Pettini M. Geometry and Topology in Hamiltonian Dynamics and Statistical Mechanics

New York: Springer Science+Business Media. – 2007. – 459 p. (Interdisciplinary Applied Mathematics Volume 33) Phase transitions are among the most impressive phenomena occurring in nature. They are an example of emergent behavior, i.e., of collective properties having no direct counterpart in the dynamics or structure of individual atoms or molecules: to give a familiar example, the molecules of ice and liquid water are identical and interact with the same laws of force, despite their remarkably different macroscopic properties. The present book is a monograph committed to a synthesis of two basic topics in physics: Hamiltonian dynamics, with all its richness unveiled since the famous numerical experiment of Fermi and coworkers at Los Alamos, and statistical mechanics, mainly for what concerns phase transition phenomena in systems described by realistic interatomic or intermolecular forces. The first part of the book is aimed at a reader who is familiar with the basics of Riemannian geometry, for example at the level of a course in general relativity. As to the second part, a knowledge of Morse theory and de Rham’s cohomology theory at an elementary level is assumed.ForewordBackground in Physics
Geometrization of Hamiltonian Dynamics
Integrability
Geometry and Chaos
Geometry of Chaos and Phase Transitions
Topological Hypothesis on the Origin of Phase Transitions
Geometry, Topology and Thermodynamics
Phase Transitions and Topology: Necessity Theorems
Phase Transitions and Topology: Exact Results
Future Developments
Elements of Geometry and Topology of Differentiable Manifolds
Elements of Riemannian Geometry
Summary of Elementary Morse TheoryAuthor Index
Subject Index