Statistical mechanics : rigorous results

This text marks the beginning of an era of vigorous mathematical progress in equilibrium statistical mechanics. It treats the infinite system limit, and discusses thermodynamic functions and states. The conceptual foundation provided by the book should be useful for the study of developments of statistical mechanics in the second half of the 20th century. Title page......Page 1 Preface......Page 5 Contents......Page 7 1.1 Thermodynamic Systems......Page 13 1.2 Classical Systems and Ensembles......Page 15 1.3 Quantum Systems and Ensembles......Page 19 1.4 Equilibrium States and Thermodynamic Functions......Page 23 2.1 Limit of an Infinite Volume......Page 25 2.2 Interactions of Quantum Lattice Systems......Page 27 2.3 Thermodynamic Limit for Quantum Lattice Systems......Page 30 2.4 Thermodynamic Limit for Classical Lattice Systems......Page 32 2.5 Some Inequalities for Quantum Systems......Page 37 3.1 Interactions of Classical Continuous Systems......Page 41 3.2 Stable Interactions......Page 45 3.3 Thermodynamic Limit for the Configurational Microcanonical Ensemble......Page 53 3.4 Thermodynamic Limit for the Classical Ensembles......Page 64 3.5 Thermodynamic Limit for the Quantum Ensembles......Page 72 Exercises......Page 80 4.1 Definitions......Page 83 4.2 The Method of Integral Equations......Page 84 4.3 Mayer Series and Virial Expansion......Page 95 4.4 The Algebraic Method......Page 98 4.5 Positive Potentials......Page 105 4.6 Quantum Systems......Page 109 4.7 Construction of a Hilbert Space......Page 112 Exercises......Page 116 5.1 The Theorem of Lee and Yang......Page 120 5.2 Absence of Phase Transitions at High Temperature......Page 124 5.3 Existence of a First-Order Phase Transition at Low Temperature......Page 125 5.4 The Griffiths Inequalities......Page 131 5.5 The Theorem of Mermin and Wagner......Page 141 5.6 One-Dimensional Systems......Page 146 Bibliographical Note......Page 154 Exercises......Page 155 6.1 Description of the State of an Infinite System......Page 157 6.2 Invariant States......Page 159 6.3 Ergodic States......Page 166 6.4 Decomposition of Invariant States into Ergodic States......Page 168 6.5 Pure Thermodynamic Phases as Ergodic States......Page 173 Bibliographical Note......Page 175 Exercises......Page 176 7.1 B*-Algebras for Classical and Quantum Statistical Mechanics......Page 180 7.2 Entropy......Page 190 7.3 Thermodynamic Limit of States......Page 196 7.4 A Variational Principle......Page 199 7.5 The Gibbs Phase Rule......Page 202 7.6 Time Evolution of Quantum Lattice Systems and the Kubo-Martin-Schwinger Boundary Condition......Page 204 Bibliographical Note......Page 208 Exercises......Page 209 A.1 Some Terminology......Page 210 A.2 Generalities on B* -Algebras......Page 211 A.3 States on B* -Algebras......Page 213 A.4 Von Neumann Algebras......Page 217 A.5 Integral Representations on Convex Compact Sets......Page 218 A.6 Groups with an Invariant Mean......Page 220 References......Page 222 Index......Page 229

this Classic Book Marks The Beginning Of An Era Of Vigorous Mathematical Progress In Equilibrium Statistical Mechanics. Its Treatment Of The Infinite System Limit Has Not Been Superseded, And The Discussion Of Thermodynamic Functions And States Remains Basic For More Recent Work. The Conceptual Foundation Provided By The Rigorous Results Remains Invaluable For The Study Of The Spectacular Developments Of Statistical Mechanics In The Second Half Of The 20th Century.

In this chapter, we review the systems and ensembles which will be used in the rest of the monograph.