## Mathematical Foundations of Quantum Statistics – Khinchin

In this post, we will see the book Mathematical Foundations of Quantum Statistics by A. I. Khinchin.

In the area of quantum statistics, I show that a rigorous and systematic mathematical basis of the computational formulas of statistical physics does not require a special unwieldy analytical apparatus (the method of Darwin-Fowler), but may be obtained from an elementary application of the well-developed limit theorems of the theory of probability. Apart from its purely scientific value, which is evident and requires no comment, the possibility of such an application is particularly satisfying to Soviet scientists, since the study of these limit theorems was founded by P. L. Chebyshev and was developed fur­ther by other Russian and Soviet mathematicians. The fact that these theorems can form the analytical basis for all the computational formulas of statistical physics once again demonstrates their value for applications.
This monograph, like my first book, is devoted entirely to the mathematical method of the theory and is in no way a complete physical treatise. In fact, no concrete physical problem is considered. The book is directed primarily towards the mathematical reader. However, I hope that the physicist who is concerned with the mathematical apparatus of his science will find something in it to interest him.

The book was translated from Russian by Irwin Shapiro and was published in  1960.

You can get the book here.

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# Contents

## INTRODUCTION

1. The most important characteristics of the mathematical apparatus
of quantum statistics 1
2. Contents of the book 4

## CHAPTER I PRELIMINARY CONCEPTS OF THE THEORY OF PROBABILITY

1. Integral-valued random variables 12
2. Limit theorems 16
3. The method of characteristic functions 21
4. The one-dimensional limit theorem 29
5. The two-dimensional limit theorem 35

## CHAPTER II PRELIMINARY CONCEPTS OF QUANTUM MECHANICS

1. Description of the state of a physical system in quantum mechanics 45
2. Physical quantities and self-adjoint linear operators 49
3. Possible values of physical quantities00 54
4. Evolution of the state of a system in time 61
5. Stationary states. The law of conservation of energy 65

## CHAPTER III GENERAL PRINCIPLES OF QUANTUM STATISTICS

1. Basic concepts of statistical methods in physics 71
2. Microcanonical averages 75
3. Complete, symmetric and antisymmetric statistics 80
4. Construction of the fundamental linear basis 85
5. Occupation numbers. Basic expressions for the structure functions 91
6. On the suitability of microcanonical averages 96

## CHAPTER IV THE FOUNDATIONS OF THE STATISTICS OF PHOTONS

1. Distinctive characteristics of the statistics of photons 102
2. Occupation numbers and their mean values 103
3. Reduction to a problem of the theory of probability 106
4. Application of a limit theorem of the theory of probability 110
5. The Planck formula 113
6. On the suitability of microcanonical averages 118

## CHAPTER V FOUNDATIONS OF THE STATISTICS OF MATERIAL PARTICLES

1. Review of fundamental concepts 123
2. Mean values of the occupation numbers 124
3. Reduction to a problem of the theory of probability 130
4. Choice of values for the parameters 𝛼 and 𝛽 135
5. Application of a limit theorem of the theory of probability 139
6. Mean values of sum functions 142
7. Correlation between occupation numbers 144
8. Dispersion of sum functions and the suitability of microcanonical
averages 147
9. Determination of the numbers g_{k} for structureless particles in the absence of external forces 149

## CHAPTER VI THERMODYNAMIC CONCLUSIONS

1. The problems of statistical thermodynamics 153
2. External parameters, external forces and their mean values 154
3. Determination of the entropy and the deduction of the second law of thermodynamics 158

Supplement I. The Statistics or Heterogeneous Systems 162
Supplement II. The Distribution of a Component and its Energy 168
Supplement III. The Principle of Canonical Averaging 172
Supplement IV. The Reduction to One-dimensional Problem in the Case of Complete Statistics 178
Supplement V. Some General Theorems of Statistical Physics. 180
Supplement VI. Symmetric Functions on Multi-dimensional Surfaces 198

1. Introduction 198
2. Preliminary formulas 199
3. Distribution of the energy of a particle. Gibbs’ theorem 203
4. Derivation of the fundamental formula 207
5. Distribution of the maximum and minimum energy of a particle 210
6. The basic limit theorem 215
7. Probability that the energy of a particle lies in a given interval 219
8. A functional limit theorem 221
9. Continuous symmetric functions004 224
10. Generalizations of Gibbs’ theorem. 226

References 231

Index 232