Geometry and Topology in Hamiltonian Dynamics and Statistical Mechanics
BeschreibungThis book covers a new explanation of the origin of Hamiltonian chaos and its quantitative characterization. The author focuses on two main areas: Riemannian formulation of Hamiltonian dynamics, providing an original viewpoint about the relationship between geodesic instability and curvature properties of the mechanical manifolds; and a topological theory of thermodynamic phase transitions, relating topology changes of microscopic configuration space with the generation of singularities of thermodynamic observables. The two areas are strongly related because the geometrization of microscopic dynamics, which is the ultimate physical source of phase transitions, naturally leads to investigate how geometry and topology of the mechanical manifolds have to change to induce a phase transition. Mathematicians and physicists working in this area will find this book of interest. TOC:Foreword.- Preface.- Introduction.- Background 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.- Appendix 1: Elements of geometry and topology of differentiable manifolds.- Appendix 2: Elements of Riemannian geometry.- Appendix 3: Summary of elementary Morse theory.- References.
InhaltsverzeichnisBackground in Physics.
Geometrization of Hamiltonian Dynamics.
Geometry and Chaos.
Geometry of Chaos and Phase Transitions.
Topological Hypothesis on the Origin.
Geometry, Topology and Thermodynamics.
Phase Transitions and Topology: Necessity Theorems.
Phase Transitions and Topology: Exact Results.
PortraitThe author is one of few pioneering individuals in this recently emerged important research area. His book will be a unique contribution to the field.
From the reviews:
"The present book is an excellent synthesis of two basic topics in classical applied mathematics: Hamiltonian dynamics, with a special view towards the Hamiltonian chaos, and statistical mechanics, mainly for what concerns phase transition phenomena in systems described by realistic intermolecular or interatomic forces. The perfect conclusion appears in a Foreword written by E.G.D. Cohen: "this book makes a courageous attempt to clarify these fundamental phenomena in a new way.""