Table of contents for Statistical physics : an introduction / Daijiro Yoshioka.


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Part I General Principles
1   Thermal Equilibrium and the Principle
of Equal Probability .................................... 3
1.1 Introduction to Thermal and Statistical Physics ............. 3
1.2  Thermal Equilibrium  ....................................  4
1.2.1 Description of a System in Equilibrium ............... 4
1.2.2 State Variables, Work, and Heat .................... 5
1.2.3 Temperature and the Zeroth Law of Thermodynamics.. 7
1.2.4 Heat Capacity and Specific Heat .............. ..... 8
1.3 Kinetic Theory of Gas Molecules ................ ......... 9
1.3.1 The Spatial Distribution of Gas Molecules ............. 10
1.3.2 Velocity Distribution of an Ideal Gas ............... 15
1.3.3  The Pressure of a Gas  ............................  18
1.4 The Principle of Equal Probability ....... ...............  20
2   Entropy ................................................ 23
2.1  The Microcanonical Distribution  ..........................  23
2.2 Number of States and Density of States .................... 26
2.3 Conditions for Thermal Equilibrium ... ................... 28
2.3.1 Equilibrium Condition
when only Energy is Exchanged ..... . .............. 28
2.3.2 Equilibrium Condition when Molecules are Exchanged . 30
2.3.3 Equilibrium Condition when Two Systems
Share a Common Volume........................... 31
2.4 Thermal Nonequilibrium and Irreversible Processes ........... 32
3   The Partition Function and the Free Energy ............. 35
3.1 A System in a Heat Bath .............................. 35
3.1.1  Canonical Distribution  ................... ........  36
3.1.2  Application  to a Molecule in  Gas ....................  37
3.2 Partition Function ....................................... 38
3.3  Free Energy  ............... ..... ....................  39
3.4  Internal Energy  ........................................ .  41
3.5 Thermodynamic Functions
and  Legendre Transformations  ............................  42
3.6 Maxwell Relations ...................................... 43
Part II Elementary Applications
4   Ideal G ases  ........................................ ....... .  47
4.1 Quantum Mechanics of a Gas Molecule .................... 47
4.2 Phase Space and the Number of Microscopic States .......... 49
4.3 Entropy of an Ideal Gas ................................. 51
4.4 Pressure of an Ideal Gas:
Quantum  Mechanical Treatment  .........................  54
4.5 Statistical-Mechanical Temperature and Pressure ............ 55
4.6  Partition  Function  of an  Ideal Gas ........................  56
4.7  Diatomic  M olecules .................. ...................  58
4.7.1 Decomposition of the Partition Function ............. 58
4.7.2  Center-of-Gravity  Part: Z(CG)  ......................  60
4.7.3  Vibrational Part: Z(V)  ............................  61
4.7.4  Rotational Part: Z ) .............................  64
5   The Heat Capacity of a Solid,
and Black-Body Radiation ................................ 67
5.1 Heat Capacity of a Solid I - Einstein Model ................ 67
5.2 Heat Capacity of a Solid II - Debye Model ................. 70
5.2.1 Collective Oscillations of the Lattice
and the Internal Energy ................... ....... 70
5.2.2 Heat Capacity at High Temperature ................. 73
5.2.3 Heat Capacity at Low Temperature .................. 74
5.2.4 Heat Capacity at Intermediate Temperature .......... 74
5.2.5 Physical Explanation for the Temperature Dependence. 75
5.3 Black-Body Radiation ................................... 76
5.3.1  W ien's Law  and  Stefan's Law  .......................  76
5.3.2  Energy of Radiation in a Cavity .....................  77
5.3.3 Spectrum of Light Emitted from a Hole .............. 78
5.3.4 The Temperature of the Universe.................... 80
6   The  Elasticity  of Rubber  ..................................  83
6.1  Characteristics of Rubber  ................... .............  83
6.2 Model of Rubber ........................................ 84
6.3 Entropy of Rubber ..................................... 85
6.4  Hooke's  Law  ............................................  86
7   Magnetic Materials ....................................... 89
