## Quantum Mechanics and Its Emergent MacrophysicsThe quantum theory of macroscopic systems is a vast, ever-developing area of science that serves to relate the properties of complex physical objects to those of their constituent particles. Its essential challenge is that of finding the conceptual structures needed for the description of the various states of organization of many-particle quantum systems. In this book, Geoffrey Sewell provides a new approach to the subject, based on a "macrostatistical mechanics," which contrasts sharply with the standard microscopic treatments of many-body problems. Sewell begins by presenting the operator algebraic framework for the theory. He then undertakes a macrostatistical treatment of both equilibrium and nonequilibrium thermodynamics, which yields a major new characterization of a complete set of thermodynamic variables and a nonlinear generalization of the Onsager theory. The remainder of the book focuses on ordered and chaotic structures that arise in some key areas of condensed matter physics. This includes a general derivation of superconductive electrodynamics from the assumptions of off-diagonal long-range order, gauge covariance, and thermodynamic stability, which avoids the enormous complications of the microscopic treatments. Sewell also unveils a theoretical framework for phase transitions far from thermal equilibrium. Throughout, the mathematics is kept clear without sacrificing rigor. Representing a coherent approach to the vast problem of the emergence of macroscopic phenomena from quantum mechanics, this well-written book is addressed to physicists, mathematicians, and other scientists interested in quantum theory, statistical physics, thermodynamics, and general questions of order and chaos. |

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### Contents

The generalised quantum mechanical framework | 7 |

21 OBSERVABLES STATES DYNAMICS | 8 |

222 The Generic Model | 10 |

223 THE ALGEBRAIC PICTURE | 13 |

INEQUIVALENT REPRESENTATIONS | 15 |

232 The Representation a | 17 |

234 Other Inequivalent Representations | 18 |

242 States and Representations | 21 |

53 INFINITE SYSTEMS | 113 |

532 Thermodynamical Stability Conditions | 118 |

541 Equilibrium States | 123 |

542 Metastable States | 124 |

55 FURTHER DISCUSSION | 125 |

Equilibrium thermodynamics and phase structure 61 INTRODUCTION | 127 |

62 PRELIMINARIES ON CONVEXITY6 | 131 |

63 THERMODYNAMIC STATES AS TANGENTS TO THE REDUCED PRESSURE FUNCTION | 135 |

243 Automorphisms and Antiautomorphisms | 24 |

244 Tensor Products | 26 |

245 Quantum Dynamical Systems | 27 |

246 Derivations of Algebras and Generators of Dynamical Groups BR | 28 |

25 ALGEBRAIC FORMULATION OF INFINITE SYSTEMS | 29 |

252 Construction of the Lattice Model cf St Rol | 32 |

253 Construction of the Continuum Model cf HHW DS Se4 | 34 |

26 THE PHYSICAL PICTURE | 39 |

263 Primary States have Short Range Correlations | 40 |

264 Decay of Time Correlations and Irreversibility | 41 |

265 Global Macroscopic Observables cf He | 42 |

266 Consideration of Pure Phases | 44 |

267 Fluctuations and Mesoscopic Observables cf GW | 45 |

27 OPEN SYSTEMS | 46 |

28 CONCLUDING REMARKS | 47 |

HILBERT SPACES | 48 |

On symmetry entropy and order | 57 |

321 Classical Preliminaries SW Khl | 58 |

322 Finite Quantum Systems | 59 |

323 Infinite Systems | 62 |

324 On Entropy and Disorder | 64 |

33 ORDER AND COHERENCE | 65 |

332 Coherence | 68 |

333 Long Range Correlations in Ginvariant Mixtures of Ordered Phases | 69 |

334 Superfluidity and Offdiagonal Long Range Order | 70 |

335 On Entropy and Order | 72 |

Reversibility irreversibilty and macroscopic causality | 75 |

411 Finite Systems | 76 |

412 Infinite Systems | 78 |

COMPLETELY POSITIVE MAPS QUANTUM DYNAMICAL SEMIGROUPS AND CONDITIONAL EXPECTATIONS 421 Complete Positivity | 79 |

422 Quantum Dynamical Semigroups | 81 |

423 Conditional Expectations | 82 |

43 INDUCED DYNAMICAL SUBSYSTEMS | 83 |

45 NOTE ON CLASSICAL MACROSCOPIC CAUSALITY From Quantum Stochastics to Classical Determinism | 86 |

EXAMPLE OF A POSITIVE MAP THAT IS NOT COMPLETELY POSITIVE | 88 |

SIMPLE MODEL OF IRREVERSIBILITY AND MIXING | 89 |

SIMPLE MODEL OF IRREVERSIBILITY AND MACROSCOPIC CAUSALITY | 94 |

C2 EQUATIONS OF MOTION | 98 |

C3 Macroscopic Description of B | 100 |

C4 The Phenomenological Law | 102 |

C5 The Fluctuation Process | 103 |

From quantum statistics to equilibrium and nonequilibrium thermodynamics prospectus | 107 |

