Foreword ...................................................... VII
Preface ........................................................ IX
1 Introduction ................................................. 1
2 Background in Physics ....................................... 17
2.1 Statistical Mechanics .................................. 18
2.1.1 Invariant Measure for the Dynamics .............. 19
2.1.2 Invariant Measure Induced on ∑E ................. 22
2.1.3 The Irreversible Approach to Equilibrium. The
Zeroth Law of Thermodynamics .................... 23
2.1.4 Ergodicity ...................................... 26
2.1.5 From Micro to Macro: The Link with
Thermodynamics .................................. 29
2.1.6 Phase Transitions ............................... 38
2.2 Hamiltonian Dynamics ................................... 55
2.2.1 Perturbative Results for Quasi-integrable
Systems ......................................... 55
2.2.2 Hamiltonian Chaos ............................... 61
2.2.3 Lyapunov Exponents .............................. 70
2.3 Dynamics and Statistical Mechanics ..................... 77
2.3.1 Numerical Hamiltonian Dynamics at Large N ....... 79
2.3.2 Numerical Investigation of Phase Transitions .... 90
3 Geometrization of Hamiltonian Dynamics ..................... 103
3.1 Geometric Formulation of the Dynamics ................. 103
3.1.1 Jacobi Metric on Configuration Space M ......... 104
3.1.2 Eisenhart Metric on Enlarged Configuration
Space M ×  .................................... 108
3.1.3 Eisenhart Metric on Enlarged Configuration
Space-Time M × 2 .............................. 110
3.2 Finslerian Geometrization of Hamiltonian Dynamics ..... 112
3.3 Sasaki Lift on TM ..................................... 115
3.4 Curvature of the Mechanical Manifolds ................. 117
3.5 Curvature and Stability of a Geodesic Flow ............ 120
3.5.1 Concluding Remark .............................. 126
4 Integrability .............................................. 129
4.1 Introduction .......................................... 129
4.2 Killing Vector Fields ................................. 131
4.3 Killing Tensor Fields ................................. 133
4.4 Explicit KTFs of Known Integrable Systems ............. 135
4.4.1 Nontrivial Integrable Models ................... 136
4.4.2 The Special Case of the N = 2 Toda Model ....... 138
4.4.3 The Generalized Henon Heiles Model ............. 140
4.5 Open Problems ......................................... 142
5 Geometry and Chaos ......................................... 145
5.1 Geometric Approach to Chaotic Dynamics ................ 145
5.2 Geometric Origin of Hamiltonian Chaos ................. 147
5.3 Effective Stability Equation in the High-Dimensional
Case .................................................. 150
5.3.1 A Geometric Formula for the Lyapunov
Exponent ....................................... 155
5.4 Some Applications ..................................... 159
5.4.1 FPU β Model .................................... 161
5.4.2 The Role of Unstable Periodic Orbits ........... 165
5.4.3 ID XY Model .................................... 171
5.4.4 Mean-Field XY Model ............................ 177
5.5 Some Remarks .......................................... 181
5.5.1 Beyond Quasi-Isotropy: Chaos and Nontrivial
Topology ....................................... 183
5.6 A Technical Remark on the Stochastic Oscillator
Equation .............................................. 185
6 Geometry of Chaos and Phase Transitions .................... 189
6.1 Chaotic Dynamics and Phase Transitions ................ 190
6.2 Curvature and Phase Transitions ....................... 196
6.2.1 Geometric Estimate of the Lyapunov Exponent .... 200
6.3 The Mean-Field XY Model ............................... 200
7 Topological Hypothesis on the Origin of Phase
Transitions ................................................ 203
7.1 From Geometry to Topology: Abstract Geometric
Models ................................................ 204
7.2 Topology Changes in Configuration Space and Phase
Transitions ........................................... 207
7.3 Indirect Numerical Investigations of the Topology
of Configuration Space ................................ 208
7.4 Topological Origin of the Phase Transition in the
Mean-Field XY Model ................................... 214
7.5 The Topological Hypothesis ............................ 218
7.6 Direct Numerical Investigations of the Topology
of Configuration Space ................................ 220
7.6.1 Monte Carlo Estimates of Geometric Integrals ... 222
7.6.2 Euler Characteristic for the Lattice Ø4
Model .......................................... 224
8 Geometry, Topology and Thermodynamics ...................... 229
8.1 Extrinsic Curvatures of Hypersurfaces ................. 232
