Geometry and topology in Hamiltonian dynamics and statistical mechanics / Marco Pettini

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.

Since the usual framework of the Hamiltonian mechanics is the symplectic geometry, the novelty of the theoretical proposal put forward in this monograph stems from a main rôle offered to the Riemannian geometry approach through two remarkable equations: 1) the Euler-Lagrange equations of the geodesics as describing the natural motion of a Hamiltonian system via the Legendre transform and 2) the Jacobi (-Levi-Civita) equation for the geodesic deviation vector field. On this way, a negatively curved (hyperbolic) compact manifold provides an interesting example of chaotic geodesic motion. In the second part, devoted to connections topology-phase transitions, some soft mathematical tools like Morse theory and de Rham’s cohomology are used to study hard physical processes. A very interesting last chapter “Future Developments” opens a door to some complex systems like polymers and proteins and unveils potential applications in quantum theory.
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.” (Zentralblatt)