinguishability as compared with distinguishability in classical physics, the differences between conserved and non-conserved elements, the different ways of counting arrangements in the three statistics (Maxwell–Boltzmann, Fermi–Dirac, Bose–Einstein), the difference between maximization of the number of arrangements of elements, and averaging in the Darwin–Fowler method.

Significant applications to solids, radiation and electrons in metals are treated in separate chapters, as well as Bose–Einstein condensation. This revised second edition contains an additional chapter on the Boltzmann transport equation along with appropriate applications. Also, more examples have been added throughout, as well as further references to literature.

Contents:

• Introduction
• Statistical Mechanics of an Ideal Gas (Maxwell)
• The A Priori Probability
• Classical Statistics (Maxwell–Boltzmann)
• Entropy
• Quantum Statistics
• Exact Form of Distribution Functions
• Application to Radiation (Light Quanta)
• Debye Theory of Specific Heat of Solids
• Electrons in Metals
• Limitations of the Preceding Theory — Improvement with Ensemble Method
• Averaging instead of Maximization, and Bose–Einstein Condensation
• The Boltzmann Transport Equation

Readership: Advanced undergraduates, graduate students and academics interested in statistical physics.

Chapter 1

Introduction

1.1 Introductory Remarks

Statistical physics embraces in particular statistical mechanics and statistical thermodynamics. The word statistical implies already that the subject deals with a large number of elements like the number of atoms or molecules in a macroscopic body. A theoretical treatment of statistical physics thus attempts to describe macroscopic phenomena in terms of microscopic processes. Elementary microscopic processes are obviously those of atoms and molecules. Statistical thermodynamics considers primarily microscopic processes in some enclosure like a box which imposes stringent boundary conditions on the dynamics of the particles. Consequently in a first approach to the subject one considers cases in which the interaction between individual atoms or molecules is of secondary importance, as in the case of a dilute gas in some container. The realization that the motion of atoms, i.e. their kinetic energy, is related to the macroscopically observed temperature was a considerable step forward in our understanding of the relationship between atomic physics and classical thermodynamics. Thus a perfect gas suggests itself naturally as a first object to consider, and then the question whether other cases, e.g. conduction electrons in a metal, can be considered similarly. And how are solids to be treated in this context? The naive picture of a solid as a lattice with atoms located at lattice sites suggests to consider these in analogy with harmonic oscillators, for instance, one oscillator at every lattice site. The simple one-dimensional harmonic oscillator serves in view of its mathematical simplicity in many areas of physics as a convenient first modelling example. Thus in a first attempt it is an obvious idea to abstract the atoms at lattice sites to such harmonic oscillators whose oscillations describe the vibrations of an atom or molecule. The harmonic oscillator played a vital role in the development of quantized statistics as conceived by Planck: It was Planck’s idea of considering the simple harmonic oscillator as statistically equivalent to a normal mode of vibration which led him to discretized energies. Thus both free particles and oscillators play a dominant role in our introduction to the subject here. However, the quantized simple one-dimensional harmonic oscillator has one limitation: Its eigenvalues are nondegenerate. Since degeneracy will be seen to be a characteristic of many particle states of statistical physics it is expedient to consider oscillators also in higher dimensions, for instance, in a model of solids. Our presentation here begins with elementary kinetic theory. We then introduce the concept of a priori probability and show that this can be identified with the degeneracy of states. In classical Maxwell–Boltzmann statistics we consider the number of arrangements W of particles among states of various degeneracies and then determine that particular arrangement which appears with maximum probability. We proceed similarly with quantum statistics, taking into account the indistinguishability of elements, the number of elements permissible per state, and whether the elements are conserved or nonconserved, and thus arrive at Bose–Einstein and Fermi–Dirac statistics. In the last chapter we consider the Darwin–Fowler method of mean values and observe that the more rigorously derived results of this method are the same as those obtained with maximization. However, before we consider quantum statistics we introduce the concept of entropy S as defined by Boltzmann by the relation S = k lnW, ...

Table of contents

• Cover Page
• Half Title Page
• Front Page
• Title Page
• Copyright Page
• Contents
• Preface to Second Edition
• Preface to First Edition
• 1 Introduction
• 2 Statistical Mechanics of an Ideal Gas (Maxwell)
• 3 The a priori Probability
• 4 Classical Statistics (Maxwell–Boltzmann)
• 5 Entropy
• 6 Quantum Statistics
• 7 Exact Form of Distribution Functions
• 8 Application to Radiation (Light Quanta)
• 9 Debye Theory of Specific Heat of Solids
• 10 Electrons in Metals
• 11 Limitations of the Preceding Theory — Improvement with Ensemble Method
• 12 Averaging instead of Maximization, and Bose–Einstein Condensation
• 13 The Boltzmann Transport Equation
• Bibliography
• Index