7.1 Origin of Permanent Magnetism ......................... 89
7.2 Statistical Mechanics of a Free Spin System ................. 91
7.2.1  M odel and  Entropy  ..............................  91
7.2.2 Free Energy, Magnetization, and Susceptibility ........ 93
7.2.3 Internal Energy and Heat Capacity .................. 95
7.3 Ising Model - Mean-Field Approximation ................... 97
7.3.1 Links ................ ......................... 97
7.3.2  Mean-Field  Approximation  .........................  99
7.3.3 Solution of the Self-Consistent Equation.............. 100
7.3.4  Entropy  and  Heat Capacity .........................103
7.3.5 Susceptibility ..................................105
7.3.6  Domain Structure .............................. 106
7.4 The One-Dimensional Ising Model ........................ 106
7.4.1  Free Energy  .................................... . 106
7.4.2  Entropy and  Heat Capacity .........................108
7.4.3 Magnetization and Susceptibility ............. ..... 110
Part III More Advanced Topics
8   First-Order Phase  Transitions  ............................ 115
8.1 The Various Phases of Matter........................... 115
8.2 System in a Heat Bath at Fixed P and T .................. 119
8.3 Coexistence of Phases ................................. 121
8.4  The Clausius-Clapeyron Law  ........................... 123
8.5 The Critical Point .................. ................... 126
8.6  The  van  der W aals Gas  ................................ 128
8.6.1 Coexistence of Gas and Liquid ...................... 130
9   Second-Order Phase Transitions .........................133
9.1 Various Phase Transitions and Order Parameters ............ 133
9.2 Landau Theory ........................................ 134
9.2.1 Free Energy ..................................... 137
9.2.2 Entropy, Internal Energy, and Heat Capacity .......... 138
9.2.3 Critical Phenomena ............................... 139
9.3  The Two-Dimensional Ising Model .........................140
10 Dense Gases - Ideal Gases at Low Temperature ........... 147
10.1 The Phase Space for N Identical Particles .................. 147
10.2 The Grand Canonical Distribution.........................149
10.3 Ideal Fermi Gases and Ideal Bose Gases .................... 151
10.3.1 Occupation Number Representation ................. 151
10.3.2  Thermodynamic Functions  ...................... .. 154
10.4  Properties of a Free-Fermion  Gas .......................... 154
10.4.1  Properties at T  =  0 .............................  157
10.4.2 Properties at Low Temperature ................... . 160
10.5  Properties of a Free-Boson  Gas ...........................  169
10.5.1  The Two Kinds of Bose Gas .......................  169
10.5.2 Properties at Low Temperature ................... . 170
10.6 Properties of Gases at High Temperature ................... 178
Part IV Appendices
A   Formulas Related to the Factorial Function ................ 185
A.1 Binomial Coefficients and Binomial Theorem ................ 185
A.2 Stirling's Formula ....................................185
A.3  n!! ................... ............................. ..... 186
B   The Gaussian Distribution Function ......................187
B.1  The Central Limit Theorem  .............................  187
B.1.1 Example .... .................................... 188
B.2 Gaussian Integrals ..................................... 188
B.3 The Fourier Transform of a Gaussian Distribution Function ... 189
C   Lagrange's Method
of Undetermined Multipliers ............................. 191
C.1 Example .............................................. 192
C.2 Generalization ......................................... 192
D   Volume of a Hypersphere ..................................193
E   Hyperbolic Functions ..................................... 195
F   Boundary Conditions ..................................... 197
F.1  Fixed  Boundary  Condition  ..............................  197
F.2  Periodic Boundary  Condition  .............................198
G   The Riemann Zeta Function ..............................201
References ................................................... . 203



Library of Congress subject headings for this publication: Statistical physics