Thermal equilibrium states and phases 51 INTRODUCTION | 109 |

52 FINITE SYSTEMS | 111 |

522 Equilibrium and Thermodynamical Stability | 112 |

64 QUANTUM STATISTICAL BASIS OF THERMODYNAMICS | 136 |

65 AN EXTENDED THERMODYNAMICS WITH ORDER PARAMETERS | 142 |

66 CONCLUDING REMARKS ON THE PAUCITY OF THERMODYNAMICAL VARIABLES | 144 |

PROOFS OF PROPOSITIONS 641 AND 642 | 145 |

FUNCTIONALS q AS SPACE AVERAGES OF LOCALLY CONSERVED QUANTUM FDZLDS | 146 |

Macrostatistics and nonequilibrium thermodynamics 71 INTRODUCTION | 149 |

72 THE QUANTUM FIELD qx | 153 |

73 THE MACROSCOPIC MODEL M | 155 |

74 RELATIONSHIP BETWEEN THE CLASSICAL FIELD q AND THE QUANTUM FIELD q | 158 |

t | 161 |

MACROSCOPIC EQUILIBRIUM CONDITIONS AND THE ONSAGER RELATIONS | 164 |

LOCAL EQUILIBRIUM AND GENERALISED ONSAGER RELATIONS | 166 |

TOWARDS A GENERALISATION OF THE THEORY TO GALILEAN CONTINUUM MECHANICS | 168 |

TEMPERED DISTRIBUTIONS | 170 |

CLASSICAL STOCHASTIC PROCESSES AND THE CONSTRUCTION OF 31nutt AS A CLASSICAL MARKOV FIELD | 176 |

B2 CLASSICAL GAUSSIAN FIELDS | 178 |

B3 Proof of Propositions 751 and 752 | 183 |

CI The Truncated Static TwoPoint Function | 184 |

C2 Quantum Statistical Formulation of sq | 186 |

C3 Formulation of T via Perturbations of pe | 187 |

C4 PROOF OF PROPOSITIONS C31 AND C32 FOR LATTICE SYSTEMS WITH FINITE RANGE INTERACTIONS | 192 |

C5 Pure Crystalline Phases | 195 |

Superconductive electrodynamics as a consequence of offdiagonal long range order gauge covariance and thermodynamical stability prospectus | 197 |

Brief historical survey of theories of superconductivity | 199 |

Offdiagonal long range order and superconductive electrodynamics 91 INTRODUCTION | 211 |

92 THE GENERAL MODEL | 213 |

93 ODLRO VERSUS MAGNETIC INDUCTION | 218 |

94 STATISTICAL THERMODYNAMICS OF THE MODEL AND THE MEISSNER EFFECT | 221 |

942 Thermodynamical Potentials | 222 |

95 FLUX QUANTISATION | 226 |

96 METASTABILITY OF SUPERCURRENTS AND SUPERSELECTION RULES | 229 |

97 NOTE ON TYPE II SUPERCONDUCTORS | 234 |

98 CONCLUDING REMARKS | 236 |

Ordered and chaotic structures far from equilibrium prospectus | 239 |

Schematic approach to a theory of nonequlibrium phase transitions order and chaos | 241 |

Laser model as a paradigm of nonequilibrium phase structures 111 INTRODUCTION | 247 |

112 THE MODEL | 248 |

113 THE MACROSCOPIC DYNAMICS | 256 |

114 THE DYNAMICAL PHASE TRANSITIONS | 260 |

115 THE MICROSCOPIC DYNAMICS | 264 |

116 A NONEQUILIBRIUM MAXIMUM ENTROPY PRINCIPLE | 269 |

117 CONCLUDING REMARKS | 271 |

275 | |

287 | |