Quantum Theory of Collective Phenomena

Quantum Theory of Collective Phenomena

by G. L. Sewell

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Product Details

ISBN-13: 9780486780443
Publisher: Dover Publications
Publication date: 06/18/2014
Series: Dover Books on Chemistry Series
Pages: 256
Product dimensions: 6.10(w) x 9.10(h) x 0.70(d)

About the Author

G. L. Sewell is Professor of Mathematical Physics at Queen Mary University, London. He is the author of Quantum Mechanics and Its Emergent Macrophysics.

Table of Contents

Part I The generalized quantum mechanical framework

1 Introductory discussion on the quantum theory of macroscopic systems 3

2 The generalized quantum mechanical framework 7

2.1 Introduction 7

2.2 Finite systems 9

2.2.1 Uniqueness of the representation 9

2.2.2 The Hilbert space description 11

Note on Poincaré cycles 12

2.2.3 The algebraic description 13

Algebra of bounded observables 13

The states 14

Mixtures and pure states 14

The dynamics 15

2.2.4 Summary of the model 15

2.3 Inequivalent representation for an infinite system 16

Discussion: representations and phases 18

2.4 General model of an infinite system 19

2.4.1 The local observables 20

Explicit constructions 21

2.4.2 The states 25

2.4.3 States, representations, and islands 26

2.4.4 Macroscopic description 28

2.4.5 The dynamics 29

Dynamics in an island 30

Absence of Poincaré cycles 32

2.4.6 Unbounded local observables 33

2.4.7 Summary of the model 34

2.5 Concluding remarks 34

Appendix A Hilbert spaces 35

Appendix B Macroscopic observables and islands of states 42

Part II Statistical thermodynamics

3 Equilibrium states 47

3.1 Introduction 47

3.2 Mathematical preliminaries: convexity, continuity, and semicontinuity 51

3.2.1 Convexity 51

Concave and affine functions 53

Some consequences of convexity 53

3.2.2 Note on convergent subsequences: the Bolzano-Weierstrass theorem 55

3.2.3 Semicontinuity 55

3.3 The model 56

3.4 Thermodynamical functions and stability conditions 58

3.4.1 Local functionals 58

3.4.2 Global functionals 60

3.4.3 Incremental functionals 62

3.4.4 Thermodynamical stability conditions 63

Note on translational invariance 63

3.5 Global stability, macroscopic degeneracy, and symmetry breakdown 64

3.5.1 Macroscopic degeneracy and symmetry breakdown in the two-dimensional Ising model 65

The Peierls argument for a finite Ising model 65

Note on the canonical free energy density 68

Polarized GTS state of the infinite Ising model 69

Macroscopic degeneracy and symmetry breakdown 70

3.6 Relationship between local and global stability 70

3.6.1 Model with states that are locally, but not globally, stable 71

3.7 The KMS fluctuation-dissipation condition 73

3.7.1 The reservoir property and detailed balance: a simple example 74

3.8 Conclusions concerning equilibrium states 77

Continuous systems 78

Notes on microcanonical description 79

Appendix A Thermodynamical stability and the KMS condition for finite systems 80

Appendix B The compactness property (P) 83

Appendix C Infinite volume limits 83

Appendix D Proof that translationally invariant LTS states are GTS in the case of finite-range interactions 86

4 Thermodynamics and phase structure 89

4.1 Introduction 89

4.2 Extensive conserved quantities 92

4.3 The Helmholtz and Gibbs potentials 93

Analyticity, singularities 94

Convexity and concavity properties 94

4.4 Equilibrium states, pure phases, and mixtures 95

4.5 Macroscopic degeneracy and thermodynamic discontinuity 97

Assumption concerning location of discontinuities 101

4.6 Phase structure 102

4.7 Thermodynamical relations 103

4.8 Concluding remarks 104

Appendix A Mathematical properties of the thermodynamic potentials 105

Appendix B The dispersion-free character of global intensive thermodynamical variables in extremal equilibrium states 108

Part III Collective phenomena

5 Phase transitions 115

5.1 Introduction 115

5.2 Order parameters, symmetry breakdown, and long-range correlations 118

Ordering and long-range correlations 120

Off-diagonal long-range order 122

Note on macroscopic wave functions 123

Further comments on ordering and long-range correlations 123

5.3 Criteria for the existence of order-disorder transitions 124

5.3.1 The absence of order-disorder transitions in some low- 125

dimensional models: energy-versus-entropy arguments 128

5.3.2 Models with order-disorder transitions 129

Systems with breakdown of discrete symmetry 129

Systems with breakdown of continuous symmetry

5.3.3 The FSS proof of ordering in the classical Heisenberg ferromagnet 130

Order as condensation of spin waves 130

Infra-red bound 131

Response to perturbations 132

Reflection positivity 134

Derivation of the inequality (36) from RP (reflection positivity) 135

5.3.4 Generalizations of the FSS and Peierls arguments 137

Systems with continuous symmetry: infra-red bounds 139

X-Y model 139

The Heisenberg antiferromagnet 140

Note on Heisenberg ferromagnet 140

Systems with discrete symmetries: generalized Peierls argument 141

5.4 Critical phenomena 142

5.4.1 Phenomenological background 144

Landau's theory 144

Critical singularities 145

Scaling laws 147

Connection between scaling and critical functions: Kadanoff's argument 148

5.4.2 Renormalization theory and non-central limits 150

Characteristic functions 150

Block variables and renormalization 150

Critical states as fixed points of the RS 152

5.4.3 RS treatment of Dyson's hierarchical model 154

The model 154

The macroscopic distribution 156

The renormalization mapping 157

Determination of the stable fixed point 158

The case where g>$$$2/4 160

The case where g<$$$2/4 161

5.4.4 Discussion of the general RS theory 162

5.5 Concluding remarks 163

Appendix A Proof that Griffiths's condition (3) implies symmetry breakdown 164

Appendix B Proof that lim$$$ is a classical characteristic function 166

6 Metastable states 169

6.1 Introduction 169

6.2 Ideal metastability 171

6.2.1 Translationally invariant states 172

Example 1. Cluster model with many-body forces 172

Example 2. Mean field theoretical models 172

Note on gravitational systems 174

6.2.2 Non-translationally invariant states: surface effects 175

6.3 Normal metastability 177

6.3.1 The double well model 178

6.3.2 States of an Ising ferromagnet with polarization opposed to an external field 181

The finite system 183

The infinite system 186

Comment on finite versus infinite systems 187

6.4 Concluding remarks 188

7 Order and phase transitions far from equilibrium 189

7.1 Introduction 189

7.2 Simple models of open systems 191

7.2.1 Damped harmonic oscillator 192

7.2.2 Model of a pumped atom 195

Dynamics of atomic observables 198

7.3 The laser model 201

The model 202

Macroscopic description 203

Microscopic equations of motion 204

Macroscopic equations of motion 205

The phase transition 207

The coherent radiation 209

7.4 Fröhlich's pumped phonon model 209

The model 210

The stationary distribution 211

Stability 212

Bose-Einstein condensation 213

7.5 Concluding remarks 214

Appendix. Statistics of the forces x(t) 214

A.I The atomic model 215

A.2 The laser model 217

References 219

Index 227

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