دانلود کتاب Geometry and topology in Hamiltonian dynamics and statistical mechanics

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Cover......Page 1
Interdisciplinary Applied Mathematics, Volume 33......Page 2
Geometry and Topologyin HamiltonianDynamics and StatisticalMechanics......Page 4
ISBN-13: 9780387308920......Page 5
Foreword......Page 7
Preface......Page 9
Contents......Page 13
1. Introduction......Page 17
2. Background in Physics......Page 33
2.1 Statistical Mechanics......Page 34
2.1.1 Invariant Measure for the Dynamics......Page 35
2.1.2 Invariant Measure Induced on Σ[sub(E)]......Page 38
2.1.3 The Irreversible Approach to Equilibrium. The Zeroth Law of Thermodynamics......Page 39
2.1.4 Ergodicity......Page 42
2.1.5 From Micro to Macro: The Link with Thermodynamics......Page 45
2.1.6 Phase Transitions......Page 54
2.2.1 Perturbative Results for Quasi-integrable Systems......Page 71
2.2.2 Hamiltonian Chaos......Page 77
2.2.3 Lyapunov Exponents......Page 86
2.3 Dynamics and Statistical Mechanics......Page 93
2.3.1 Numerical Hamiltonian Dynamics at Large N......Page 95
2.3.2 Numerical Investigation of Phase Transitions......Page 106
3.1 Geometric Formulation of the Dynamics......Page 118
3.1.1 Jacobi Metric on Configuration Space M......Page 119
3.1.2 Eisenhart Metric on Enlarged Configuration Space M × R......Page 123
3.1.3 Eisenhart Metric on Enlarged Configuration Space-Time M × R[sup(2)]......Page 125
3.2 Finslerian Geometrization of Hamiltonian Dynamics......Page 127
3.3 Sasaki Lift on TM......Page 130
3.4 Curvature of the Mechanical Manifolds......Page 132
3.5 Curvature and Stability of a Geodesic Flow......Page 135
3.5.1 Concluding Remark......Page 141
4.1 Introduction......Page 143
4.2 Killing Vector Fields......Page 145
4.3 Killing Tensor Fields......Page 147
4.4 Explicit KTFs of Known Integrable Systems......Page 149
4.4.1 Nontrivial Integrable Models......Page 150
4.4.2 The Special Case of the N = 2 Toda Model......Page 152
4.4.3 The Generalized Hénon–Heiles Model......Page 154
4.5 Open Problems......Page 156
5.1 Geometric Approach to Chaotic Dynamics......Page 158
5.2 Geometric Origin of Hamiltonian Chaos......Page 160
5.3 Effective Stability Equation in the High-Dimensional Case......Page 163
5.3.1 A Geometric Formula for the Lyapunov Exponent......Page 168
5.4 Some Applications......Page 172
5.4.1 FPU β Model......Page 174
5.4.2 The Role of Unstable Periodic Orbits......Page 178
5.4.3 1D XY Model......Page 184
5.4.4 Mean-Field XY Model......Page 190
5.5 Some Remarks......Page 194
5.5.1 Beyond Quasi-Isotropy: Chaos and Nontrivial Topology......Page 196
5.6 A Technical Remark on the Stochastic Oscillator Equation......Page 198
6. Geometry of Chaos and Phase Transitions......Page 202
6.1 Chaotic Dynamics and Phase Transitions......Page 203
6.2 Curvature and Phase Transitions......Page 209
6.3 The Mean-Field XY Model......Page 213
7. Topological Hypothesis on the Origin of Phase Transitions......Page 216
7.1 From Geometry to Topology: Abstract Geometric Models......Page 217
7.2 Topology Changes in Configuration Space and Phase Transitions......Page 220
7.3 Indirect Numerical Investigations of the Topology of Configuration Space......Page 221
7.4 Topological Origin of the Phase Transition in the Mean-Field XY Model......Page 227
7.5 The Topological Hypothesis......Page 231
7.6 Direct Numerical Investigations of the Topology of Configuration Space......Page 233
