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Introductory Statistical Mechanics
Article in European Journal of Physics · March 2000
DOI: 10.1088/0143-0807/21/2/701
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Introductory Statistical
Mechanics
Second Edition
ROGER BOWLEY
Department of Physics, University of Nottingham
and
MARIANA/SANCHEZ
CLARENDON PRESS • OXFORD
Contents
The first law of thermodynamics 1
1.1 Fundamental definitions 1
1.2 Thermometers 4
1.3 Different aspects of equilibrium 7
1.3.1 Mechanical equilibrium 7
1.3.2 Thermal equilibrium 7
1.3.3 Chemical equilibrium 8
1.4 Functions of state 9
1.5 Internal energy 13
1.6 Reversible changes /' 16
1.7 Enthalpy 19
1.8 Heat capacities 20
1.9 Reversible adiabatic changes in an ideal gas 22
1.10 Problems 23
Entropy and the second law of thermodynamics 25
2.1 A first look at the entropy 25
2.2 The second law of thermodynamics 27
2.3 The Carnot cycle 28
2.4 The equivalence of the absolute and the perfect gas scale
of temperature 33
2.5 Definition of entropy 34
2.6 Measuring the entropy 38
2.7 The law of increase of entropy 42
2.8 Calculations of the increase in the entropy in irreversible
processes 44
2.8.1 Two systems at different temperatures thermalize 44
2.8.2 Extending a spring 45
2.8.3 Expanding a gas into a vacuum 46
2.9 The approach to equilibrium 46
2.10 Questions left unanswered 48
2.11 Problems 48
Probability and statistics 52
3.1 Ideas about probability 52
3.2 Classical probability 53
3.3 Statistical probability 54
3.4 The axioms of probability theory 56
xii Contents
3.5 Independent events
3.6 Counting the number of events
3.6.1 Arrangements
3.6.2 Permutations of r objects from n
3.6.3 Combinations of r objects from n
3.7 Statistics and distributions
3.8 Problems
The ideas of statistical mechanics
4.1 Introduction
4.2 Definition of the quantum state of the system
4.3 A simple model of spins on lattice sites
4.4 Equations of state
4.4.1 Spin system
4.4.2 Vacancies in a crystal
4.4.3 A model for a rubber band
4.5 The second law of thermodynamics
4.6 Logical positivism
4.7 Problems /'
The canonical ensemble
5.1 A system in contact with a heat bath
5.2 The partition function
5.3 Definition of the entropy in the canonical ensemble
5.4 The bridge to thermodynamics through Z
5.5 The condition for thermal equilibrium
5.6 Thermodynamic quantities from ln(Z)
5.7 Two-level system
5.8 Single particle in a one-dimensional box
5.9 Single particle in a three-dimensional box
5.10 Expressions for heat and work
5.11 Rotational energy levels for diatomic molecules
5.12 Vibrational energy levels for diatomic molecules
5.13 Factorizing the partition function
5.14 Equipartition theorem
5.15 Minimizing the free energy
5.15.1 Minimizing the Helmholtz free energy
5.15.2 Minimizing the Gibbs free energy
5.16 Problems
Identical particles
6.1 Identical particles
6.2 Symmetric and antisymmetric wavefunctions
6.3 Bose particles or bosons
6.4 Fermi particles or fermions
6.5 Calculating the partition function for identical particles
Contents xiii
6.5.1 Bosons 134
6.5.2 Fermions 134
6.5.3 A large number of energy levels 135
6.6 Spin . 138
6.7 Identical particles localized on lattice sites 139
6.8 Identical particles in a molecule 140
6.9 Problems 142
7 Maxwell distribution of molecular speeds 144
7.1 The probability that a particle is in a quantum state 144
7.2 Density of states hi k space 146
7.3 Single-particle density of states in energy 150
7.4 The distribution of speeds of particles in a classical gas 151
7.5 Molecular beams 154
7.6 Problems 158
8 Planck's distribution 160
8.1 Black-body radiation 160
8.2 The Rayleigh-Jeans theory .t ' 165
;
8.3 Planck's distribution 167
8.4 Waves as particles 170
8.5 Derivation of the Planck distribution 172
8.6 The free energy 175
8.7 Einstein's model of vibrations in a solid * 176
8.8 Debye's model of vibrations in a solid 178
3.9 Solid and vapour in equilibrium 182
8.10 Cosmic background radiation 183
8.11 Problems 185
9 Systems with variable numbers of particles 188
9.1 Systems with variable number of particles 188
9.2 The condition for chemical equilibrium 191
9.3 The approach to chemical equilibrium 193
9.4 The chemical potential 193
9.4.1 Method of measuring y, 193
9.4.2 Methods of calculating \i 195
9.5 Reactions 198
9.6 External chemical potential 201
9.7 The grand canonical ensemble 202
9.8 Absorption of atoms on surface sites 205
9.9 The grand potential 205
9.10 Problems 207
10 Fermi and Bose particles • 210
10.1 Introduction 210
10.2 The statistical mechanics of identical particles 212
xiw Contents
•
10.2.1 Fermi particle
10.2.2 Bose particle
10.3 The thermodynamic properties of a Fermi gas
10.3.1 High-temperature region
10.3.2 Properties at the absolute zero
10.3.3 Thermal properties of a Fermi gas at low temper-
atures
10.4 Examples of Fermi systems
10.4.1 Dilute 3He solutions in superfiuid 4He
10.4.2 Electrons in metals
10.4.3 Electrons in stars
10.4.4 Electrons in white dwarf stars
10.5 A non-interacting Bose gas
10.6 Problems
11 Phase transitions
11.1 Phases
11.2 Thermodynamic potential
11.3 Approximation /'
11.4 First-order phase transition
11.5 Clapeyron equation
11.6 Phase separation
11.7 Phase separation in mixtures
11.8 Liquid-gas system
11.9 Problems
12 Continuous phase transitions
12.1 Introduction
12.2 Ising model
12.2.1 Mean field theory
12.3 Order parameter
12.4 Landau theory
12.4.1 Case I: b > 0, second-order transition
12.4.2 Case II: b < 0, first-order transition
12.5 Symmetry-breaking field
12.6 Critical exponents
12.7 Problems
13 Ginzburg-Landau theory
13.1 Ginzburg-Landau theory
13.2 Ginzburg criterion
13.3 Surface tension
13.4 Nucleation of droplets
13.5 Superfluidity
13.6 Order parameter
13.7 Circulation
Contents xv
13.8 Vortices 284
13.9 Charged superfluids 285
13.10 Quantization of flux * 286
13.11 Problems 287
A Thermodynamics in a magnetic field 289
B Useful integrals 292
C The quantum treatment of a diatomic molecule 296
'i
D Travelling waves 299
D.I Travelling waves in one dimension 299
D.2 Travelling waves in three dimensions 300
E Partial differentials and thermodynamics 302
E.I Mathematical relations 302
E.2 Maxwell relations 303
E.3 TdS relations 304
E.4 Extensive quantities 305
/
F Van der Waals equation 307
G Answers to problems 312
H Physical constants 341
Bibliography 342
Index 347
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