Geometry and Topology in Hamiltonian Dynamics and Statistical Mechanics (Interdisciplinary Applied Mathematics Book 33)

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 â€" 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. Itisaspecialpleasureformetowritethisforewordforaremarkablebookbya remarkableauthor.MarcoPettiniisadeepthinker,whohasspentmanyyears probing the foundations of Hamiltonian chaos and statistical mechanics, in particular phase transitions, from the point of view of geometry and topology. Itisinparticularthequalityofmindoftheauthorandhisdeepphysical,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. Although topology has penetrated a number of ?elds of physics, a broad participationoftopologyintheclari?cationandprogressoffundamentalpr- lems in the above-mentioned ?elds has been lacking. The new perspectives topology gives to the above-mentioned problems are bound to help in their clari?cation and to spread to other ?elds of science. The sparsity of geometric thinking and of its use to solve fundamental problems, when compared with purely analytical methods in physics, could be relieved and made highly productive using the material discussed in this book. It is unavoidable that the physicist reader may have then to learn some new mathematics and be challenged to a new way of thinking, but with the author as a guide, he is assured of the best help in achieving this that is presently available.

This book explores the foundations of hamiltonian dynamical systems and statistical mechanics, in particular phase transitionsc, 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 Arcetri 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 oe want to use new tools for new problems. — Professor E. G. D. Cohen, Rockefeller University, New York.

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"This book explores the foundations of Hamiltonian dynamical systems and statistical mechanics, in particular phase transitions, 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 an 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 guides the reader, whether mathematician or physicist, through the background needed to understand and use these techniques."--Jacket This 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 book contains numerous illustrations throughout and it will interest both mathematicians and physicists. 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