Book contents
- Frontmatter
- Contents
- Foreword to the first edition
- Preface to the first edition
- Preface to the second edition
- Introduction
- 1 Theory of Gibbs states
- 2 Gibbs states: complements
- 3 Translation invariance. Theory of equilibrium states
- 4 Connection between Gibbs states and Equilibrium states
- 5 One-dimensional systems
- 6 Extension of the thermodynamic formalism
- 7 Statistical mechanics on Smale spaces
- Appendix A.1 Miscellaneous definitions and results
- Appendix A.2 Topological dynamics
- Appendix A.3 Convexity
- Appendix A.4 Measures and abstract dynamical systems
- Appendix A.5 Integral representations on convex compact sets
- Appendix B Open problems
- Appendix C Flows
- Appendix D Update of open problems
- References
- Index
Introduction
Published online by Cambridge University Press: 15 January 2010
- Frontmatter
- Contents
- Foreword to the first edition
- Preface to the first edition
- Preface to the second edition
- Introduction
- 1 Theory of Gibbs states
- 2 Gibbs states: complements
- 3 Translation invariance. Theory of equilibrium states
- 4 Connection between Gibbs states and Equilibrium states
- 5 One-dimensional systems
- 6 Extension of the thermodynamic formalism
- 7 Statistical mechanics on Smale spaces
- Appendix A.1 Miscellaneous definitions and results
- Appendix A.2 Topological dynamics
- Appendix A.3 Convexity
- Appendix A.4 Measures and abstract dynamical systems
- Appendix A.5 Integral representations on convex compact sets
- Appendix B Open problems
- Appendix C Flows
- Appendix D Update of open problems
- References
- Index
Summary
Generalities
The formalism of equilibrium statistical mechanics — which we shall call thermodynamic formalism — has been developed since G. W. Gibbs to describe the properties of certain physical systems. These are systems consisting of a large number of subunits (typically 1027) like the molecules of one liter of air or water. While the physical justification of the thermodynamic formalism remains quite insufficient, this formalism has proved remarkably successful at explaining facts.
In recent years it has become clear that, underlying the thermodynamic formalism, there are mathematical structures of great interest: the formalism hints at the good theorems, and to some extent at their proofs. Outside of statistical mechanics proper, the thermodynamic formalism and its mathematical methods have now been used extensively in constructive quantum field theory and in the study of certain differentiable dynamical systems (notably Anosov diffeomorphisms and flows). In both cases the relation is at an abstract mathematical level, and fairly inobvious at first sight. It is evident that the study of the physical world is a powerful source of inspiration for mathematics. That this inspiration can act in such a detailed manner is a more remarkable fact, which the reader will interpret according to his own philosophy.
The main physical problem which equilibrium statistical mechanics tries to clarify is that of phase transitions. When the temperature of water is lowered, why do its properties change first smoothly, then suddenly as the freezing point is reached?
- Type
- Chapter
- Information
- Thermodynamic FormalismThe Mathematical Structure of Equilibrium Statistical Mechanics, pp. 1 - 10Publisher: Cambridge University PressPrint publication year: 2004