This text by one of the originators of the cluster variation method of statistical mechanics is aimed at second- and third-year graduate students studying such topics as the theory of complex analysis, classical mechanics, classical electrodynamics, and quantum mechanics. The central theme is that, given the Hamiltonian for a system, it is possible to calculate the thermodynamics correlation function, either numerically or through the use of infinite series. The book is self-contained, with all required mathematics included either in the text or in Appendixes. The text includes many exercises designed for self-study.
Author(s): Tomoyasu Tanaka
Edition: First Edition
Year: 2002
Language: English
Pages: 312
Cover......Page 1
Half-title......Page 3
Title......Page 5
Copyright......Page 6
Dedication......Page 7
Contents......Page 9
Preface......Page 13
Acknowledgements......Page 17
1.2 The zeroth law of thermodynamics......Page 19
1.3 The thermal equation of state......Page 20
1.4 The classical ideal gas......Page 22
1.6 The first law of thermodynamics......Page 25
1.7 The heat capacity......Page 26
1.8 The isothermal and adiabatic processes......Page 28
1.10 The second law of thermodynamics......Page 30
1.11 The Carnot cycle......Page 32
1.12 The thermodynamic temperature......Page 33
1.13 The Carnot cycle of an ideal gas......Page 37
1.14 The Clausius inequality......Page 40
1.15 The entropy......Page 42
1.16 General integrating factors......Page 44
1.17 The integrating factor and cyclic processes......Page 46
Process (ii) Adiabatic for each subsystem individually......Page 48
1.19 Employment of the second law of thermodynamics......Page 49
1.20 The universal integrating factor......Page 50
2.1 Thermodynamic potentials......Page 56
2.2 Maxwell relations......Page 59
2.3 The open system......Page 60
2.4 The Clausius–Clapeyron equation......Page 62
2.5 The van der Waals equation......Page 64
2.6 The grand potential......Page 66
3.1 Microstate and macrostate......Page 68
3.3 The number of microstates......Page 70
3.4 The most probable distribution......Page 71
3.5 The Gibbs paradox......Page 73
3.6 Resolution of the Gibbs paradox: quantum ideal gases......Page 74
3.7 Canonical ensemble......Page 76
3.8 Thermodynamic relations......Page 79
3.10 The grand canonical distribution......Page 81
3.11 The grand partition function......Page 82
3.12 The ideal quantum gases......Page 84
4.1.1 Coordinate representation......Page 87
4.1.2 The momentum representation......Page 91
4.1.3 The eigenrepresentation......Page 92
4.2 The unitary transformation......Page 94
4.3 Representations of operators......Page 95
4.4 Number representation for the harmonic oscillator......Page 96
4.5 Coupled oscillators: the linear chain......Page 100
4.6 The second quantization for bosons......Page 102
4.7 The system of interacting fermions......Page 106
4.8.1 The Fermi hole......Page 109
4.8.2 The hydrogen molecule......Page 110
4.9 The Heisenberg exchange Hamiltonian......Page 112
4.10 The electron–phonon interaction in metal......Page 113
4.11 The dilute Bose gas......Page 117
4.12 The spin-wave Hamiltonian......Page 119
5.1 The canonical partition function......Page 124
5.2 The trace invariance......Page 125
5.3 The perturbation expansion......Page 126
5.4 Reduced density matrices......Page 128
5.5.1 One-site density matrix......Page 129
5.5.2 Two-site density matrix......Page 131
5.6 The four-site reduced density matrix......Page 132
5.6.1 The reduced density matrix for a square cluster......Page 135
5.7 The probability distribution functions for the Ising model......Page 139
5.7.2 The two-site distribution function......Page 140
5.7.4 The four-site (square) distribution function......Page 141
5.7.6 The six-site (regular octahedron) distribution function [Exercise 5.7]......Page 142
6.1 The variational principle......Page 145
6.2 The cumulant expansion......Page 146
6.3 The cluster variation method......Page 148
6.4 The mean-field approximation......Page 149
6.5 The Bethe approximation......Page 152
6.6 Four-site approximation......Page 155
6.7 Simplified cluster variation methods......Page 159
6.8.1 One-site density matrix......Page 162
6.9 The point and pair approximations in the CFF......Page 163
6.10 The tetrahedron approximation in the CFF......Page 165
7.1 Singularity of the correlation functions......Page 171
7.2.1 The mean-field approximation......Page 172
7.2.2 The pair approximation......Page 173
7.3 An infinite-series representation of the partition function......Page 174
7.4 The method of Padé approximants......Page 176
7.5.1 Mean-field approximation......Page 179
7.5.3 Tetrahedron approximation......Page 180
7.5.4 Tetrahedron-plus-octahedron approximation......Page 181
7.6 High temperature specific heat......Page 183
7.6.3 Tetrahedron-plus-octahedron approximation......Page 184
7.7.1 Mean-field approximation......Page 185
7.7.3 Tetrahedron approximation......Page 186
7.8 Low temperature specific heat......Page 187
7.8.2 Pair approximation......Page 188
7.8.4 Tetrahedron-plus-octahedron approximation......Page 189
7.9 Infinite series for other correlation functions......Page 190
8.1 The Wentzel criterion......Page 193
8.2 The BCS Hamiltonian......Page 196
8.3 The s–d interaction......Page 202
8.4 The ground state of the Anderson model......Page 208
8.5 The Hubbard model......Page 215
8.6 The first-order transition in cubic ice......Page 221
9.1 The basic generating equations......Page 230
9.2 Linear identities for odd-number correlations......Page 231
9.3 Star-triangle-type relationships......Page 234
9.4 Exact solution on the triangular lattice......Page 236
9.6 Systematic naming of correlation functions on the lattice......Page 239
Site-number representation......Page 241
Bond representation......Page 242
Vertex-number representation......Page 243
9.6.2 Is the vertex-number representation an over-characterization?......Page 244
9.6.3 Computer evaluation of the correlation functions......Page 245
10.1 The radial distribution function......Page 248
10.2 Lattice structure of the superionic conductor AlphaAgI......Page 250
10.3 The mean-field approximation......Page 252
10.4 The pair approximation......Page 253
10.5 Higher order correlation functions......Page 255
10.5.1 AgI......Page 256
10.5.2 Ag2S......Page 257
10.6 Oscillatory behavior of the radial distribution function......Page 258
10.7 Summary......Page 262
11.1 The high temperature series expansion of the partition function......Page 264
11.2.1 Lattice terminals......Page 266
11.2.2 The Pfaffian......Page 268
11.3 Exact partition function......Page 271
11.4 Critical exponents......Page 277
A1.1 The Stirling formula......Page 279
A1.2 Surface area of the N-dimensional sphere......Page 281
Appendix 2 The critical exponent in the tetrahedron approximation......Page 283
A3.1 Characteristic matrices......Page 287
A3.2 Properties of characteristic matrices......Page 290
A3.3 Susceptibility determinants......Page 291
Appendix 4 A unitary transformation applied to the Hubbard Hamiltonian......Page 296
A5.1.2 Odd correlation functions......Page 299
A5.2 Some of the Ising identities for the odd correlation functions......Page 301
References......Page 303
Bibliography......Page 307
Index......Page 309