8.1.1 Two Useful Derivation Formulas ................. 235
8.2 Geometry, Topology and Thermodynamics ................. 237
8.2.1 An Alternative Derivation ...................... 242
9 Phase Transitions and Topology: Necessity Theorems ......... 245
9.1 Basic Definitions ..................................... 249
9.2 Main Theorems: Theorem 1 .............................. 254
9.3 Proof of Lemma 2, Smoothness of the Structure
Integral .............................................. 258
9.4 Proof of Lemma 9.18, Upper Bounds ..................... 259
9.4.1 Part A ......................................... 262
9.4.2 Part В ......................................... 269
9.5 Main Theorems: Theorem 2 .............................. 281
10 Phase Transitions and Topology: Exact Results .............. 297
10.1 The Mean-Field XY Model ............................... 298
10.1.1 Canonical Ensemble Thermodynamics ............. 298
10.1.2 Microcanonical Ensemble Thermodynamics ........ 300
10.1.3 Analytic Computation of the Euler
Characteristic ................................ 302
10.2 The One-Dimensional XY Model .......................... 309
10.2.1 The Role of the External Field h ............... 314
10.3 Two-Dimensional Toy Model of Topological Changes ...... 315
10.4 Technical Remark on the Computation of the Indexes
of the Critical Points ................................ 318
10.4.1 Mean-Field XY Model ............................ 318
10.4.2 One-Dimensional XY Model ....................... 323
10.5 The fc-Trigonometric Model ............................ 325
10.5.1 Canonical Ensemble Thermodynamics .............. 327
10.5.2 Microcanonical Thermodynamics .................. 332
10.5.3 Topology of Configuration Space ................ 335
10.5.4 Topology of the Order Parameter Space .......... 340
10.6 Comments on Other Exact Results ....................... 342
11 Future Developments ........................................ 347
11.1 Theoretical Developments .............................. 348
11.2 Transitional Phenomena in Finite Systems .............. 351
11.3 Complex Systems ....................................... 352
11.4 Polymers and Proteins ................................. 353
11.5 A Glance at Quantum Systems ........................... 358
Appendix A: Elements of Geometry and Topology of
Differentiable Manifolds ...................................... 361
A.l Tensors ............................................... 361
A.1.1 Symmetrizer and Antisymmetrizer ................ 364
A.2 Grassmann Algebra ..................................... 365
A.3 Differentiable Manifolds .............................. 367
A.3.1 Topological Spaces ............................. 367
A.3.2 Manifolds ...................................... 368
A.4 Calculus on Manifolds ................................. 370
A.4.1 Vectors ........................................ 371
A.4.2 Flows and Lie Derivatives ...................... 373
A.4.3 Tensors and Forms on Manifolds ................. 374
A.4.4 Exterior Derivatives ........................... 378
A.4.5 Interior Product ............................... 379
A.4.6 Integration of Forms on Manifolds .............. 380
A.5 The Fundamental Group ................................. 381
A.6 Homology and Cohomology ............................... 385
A.6.1 Homology Groups ................................ 385
A.6.2 Cohomology Groups .............................. 389
A.6.3 Betti Numbers .................................. 393
Appendix B: Elements of Riemannian Geometry ................... 397
B.l Riemannian Manifolds .................................. 397
B.l.l Riemannian Metrics on Differentiable
Manifolds ...................................... 398
B.2 Linear Connections and Covariant Differentiation ...... 400
B.2.1 Geodesies ...................................... 404
B.2.2 The Exponential Map ............................ 405
B.3 Curvature ............................................. 406
B.3.1 Sectional Curvature ............................ 408
B.3.2 Isotropic Manifolds ............................ 410
B.4 The Jacobi-Levi-Civita Equation for Geodesic Spread ... 412
B.5 Topology and Curvature ................................ 416
B.5.1 The Gauss-Bonnet Theorem ....................... 417
B.5.2 Hopf-Rinow Theorem ............................. 418
Appendix C: Summary of Elementary Morse Theory ................ 421
C.l The Non-Critical Neck Theorem ......................... 423
C.l.l Critical Points and Topological Changes ........ 424
C.1.2 Morse Inequalities ............................. 428
References .................................................... 431
Author Index .................................................. 441
Subject Index ................................................. 443
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