Bogolyubov / Ter Haar A Method for Studying Model Hamiltonians

A Minimax Principle for Problems in Statistical Physics
1. Auflage 2013
ISBN: 978-1-4831-4877-9
Verlag: Elsevier Science & Techn.
Format: PDF
Kopierschutz: 1 - PDF Watermark

E-Book, Englisch, 180 Seiten, Web PDF

A Method for Studying Model Hamiltonians: A Minimax Principle for Problems in Statistical Physics centers on methods for solving certain problems in statistical physics which contain four-fermion interaction. Organized into four chapters, this book begins with a presentation of the proof of the asymptotic relations for the many-time correlation functions. Chapter 2 details the construction of a proof of the generalized asymptotic relations for the many-time correlation averages. Chapter 3 explains the correlation functions for systems with four-fermion negative interaction. The last chapter shows the model systems with positive and negative interaction components.

Autoren/Hrsg.

Bogolyubov, N. N.

Ter Haar, D.

Inhaltsverzeichnis

1;Front Cover;1
2;A Method for Studying Model Hamiltonians: A Minimax Principle for Problems in Statistical Physics;4
3;Copyright Page;5
4;Table of Contents;6
5;SERIES EDITOR'S PREFACE;8
6;PREFACE;10
7;INTRODUCTION;12
7.1;§ 1. General Remarks;12
7.2;§ 2. Remarks an Quasi-Averages;27
8;CHAPTER 1. PROOF OF THE ASYMPTOTIC RELATIONS FOR THE MANY-TIME CORRELATION FUNCTIONS;36
8.1;§ 1. General Treatment of the Problem. Some Preliminary Results and Formulation of the Problem;36
8.2;§ 2. Equations of Motion and Auxiliary Operator Inequalities;44
8.3;§ 3. Additional Inequalities;48
8.4;§ 4. Bounds for the Difference of the Single-time Averages;51
8.5;§ 5. Remark (I);58
8.6;§ 6. Proof of the Closeness of Averages Constructed on the Basis of Model and Trial Hamiltonians for "Normal" Ordering of the Operators in the Averages;61
8.7;§ 7. Proof of the Closeness of the Averages for Arbitrary Ordering of the Operators in the Averages;65
8.8;Remark (II);67
8.9;§ 8. Estimates of the Asymptotic Closeness of the Many-time Correlation Averages;68
9;CHAPTER 2. CONSTRUCTION OF A PROOF OF THE GENERALIZED ASYMPTOTIC RELATIONS FOR THE MANY-TIME CORRELATION AVERAGES;76
9.1;§ 1. Selection Rules and Calculation of the Averages;76
9.2;§ 2. Generalized Convergence;81
9.3;§ 3. Remark;85
9.4;§ 4. Proof of the Asymptotic Relations;87
9.5;§ 5. Remark on the Construction of Uniform Bounds;90
9.6;§ 6. Generalized Asymptotic Relations for the Green's Functions;93
9.7;§ 7. The Existence of Generalized Limits;96
10;CHAPTER 3. CORRELATION FUNCTIONS FOR SYSTEMS WITH FOURFERMION NEGATIVE INTERACTION;100
10.1;§ 1. Calculation of the Free Energy for Model Systems with Attraction;100
10.2;§ 2. Further Properties of the Expressions for the Free Energy;112
10.3;§ 3. Construction of Asymptotic Relations for the Free Energy;116
10.4;§ 4. On the Uniform Convergence with Respect to . of the Free Energy Function and on Bounds for the Quantities dv;122
10.5;§ 5. Properties of Partial Derivatives of the Free Energy Function. Theorem 3.III;125
10.6;§ 6. Rider to Theorem 3.Ill and Construction of an Auxiliary Inequality;128
10.7;§ 7. On the Difficulties of Introducing Quasi-averages;131
10.8;§ 8. A New Method of Introducing Quasi-averages;135
10.9;§ 9. The Question of the Choice of Sign for the Source-terms;141
10.10;§ 10. The Construction of Upper-bound Inequalities in the Case when C=0;142
11;CHAPTER 4. MODEL SYSTEMS WITH POSITIVE AND NEGATIVE INTERACTION COMPONENTS;148
11.1;§ 1. Hamiltonian with Negative Coupling Constants (Repulsive Interaction);148
11.2;§ 2. Features of the Asymptotic Relations for the Free Energies in the Case of Systems with Positive Interaction;152
11.3;§ 3. Bounds for the Free Energies and Correlation Functions;154
11.4;§ 4. Examination of an Auxiliary Problem;157
11.5;§ 5. Solution of the Question of Uniqueness;161
11.6;§ 6. Hamiltonians with Coupling Constants of Different Signs. The Minimax Principle;165
12;REFERENCES;176
13;INDEX;180