Mathematical Foundations of Statistical Mechanics: (Dover Books on Mathema 1.4tics)

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.

Table of Contents:
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