Kurth / Sneddon / Ulam Axiomatics of Classical Statistical Mechanics

1. Auflage 2014
ISBN: 978-1-4831-9478-3
Verlag: Elsevier Science & Techn.
Format: PDF
Kopierschutz: 1 - PDF Watermark

E-Book, Englisch, Band Volume 11, 190 Seiten, Web PDF

Reihe: International Series in Pure and Applied Mathematics

Axiomatics of Classical Statistical Mechanics provides an understanding of classical statistical mechanics as a deductive system. This book presents the mechanical systems of a finite number of degrees of freedom. Organized into seven chapters, this book begins with an overview of the average behavior of mechanical systems. This text then examines the concept of a mechanical system and explains the equations of motion of the system. Other chapters consider an ensemble of mechanical systems wherein a Hamiltonian function and a truncated canonical probability density corresponds to each system. This book discusses as well the necessary and sufficient conditions that are given for the existence of statistically stationary states and for the approach of mechanical systems towards these states. The final chapter deals with the fundamental laws of thermodynamics. This book is a valuable resource for mathematicians.

Autoren/Hrsg.

Kurth, Rudolf

Sneddon, I. N.

Ulam, S.

Stark, M.

Inhaltsverzeichnis

1;Front Cover;1
2;Axiomatics of Classical Statistical Mechanics;4
3;Copyright Page;5
4;Table of Contents;10
5;By the same author;6
6;PREFACE;8
7;CHAPTER I. INTRODUCTION;12
7.1;§ 1. Statement of the problem;12
8;CHAPTER II. MATHEMATICAL TOOLS;14
8.1;§2. Sets;14
8.2;§3. Mapping;18
8.3;§ 4. Point sets in the w-dimensional vector space Rn;20
8.4;§ 5. Topological mapping in vector spaces;27
8.5;§ 6. Systems of ordinary differential equations;29
8.6;§ 7. The Lebesgue measure;35
8.7;§ 8. The Lebesgue integral;40
8.8;§ 9. Hubert spaces;49
8.9;References;56
9;CHAPTER III. THE PHASE FLOWS OF MECHANICAL SYSTEMS;58
9.1;§ 10. Mechanical systems;58
9.2;§ 11. Phase flow; Liouville's Theorem;63
9.3;§ 12. Stationary measure-conserving phase flow; Poincaré's,Hopf's, and Jacobi's Theorems;68
9.4;§ 13. The theorems of v. Neumann and Birkhoff; the ergodic hypothesis;77
9.5;References;87
10;CHAPTER IV. THE INITIAL DISTRIBUTION OF PROBABILITY IN THE PHASE SPACE;88
10.1;§ 14. A formal description of the concept of probability;88
10.2;§ 15. On the application of the concept of probability;91
10.3;References;98
11;CHAPTER V. PROBABILITY DISTRIBUTIONS WHICH DEPEND ON TIME;99
11.1;§ 16. Mechanical systems with general equations of motion;99
11.2;§ 17. Hamiltonian and Newtonian systems;104
11.3;§ 18. The initial value problem;114
11.4;§ 19. The approach of mechanical systems towards states of statistical equilibrium;121
11.5;References;130
12;CHAPTER VI. TIME-INDEPENDENT PROBABILITY DISTRIBUTIONS;132
12.1;§ 20. Fluctuations in statistical equilibrium;132
12.2;§ 21. Gibbs's canonic probability distribution;140
12.3;References;152
13;CHAPTER VII. STATISTICAL THERMODYNAMICS;153
13.1;§ 22. The equation of state;153
13.2;§ 23. The fundamental laws of thermodynamics;164
13.3;§ 24. Entropy and probability;176
13.4;References;186
14;SUBJECT INDEX;188