## A Method For Studying Model Hamiltonians – A Minimax Principle For Problems In Statistical Physics – Bogolyubov Jr

In this post, we will see the book A Method For Studying Model Hamiltonians – A Minimax Principle For Problems In Statistical Physics by N. N. Bogolyubov Jr..

In this book methods are proposed for solving certain problems in statistical physics which contain four-fermion interaction.

It has been possible, by means of “approximating (trial) Hamiltonians”, to distinguish a whole class of exactly soluble model systems. An essential difference between the two types of problem with positive and negative four-fermion interaction is discovered and examined. The determination of exact solutions for the free energies, single-time and many-time correlation functions, T-products and Green’s functions is treated for each type of problem.

The more general problem for which the Hamiltonian contains some terms with positive and others with negative four-fermion interaction is also investigated. On the basis of analysing and general­izing the results of Chapters 1 to 3, it becomes possible to formulate and develop a new principle, the minimax principle, for problems in statistical physics (Chapter 4).

The book was translated from Russian by P. J. Shepherd and was published in 1972.

You can get the book here.

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# Contents

INTRODUCTION 1

§ I. GENERAL REMARKS 1
§ II. REMARKS ON QUASI AVERAGES 16

## CHAPTER 1 PROOF OF THE ASYMPTOTIC RELATIONS FOR THE MANY-TIME CORRELATION FUNCTIONS

§ 1. GENERAL TREATMENT OF THE PROBLEM. SOME PRELIMINARY RESULTS AND FORMULATION OF THE PROBLEM 25

§ 2. EQUATIONS OF MOTION AND AUXILIARY OPERATOR INEQUALITIES 33

§ 4. BOUNDS FOR THE DIFFERENCE OF THE SINGLE-TIME AVERAGES 40

§ 5. REMARK (1) 47

§ 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 50

§ 7. PROOF OF THE CLOSENESS OF THE AVERAGES FOR ARBITRARY ORDERING OF THE OPERATORS IN THE AVERAGES 54

§ 8. ESTIMATES OF THE ASYMPTOTIC CLOSENESS OF THE MANY-TIME CORRELATION AVERAGES 57

## CHAPTER 2 CONSTRUCTION OF A PROOF OF THE GENERALIZED ASYMPTOTIC RELATIONS FOR THE MANY-TIME CORRELATION AVERAGES 65

§ 1. SELECTION RULES AND CALCULATION OF THE AVERAGES 65

§ 2. GENERALIZED CONVERGENCE 70

§ 3. REMARK 74

§ 4. PROOF OF THE ASYMPTOTIC RELATIONS 76

§ 5. REMARK ON THE CONSTRUCTION OF UNIFORM BOUNDS 79

§ 6. GENERALIZED ASYMPTOTIC RELATIONS FOR THE GREEN’S FUNCTIONS 82

§ 7. THE EXISTENCE OF GENERALIZED LIMITS 85

## CHAPTER 3 CORRELATION FUNCTIONS FOR SYSTEMS WITH FOUR-FERMION NEGATIVE INTERACTION 90

§ 1. CALCULATION OF THE FREE ENERGY FOR MODEL SYSTEMS WITH ATTRACTION 90

§ 2. FURTHER PROPERTIES OF THE EXPRESSIONS FOR THE FREE ENERGY 101

§ 3. CONSTRUCTION OF ASYMPTOTIC RELATIONS FOR THE FREE ENERGY 105

§ 4. ON THE UNIFORM CONVERGENCE WITH RESPECT TO 𝛳 OF THE FREE ENERGY FUNCTION AND ON BOUNDS FOR THE QUANTITIES 𝛿_{V} 111

§ 5. PROPERTIES OF PARTIAL DERIVATIVES OF THE FREE ENERGY FUNCTION. THEOREM 3.III 114

§ 6. RIDER TO THEOREM 3.III AND CONSTRUCTION OF AN AUXILIARY INEQUALITY 117

§ 7. ON THE DIFFICULTIES OF INTRODUCING QUASI-AVERAGES 120

§ 8. A NEW METHOD OF INTRODUCING QUASI-AVERAGES 124

§ 9. THE QUESTION OF THE CHOICE OF SIGN FOR THE SOURCE-TERMS 30

§ 10. THE CONSTRUCTION OF UPPER-BOUND INEQUALITIES IN THE CASE WHEN C = 0 131

## CHAPTER 4 MODEL SYSTEMS WITH POSITIVE AND NEGATIVE INTERACTION COMPONENTS 137

§ 1. HAMILTONIAN WITH NEGATIVE COUPLING CONSTANTS (REPULSIVE INTERACTION) 137

§ 2. FEATURES OF THE ASYMPTOTIC RELATIONS FOR THE FREE ENERGIES IN THE CASE OF SYSTEMS WITH POSITIVE INTERACTION 141

§ 3. BOUNDS FOR THE FREE ENERGIES AND CORRELATION FUNCTIONS 143

§ 4. EXAMINATION OF AN AUXILIARY PROBLEM 146

§ 5. SOLUTION OF THE QUESTION OF UNIQUENESS 150

§ 6. HAMILTONIANS WITH COUPLING CONSTANTS OF DIFFERENT SIGNS. THE MINIMAX PRINCIPLE 154

REFERENCES 164

INDEX 169