7.6.1 Monte Carlo Estimates of Geometric Integrals......Page 235
7.6.2 Euler Characteristic for the Lattice φ[sup(4)] Model......Page 237
8. Geometry, Topology and Thermodynamics......Page 241
8.1 Extrinsic Curvatures of Hypersurfaces......Page 244
8.1.1 Two Useful Derivation Formulas......Page 247
8.2 Geometry, Topology and Thermodynamics......Page 249
8.2.1 An Alternative Derivation......Page 254
9. Phase Transitions and Topology: Necessity Theorems......Page 256
9.1 Basic Definitions......Page 260
9.2 Main Theorems: Theorem 1......Page 265
9.3 Proof of Lemma 2, Smoothness of the Structure Integral......Page 269
9.4 Proof of Lemma 9.18, Upper Bounds......Page 270
9.4.1 Part A......Page 273
9.4.2 Part B......Page 280
9.5 Main Theorems: Theorem 2......Page 292
10. Phase Transitions and Topology: Exact Results......Page 308
10.1.1 Canonical Ensemble Thermodynamics......Page 309
10.1.2 Microcanonical Ensemble Thermodynamics......Page 311
10.1.3 Analytic Computation of the Euler Characteristic......Page 313
10.2 The One-Dimensional XY Model......Page 320
10.2.1 The Role of the External Field h......Page 325
10.3 Two-Dimensional Toy Model of Topological Changes......Page 326
10.4.1 Mean–Field XY Model......Page 329
10.4.2 One-Dimensional XY Model......Page 334
10.5 The k-Trigonometric Model......Page 336
10.5.1 Canonical Ensemble Thermodynamics......Page 338
10.5.2 Microcanonical Thermodynamics......Page 343
10.5.3 Topology of Configuration Space......Page 346
10.5.4 Topology of the Order Parameter Space......Page 351
10.6 Comments on Other Exact Results......Page 353
11. Future Developments......Page 358
11.1 Theoretical Developments......Page 359
11.2 Transitional Phenomena in Finite Systems......Page 362
11.3 Complex Systems......Page 363
11.4 Polymers and Proteins......Page 364
11.5 A Glance at Quantum Systems......Page 369
A.1 Tensors......Page 372
A.1.1 Symmetrizer and Antisymmetrizer......Page 375
A.2 Grassmann Algebra......Page 376
A.3.1 Topological Spaces......Page 378
A.3.2 Manifolds......Page 379
A.4 Calculus on Manifolds......Page 381
A.4.1 Vectors......Page 382
A.4.2 Flows and Lie Derivatives......Page 384
A.4.3 Tensors and Forms on Manifolds......Page 385
A.4.4 Exterior Derivatives......Page 389
A.4.5 Interior Product......Page 390
A.4.6 Integration of Forms on Manifolds......Page 391
A.5 The Fundamental Group......Page 392
A.6.1 Homology Groups......Page 396
A.6.2 Cohomology Groups......Page 400
A.6.3 Betti Numbers......Page 404
B.1 Riemannian Manifolds......Page 408
B.1.1 Riemannian Metrics on Differentiable Manifolds......Page 409
B.2 Linear Connections and Covariant Differentiation......Page 411
B.2.1 Geodesics......Page 415
B.2.2 The Exponential Map......Page 416
B.3 Curvature......Page 417
B.3.1 Sectional Curvature......Page 419
B.3.2 Isotropic Manifolds......Page 421
B.4 The Jacobi–Levi-Civita Equation for Geodesic Spread......Page 423
B.5 Topology and Curvature......Page 427
B.5.1 The Gauss–Bonnet Theorem......Page 428
B.5.2 Hopf-Rinow Theorem......Page 429
Appendix C: Summary of Elementary Morse Theory......Page 432
C.1 The Non-Critical Neck Theorem......Page 434
C.1.1 Critical Points and Topological Changes......Page 435
C.1.2 Morse Inequalities......Page 439
References......Page 441
Author Index......Page 450
C......Page 451
D......Page 452
G......Page 453
I......Page 454
M......Page 455
O......Page 456
R......Page 457
T......Page 458
Z......Page 459