Geometry and Topology in Hamiltonian Dynamics and Statistical Mechanics (eBook)

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.

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