Geometry and Topology in Hamiltonian Dynamics and Statistical Mechanics

Marco Pettini (auth.)9780387308920, 038730892X, 0387499571, 9780387499574

This book explores the foundations of hamiltonian dynamical systems and statistical mechanics, in particular phase transition, from the point of view of geometry and topology. A broad participation of topology in these fields has been lacking and this book will provide a welcome overview of the current research in the area, in which the author himself is a pioneer. Using geometrical thinking to solve fundamental problems in these areas, compared to the purely analytical methods usually used in physics could be highly productive. The author skillfully guides the reader, whether mathematician or physicists through the background needed to understand and use these techniques.

Dr. Marco Pettini is affiliated with the Istituto Nazionale di Astrofisica — Osservatorio Astrofisico di Arretri in Firenze, Italy.

From the foreword:

“It is in particular the quality of mind of the author and his deep physical, as well as mathematical insights, which make this book so special and inspiring. It is a “must” for those who want to venture into a new approach to old problems or want to use new tools for new problems.” — Professor E. G. D. Cohen, Rockefellar University, New York.

Table of contents :
Front Matter….Pages I-XVI
Introduction….Pages 1-16
Background in Physics….Pages 17-101
Geometrization of Hamiltonian Dynamics….Pages 103-127
Integrability….Pages 129-143
Geometry and Chaos….Pages 145-188
Geometry of Chaos and Phase Transitions….Pages 189-202
Topological Hypothesis on the Origin….Pages 203-227
Geometry, Topology and Thermodynamics….Pages 229-243
Phase Transitions and Topology: Necessity Theorems….Pages 245-296
Phase Transitions and Topology: Exact Results….Pages 297-346
Future Developments….Pages 347-360
Back Matter….Pages 361-452