Advanced statistical mechanics

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1. Verfasser: McCoy, Barry M. (VerfasserIn)
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Sprache:English
Veröffentlicht: Oxford [u.a.] Oxford Univ. Press 2010
Ausgabe:1. publ.
Schriftenreihe:International series of monographs on physics 146
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Datensatz im Suchindex

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adam_text Contents PART I GENERAL THEORY Basic principles 3 1.1 Thermodynamics 3 1.1.1 Macroscopic, extensive and intensive 3 1.1.2 Equilibrium 5 1.1.3 The four laws of thermodynamics 6 1.2 Statistical mechanics 9 1.2.1 Statistical philosophy 9 1.2.2 The microcanonical ensemble 10 1.2.3 The canonical ensemble 11 1.2.4 The grand canonical ensemble 15 1.2.5 Phases and ergodic components 17 1.3 Quantum statistical mechanics 17 1.3.1 The relation of classical to quantum statistical mechanics 18 1.4 Quantum field theory 19 References 21 Reductionism, phenomena and models 22 2.1 Reductionism 22 2.2 Phenomena 24 2.2.1 Monatomic insulators 24 2.2.2 Diatomic insulators 25 2.2.3 Liquid crystals 28 2.2.4 Water 28 2.2.5 Metals 29 2.2.6 Helium 29 2.2.7 Magnetic transitions 30 2.3 Models 33 2.3.1 Continuum models 34 2.3.2 Lattice models 37 2.4 Discussion 41 2.5 Appendix: Bravais lattices 42 References 44 Stability, existence and uniqueness 45 3.1 Classical stability 49 3.1.1 Catastrophic potentials 49 3.1.2 Conditions for stability 49 3.1.3 Superstability 57 χ Contents 3.1.4 Multispecies interactions 59 3.2 Quantum stability 61 3.2.1 Stability of matter 61 3.2.2 Proofs of theorems 1 and 2 63 3.3 Existence and uniqueness of the thermodynamic limit 66 3.3.1 Box boundary conditions 67 3.3.2 Periodic boundary conditions 69 3.3.3 Existence and uniqueness in the canonical ensemble 69 3.3.4 Existence and uniqueness in the grand canonical ensemble 77 3.3.5 Continuity of the pressure 78 3.4 First order phase transitions, zeros and analyticity 80 3.5 Discussion 82 3.6 Open questions 84 3.7 Appendix A: Properties of functions of positive type 85 3.8 Appendix B: Fourier transforms 86 References 90 4 Theorems on order 92 4.1 Densest packing of hard spheres and ellipsoids 93 4.2 Lack of order in the isotropie Heisenberg model in D = 1, 2 97 4.3 Lack of crystalline order in D = 1, 2 103 4.4 Existence of ferromagnetic and antiferromagnetic order in the classical Heisenberg model (n vector model) in D = 3 110 4.4.1 The mechanism for ferromagnetic order 111 4.4.2 Proof of the bound (4.123) 113 4.4.3 Antiferromagnetism 117 4.5 Existence of antiferromagnetic order in the quantum Heisenberg model for Τ > 0 and D = 3 118 4.6 Existence of antiferromagnetic order in the quantum Heisenberg model for Τ = 0 and D = 2 120 4.7 Missing theorems 120 References 122 5 Critical phenomena and scaling theory 124 5.1 Thermodynamic critical exponents and inequalities for Ising-like systems 125 5.2 Scalingjiheory for Ising-like systems 128 5.2.1 Scaling for Я = 0 129 5.2.2 Scaling for Η φ 0 132 5.2.3 Summary of critical exponent equalities 136 5.3 Scaling for general systems 136 5.3.1 The classical η vector and quantum Heisenberg models 137 5.3.2 Lennard-Jones fluids 142 5.4 Universality 142 5.5 Missing theorems 143 References 145 Contents xi PART II SERIES AND NUMERICAL METHODS Mayer virial expansions and Groeneveld s theorems 149 6.1 The second virial coefficient 156 6.2 Mayers first theorem 15g 6.3 Mayers second theorem 160 6.3.1 Step 1 160 6.3.2 Step 2 162 6.3.3 Step 3 164 6.4 Non-negative potentials and Groeneveld s theorems 167 6.5 Convergence of virial expansions 173 6.6 Counting of Mayer graphs 176 6.7 Appendix: The irreducible Mayer graphs of four and five points 178 References 180 Ree— Hoover virial expansion and hard particles 181 7.1 The Ree-Hoover expansion 182 7.2 The Tonks Gas 186 7.3 Hard sphere virial coefficients B2-B4 in two and higher dimen¬ sions 189 7.3.1 Evaluation