Pettini Geometry and Topology in Hamiltonian Dynamics and Statistical Mechanics

1. Auflage 2007
ISBN: 978-0-387-49957-4
Verlag: Springer-Verlag
Format: PDF
Kopierschutz: Wasserzeichen (»Systemvoraussetzungen)

E-Book, Englisch, 456 Seiten

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.

The author is one of few pioneering individuals in this recently emerged important research area. His book will be a unique contribution to the field.

Autoren/Hrsg.

Pettini, Marco

Inhaltsverzeichnis

1;Foreword;7
2;Preface;9
3;Contents;13
4;Chapter 1 Introduction;17
5;Chapter 2 Background in Physics;33
5.1;2.1 Statistical Mechanics;34
5.2;2.2 Hamiltonian Dynamics;71
5.3;2.3 Dynamics and Statistical Mechanics;93
6;Chapter 3 Geometrization of Hamiltonian Dynamics;118
6.1;3.1 Geometric Formulation of the Dynamics;118
6.2;3.2 Finslerian Geometrization of Hamiltonian Dynamics;127
6.3;3.3 Sasaki Lift on TM;130
6.4;3.4 Curvature of the Mechanical Manifolds;132
6.5;3.5 Curvature and Stability of a Geodesic Flow;135
7;Chapter 4 Integrability;143
7.1;4.1 Introduction;143
7.2;4.2 Killing Vector Fields;145
7.3;4.3 Killing Tensor Fields;147
7.4;4.4 Explicit KTFs of Known Integrable Systems;149
7.5;4.5 Open Problems;156
8;Chapter 5 Geometry and Chaos;158
8.1;5.1 Geometric Approach to Chaotic Dynamics;158
8.2;5.2 Geometric Origin of Hamiltonian Chaos;160
8.3;5.3 Effective Stability Equation in the High-Dimensional Case;163
8.4;5.4 Some Applications;172
8.5;5.5 Some Remarks;194
8.6;5.6 A Technical Remark on the Stochastic Oscillator Equation;198
9;Chapter 6 Geometry of Chaos and Phase Transitions;202
9.1;6.1 Chaotic Dynamics and Phase Transitions;203
9.2;6.2 Curvature and Phase Transitions;209
9.3;6.3 The Mean-Field XY Model;213
10;Chapter 7 Topological Hypothesis on the Origin of Phase Transitions;216
10.1;7.1 From Geometry to Topology: Abstract Geometric Models;217
10.2;7.2 Topology Changes in Configuration Space and Phase Transitions;220
10.3;7.3 Indirect Numerical Investigations of the Topology of Configuration Space;221
10.4;7.4 Topological Origin of the Phase Transition in the Mean- Field XY Model;227
10.5;7.5 The Topological Hypothesis;231
10.6;7.6 Direct Numerical Investigations of the Topology of Configuration Space;233
11;Chapter 8 Geometry, Topology and Thermodynamics;241
11.1;8.1 Extrinsic Curvatures of Hypersurfaces;244
11.2;8.2 Geometry, Topology and Thermodynamics;249
12;Chapter 9 Phase Transitions and Topology: Necessity Theorems;256
12.1;9.1 Basic Definitions;260
12.2;9.2 Main Theorems: Theorem 1;265
12.3;9.3 Proof of Lemma 2, Smoothness of the Structure Integral;269
12.4;9.4 Proof of Lemma 9.18, Upper Bounds;270
12.5;9.5 Main Theorems: Theorem 2;292
13;Chapter 10 Phase Transitions and Topology: Exact Results;308
13.1;10.1 The Mean-Field XY Model;309
13.2;10.2 The One-Dimensional XY Model;320
13.3;10.3 Two-Dimensional Toy Model of Topological Changes;326
13.4;10.4 Technical Remark on the Computation of the Indexes of the Critical Points;329
13.5;10.5 The k-Trigonometric Model;336
13.6;10.6 Comments on Other Exact Results;353
14;Chapter 11 Future Developments;358
14.1;11.1 Theoretical Developments;359
14.2;11.2 Transitional Phenomena in Finite Systems;362
14.3;11.3 Complex Systems;363
14.4;11.4 Polymers and Proteins;364
14.5;11.5 A Glance at Quantum Systems;369
15;Appendix A Elements of Geometry and Topology of Differentiable Manifolds;372
15.1;A.1 Tensors;372
15.2;A.2 Grassmann Algebra;376
15.3;A.3 Differentiable Manifolds;378
15.4;A.4 Calculus on Manifolds;381
15.5;A.5 The Fundamental Group;392
15.6;A.6 Homology and Cohomology;396
16;Appendix B Elements of Riemannian Geometry;408
16.1;B.1 Riemannian Manifolds;408
16.2;B.2 Linear Connections and Covariant Differentiation;411
16.3;B.3 Curvature;417
16.4;B.4 The Jacobi–Levi-Civita Equation for Geodesic Spread;423
16.5;B.5 Topology and Curvature;427
17;Appendix C Summary of Elementary Morse Theory;432
17.1;C.1 The Non-Critical Neck Theorem;434
18;References;441
19;Author Index;450
20;Subject Index;451