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..

# About the book

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.

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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

§ 3. ADDITIONAL INEQUALITIES 37

§ 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