of B2 189 7.3.2 Evaluation of B3 191 7.3.3 Evaluation of B4 194 7.4 Monte-Carlo evaluations of B¿-B q 195 7.5 Hard sphere virial coefficients for к > 11 196 7.6 Radius of convergence and approximate equations of state 198 7.7 Parallel hard squares, parallel hard cubes and hard hexagons on a lattice 202 7.8 Convex nonspherical hard particles 204 7.9 Open questions 205 References 208 High density expansions 210 8.1 Molecular dynamics 211 8.2 Hard spheres and discs 212 8.2.1 Behavior near close packing 213 8.2.2 Freezing of hard spheres 214 8.2.3 The phase transition for hard discs 219 8.3 The inverse power law potential 222 8.3.1 Scaling behavior 223 8.3.2 Numerical computations 224 8.4 Hard spheres with an additional square well 225 8.5 Lennard-Jones potentials 227 8.6 Conclusions 228 References 230 xü Contents 9 High temperature expansions for magnets at H = 0 232 9.1 Classical η vector model for D = 2, 3 234 9.1.1 Results for D = 2 237 9.1.2 A qualitative interpretation of the D = 2 data 240 9.1.3 Results for D = 3 242 9.1.4 Critical exponents 243 9.1.5 The ratio method 248 9.1.6 Estimates from differential approximates 254 9.2 Quantum Heisenberg model 255 9.2.1 Results for D = 2 257 9.2.2 Results for D = 3 258 9.2.3 Analysis of results 259 9.3 Discussion 261 9.4 Statistical mechanics versus quantum field theory 265 9.5 Appendix: The expansion coefficients for the susceptibility on the square lattice 267 References 272 PART III EXACTLY SOLVABLE MODELS 10 The Ising model in two dimensions: summary of results 277 10.1 The homogeneous lattice at Я = 0 280 10.1.1 Partition function on the torus 280 10.1.2 Zeros of the partition function 281 10.1.3 Bulk free energy per site 283 10.1.4 Partition function at Τ = Tc 286 10.1.5 Spontaneous magnetization 286 10.1.6 Row and diagonal spin correlation functions 287 10.1.7 The correlation C(M, N) for general Μ, Ν 295 10.1.8 Scaling limit 297 10.1.9 Magnetic susceptibility of the bulk 302 10.1.10 The diagonal susceptibility 306 10.2 Boundary properties of the homogeneous lattice at Η = 0 309 10.2.1 Boundary free energy at Hb = 0 309 10.2.2 Boundary magnetization Mi(Hb) 310 10.2.3 Boundary spin correlations 312 10.2.4 Analytic continuation and hysteresis 314 10.3 The layered random lattice 316 10.4 The Ising model for Я φ 0 319 10.4.1 The circle theorem 319 10.4.2 The imaginary magnetic field H/kBT = ¿π/2 319 10.4.3 Expansions for small Я 321 10.4.4 Τ = TC with Я > 0 322 10.4.5 Extended analyticity 323 References 324 Contents xiii 11 The Pfaffian solution of the Ising model 328 11.1 Dimers 329 11.1.1 Dimers on lattices with free boundary conditions 330 11.1.2 Dimers on a cylinder 337 11.1.3 Dimers on lattices of genus g > 1 338 11.1.4 Explicit evaluation of the Pfaffians 339 11.1.5 Thermodynamic limit 344 11.1.6 Other lattices and boundary conditions 345 11.2 The Ising partition function 347 11.2.1 Toroidal (periodic) boundary conditions 347 11.2.2 Cylindrical boundary conditions 354 11.3 Correlation functions 355 11.3.1 The correlation (cta/.tv^m.tv) 355 11.3.2 The diagonal correlation (cto^ctív.w) 359 11.3.3 Correlations near the boundary 360 References 361 12 Ising model spontaneous magnetization and form factors 363 12.1 Wiener-Hopf sum equations 364 12.1.1 Fourier transforms 364 12.1.2 Splitting and factorization 366 12.1.3 Solution 367 12.2 Spontaneous magnetization and Szegö s theorem 368 12.2.1 Proof of Szegö s theorem 369 12.2.2 The spontaneous magnetization 374 12.3 Form factor expansions of C{N, TV) and C(0. Лг) 375 12.3.1 Expansion for Τ < Tc 375 12.3.2 Expansion for Τ > Tc 386 12.4 Asymptotic expansions of С (TV. TV) and C(0. TV) for TV -+ oc 392 12.4.1 Large TV for Τ < Tc 392 12.4.2 Large TV for Τ > Tc 393 12.4.3 Large TV for Τ = Tc 393 12.5 Evaluation of diagonal form factor integrals 398 12.5.1 Differential equations 399 12.5.2 Factorization and direct sums 399 12.5.3 Homomorphisms of operators 402 12.5.4 Symmetric powers 402 12.5.5 Results 