Advanced statistical mechanics

Contents......Page 10
PART I: GENERAL THEORY......Page 18
1.1 Thermodynamics......Page 20
1.2 Statistical mechanics......Page 26
1.3 Quantum statistical mechanics......Page 34
1.4 Quantum field theory......Page 36
References......Page 38
2.1 Reductionism......Page 39
2.2 Phenomena......Page 41
2.3 Models......Page 50
2.4 Discussion......Page 58
2.5 Appendix: Bravais lattices......Page 59
References......Page 61
3 Stability, existence and uniqueness......Page 62
3.1 Classical stability......Page 66
3.2 Quantum stability......Page 78
3.3 Existence and uniqueness of the thermodynamic limit......Page 83
3.4 First order phase transitions, zeros and analyticity......Page 97
3.5 Discussion......Page 99
3.6 Open questions......Page 101
3.7 Appendix A: Properties of functions of positive type......Page 102
3.8 Appendix B: Fourier transforms......Page 103
References......Page 107
4 Theorems on order......Page 109
4.1 Densest packing of hard spheres and ellipsoids......Page 110
4.2 Lack of order in the isotropic Heisenberg model in D = 1, 2......Page 114
4.3 Lack of crystalline order in D = 1, 2......Page 120
4.4 Existence of ferromagnetic and antiferromagnetic order in the classical Heisenberg model (n vector model) in D = 3......Page 127
4.5 Existence of antiferromagnetic order in the quantum Heisenberg model for T > 0 and D = 3......Page 135
4.7 Missing theorems......Page 137
References......Page 139
5 Critical phenomena and scaling theory......Page 141
5.1 Thermodynamic critical exponents and inequalities for Ising-like systems......Page 142
5.2 Scaling theory for Ising-like systems......Page 145
5.3 Scaling for general systems......Page 153
5.4 Universality......Page 159
5.5 Missing theorems......Page 160
References......Page 162
PART II: SERIES AND NUMERICAL METHODS......Page 164
6 Mayer virial expansions and Groeneveld\'s theorems......Page 166
6.1 The second virial coefficient......Page 173
6.2 Mayers\' first theorem......Page 175
6.3 Mayers\' second theorem......Page 177
6.4 Non-negative potentials and Groeneveld\'s theorems......Page 184
6.5 Convergence of virial expansions......Page 190
6.6 Counting of Mayer graphs......Page 193
6.7 Appendix: The irreducible Mayer graphs of four and five points......Page 195
References......Page 197
7 Ree–Hoover virial expansion and hard particles......Page 198
7.1 The Ree–Hoover expansion......Page 199
7.2 The Tonks Gas......Page 203
7.3 Hard sphere virial coefficients B[sub(2)]–B[sub(4)] in two and higher dimensions......Page 206
7.4 Monte-Carlo evaluations of B[sub(5)]–B[sub(10)]......Page 212
7.5 Hard sphere virial coefficients for k > &#;8805 11......Page 213
7.6 Radius of convergence and approximate equations of state......Page 215
7.7 Parallel hard squares, parallel hard cubes and hard hexagons on a lattice......Page 219
7.8 Convex nonspherical hard particles......Page 221
7.9 Open questions......Page 222
References......Page 225
8 High density expansions......Page 227
8.1 Molecular dynamics......Page 228
8.2 Hard spheres and discs......Page 229
8.3 The inverse power law potential......Page 239
8.4 Hard spheres with an additional square well......Page 242
8.5 Lennard-Jones potentials......Page 244
8.6 Conclusions......Page 245
References......Page 247
9 High temperature expansions for magnets at H = 0......Page 249
9.1 Classical n vector model for D = 2, 3......Page 251
9.2 Quantum Heisenberg model......Page 272
9.3 Discussion......Page 278
9.4 Statistical mechanics versus quantum field theory......Page 282
9.5 Appendix: The expansion coefficients for the susceptibility on the square lattice......Page 284
References......Page 289
PART III: EXACTLY SOLVABLE MODELS......Page 292
10 The Ising model in two dimensions: summary of results......Page 294
10.1 The homogeneous lattice at H = 0......Page 297
10.2 Boundary properties of the homogeneous lattice at H = 0......Page 326
10.3 The layered random lattice......Page 333
10.4 The Ising model for H ≠ 0......Page 336
References......Page 341
11 The Pfaffian solution of the Ising model......Page 345
11.1 Dimers......Page 346
11.2 The Ising partition function......Page 364
11.3 Correlation functions......Page 372
References......Page 378
12 Ising model spontaneous magnetization and form factors......Page 380
12.1 Wiener–Hopf sum equations......Page 381
12.2 Spontaneous magnetization and Szegö\'s theorem......Page 385
12.3 Form factor expansions of C(N, N) and C(0, N)......Page 392
12.4 Asymptotic expansions of C(N, N) and C(0, N) for N → ∞......Page 409
12.5 Evaluation of diagonal form factor integrals......Page 415
References......Page 424
13.1 Historical overview......Page 425
13.2 Transfer matrices......Page 429
13.3 Integrability......Page 434
13.4 Star–triangle equation for vertex models......Page 435
13.5 Star–triangle equation for spin models......Page 457
13.6 Star–triangle equation for face models......Page 469
13.7 Hamiltonian limits......Page 481
13.8 Appendix: Properties of theta functions......Page 489
References......Page 494
14 The eight-vertex and XYZ model......Page 497
14.1 Historical overview......Page 498
14.2 The matrix TQ equation for the eight-vertex model......Page 501
14.3 Eigenvalues and free energy......Page 531
14.4 Excitations, order parameters and correlation functions of the eight- and six-vertex model......Page 554
14.5 Appendix: Properties of the modified theta functions......Page 569
References......Page 574
15.1 The hard hexagon and RSOS models......Page 579
15.2 The chiral Potts model......Page 592
15.3 Open questions......Page 617
References......Page 622
PART IV: CONCLUSION......Page 628
16.1 Does history matter?......Page 630
16.2 Size is important......Page 632
16.3 The paradox of integrability......Page 633
16.4 Conclusion......Page 634
References......Page 635
D......Page 636
H......Page 637
I......Page 638
Q......Page 639
T......Page 640
Z......Page 641