Mathematical Foundations of Statistical Mechanics

The translation of this important book brings to the English-speaking mathematician and mathematical physicist a thoroughly up-to-date introduction to statistical mechanics.
It offers a precise and mathematically rigorous formulation of the problems of statistical mechanics, as opposed to the non-rigorous discussion presented in most other works. It provides analytical tools needed to replace many of the cumbersome concepts and devices commonly used for establishing basic formulae, and it furnishes the mathematician with a logical step-by-step introduction, which will enable him to master the elements of statistical mechanics in the shortest possible time.
After a historical sketch, the author discusses the geometry and kinematics of the phase space, with the theorems of Liouville and Birkhoff; the ergodic problem (in the sense of replacing time averages by phase averages); the theory of probability; central limit theorem; ideal monatomic gas; foundation of thermodynamics, and dispersion and distribution of sum functions.
"An excellent introduction to the difficult and important discipline of Statistical Mechanics. It is clear, concise, and rigorous. There is a very good chapter on the ergodic theorem (with a complete proof!) and . . . a highly lucid chapter on statistical foundations of thermodynamics . . . useful to teachers . . . and to mathematicians." ― M. Kac, Quarterly of Applied Mathematics.

Preface
Chapter I. Introduction
1. A brief historical sketch
2. Methodological characterization
Chapter II. Geometry and Kinematics of the Phase Space
3. The phase space of a mechanical system
4. Theorem of Liouville
5. Theorem of Birkhoff
6. Case of metric indecomposability
7. Structure functions
8. Components of mechanical systems
Chapter III. Ergodic Problem
9. Interpretation of physical quantities in statistical mechanics
10. Fixed and free integrals
11. Brief historical sketch
12. On metric indecomposability of reduced manifolds
13. The possibility of a formulation without the use of metric indecomposability
Chapter IV. Reduction to the Problem of the Theory of Probability
14. Fundamental distribution law
15. The distribution law of a component and its energy
16. Generating functions
17. Conjugate distribute functions
18. Systems consisting of a large number of components
Chapter V. Application of the Central Limit Theorem
19. Approximate expressions of structure functions
20. The small component and its energy. Boltzmann's law
21. Mean values of the sum functions
22. Energy distribution law of the large component
23. Example of monatomic ideal gas
24. The theorem of equipartition of energy
25. A system in thermal equilibrium. Canonical distribution of Gibbs
Chapter VI. Ideal Monatomic Gas
26. Velocity distribution. Maxwell's law
27. The gas pressure
28. Physical interpretation of the parameter
29. Gas pressure in an arbitrary field of force
Chapter VII. The Foundation of Thermodynamics
30. External parameters and the mean values of external forces
31. The volume of the gas as an external parameter
32. The second law of thermodynamics
33. The properties of entropy
34. Other thermodynamical functions
Chapter VIII. Dispersion and the Distributions of Sum Functions
35. The intermolecular correlation
36. Dispersion and distribution laws of the sum functions
Appendix
  The proof of the central limit theorem of the theory of probability
Notations
Index