404 12.5.6 Discussion 406 References 407 13 The star-triangle (Yang-Baxter) equation 408 13.1 Historical overview 408 13.2 Transfer matrices 412 13.2.1 Explicit forms of the transfer matrix 415 13.2.2 The physical regime 416 xiv Contents 13.3 Integrability 417 13.4 Star-triangle equation for vertex models 418 13.4.1 Boltzmann weights for two-state vertex models 419 13.4.2 Vertex-spin correspondence 436 13.4.3 Inhomogeneous lattices 439 13.5 Star-triangle equation for spin models 440 13.5.1 Chiral Potts model 440 13.5.2 Proof of the star-triangle equation 448 13.5.3 Determination of Rpqr 451 13.6 Star-triangle equation for face models 452 13.6.1 SOS and RSOS models 452 13.6.2 The hard hexagon model 457 13.7 Hamiltonian limits 464 13.7.1 Spin chains for the eight- and six-vertex models 466 13.7.2 Spin chain for the chiral Potts model 469 13.8 Appendix: Properties of theta functions 472 References 477 14 The eight-vertex and XYZ model 480 14.1 Historical overview 481 14.2 The matrix TQ equation for the eight-vertex model 484 14.2.1 Modified theta functions 486 14.2.2 Formal construction of the matrices Qi2{v) 488 14.2.3 Explicit construction of Qr{v) and Ql{v) 489 14.2.4 The interchange relation 498 14.2.5 Nonsingularity and nondegeneracy 508 14.2.6 Quasiperiodicity 509 14.3 Eigenvalues and free energy 514 14.3.1 The form of the eigenvalues 514 14.3.2 Numerical study of the eigenvalues of Qj2{v) 516 14.3.3 Bethe s equation 526 14.3.4 Computation of the free energy 528 14.4 Excitations, order parameters and correlation functions of the eight- and six-vertex model 537 14.4.1 Eight-vertex polarization P8 and XYZ order 538 14.4.2 Eight-vertex magnetization M8 541 14.4.3 Correlations for the XY model 542 14.4.4 XYZ correlations 550 14.5 Appendix: Properties of the modified theta functions 552 References 557 15 The hard hexagon, RSOS and chiral Potts models 562 15.1 The hard hexagon and RSOS models 562 15.1.1 Historical overview 552 15.1.2 Hard hexagons for 0 < ζ < zc 565 15.1.3 Hard hexagons for zc < ζ < oc 571 Contents xv 15.1.4 Discussion 574 15.2 The chiral Potts model 575 15.2.1 Historical overview 575 15.2.2 Real and positive Boltzmann weights 578 15.2.3 The superintegrable chiral Potts model and Onsager s al¬ gebra 585 15.2.4 The functional equation for the superintegrable case for N = 3 588 15.2.5 Superintegrable ground state energy for small λ 590 15.2.6 Single particle excitations and level crossing 595 15.2.7 Order parameter 598 15.2.8 The phase diagram of the spin chain 599 15.3 Open questions 600 15.3.1 Q operators 601 15.3.2 Degenerate subspaces for the eight-vertex model 602 15.3.3 Symmetry algebra for the eight-vertex model at roots of unity 603 15.3.4 Chiral Potts correlations 603 References 605 PART IV CONCLUSION 16 Reductionism versus complexity 613 16.1 Does history matter? 613 16.2 Size is important 615 16.3 The paradox of integrability 616 16.4 Conclusion 617 References 618 Index 619
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series International series of monographs on physics
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Oxford science publications
spellingShingle McCoy, Barry M.
Advanced statistical mechanics
International series of monographs on physics
Statistical mechanics
Statistische Mechanik (DE-588)4056999-8 gnd
subject_GND (DE-588)4056999-8
title Advanced statistical mechanics
title_auth Advanced statistical mechanics
title_exact_search Advanced statistical mechanics
title_full Advanced statistical mechanics Barry M. McCoy
title_fullStr Advanced statistical mechanics Barry M. McCoy
title_full_unstemmed Advanced statistical mechanics Barry M. McCoy
title_short Advanced statistical mechanics
title_sort advanced statistical mechanics
topic Statistical mechanics
Statistische Mechanik (DE-588)4056999-8 gnd
topic_facet Statistical mechanics
Statistische Mechanik
url http://media.obvsg.at/AC08027992-1001
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