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Geometry and Topology in Hamiltonian Dynamics and Statistical Mechanics

:	Marco Pettini
:	Geometry and Topology in Hamiltonian Dynamics and Statistical Mechanics
:	Springer-Verlag
:	9780387499574
:	1
:	CHF 89.50
:

:	Analysis
:	English

:	456
:	Wasserzeichen
:	PC/MAC/eReader/Tablet
:	PDF

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.

	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