Chaos, Scattering and Statistical Mechanics

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ISBN: 9780521395113 | 0521395119
Cover: Hardcover
Copyright: 6/28/1998

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This book describes recent advances in the application of chaos theory to classical scattering and nonequilibrium statistical mechanics generally, and to transport by deterministic diffusion in particular. The author presents the basic tools of dynamical systems theory, such as dynamical instability, topological analysis, periodic-orbit methods, Liouvillian dynamics, dynamical randomness and large-deviation formalism. These tools are applied to chaotic scattering and to transport in systems near equilibrium and maintained out of equilibrium. This book will be bought by researchers interested in chaos, dynamical systems, chaotic scattering, and statistical mechanics in theoretical, computational and mathematical physics and also in theoretical chemistry.

Preface

xvii

Introduction

(11)

Chapter 1 Dynamical systems and their linear stability

(31)

1.1 Dynamics in phase space

(3)

1.1.1 The group of time evolutions

(1)

1.1.2 The Poincare map

(1)

1.1.3 Hamiltonian systems

(1)

1.1.4 Billiards

(1)

1.2 Linear stability and the tangent space

(8)

1.2.1 The fundamental matrix

(1)

1.2.2 Lyapunov exponents

(1)

1.2.3 Decomposition of the tangent space into orthogonal directions

(2)

1.2.4 Homological decomposition of the multiplicative cocycle

(1)

1.2.5 The local stretching rates

(2)

1.2.6 Stable and unstable manifolds

(1)

1.3 Linear stability of Hamiltonian systems

(8)

1.3.1 Symplectic dynamics and the pairing rule of the Lyapunov exponents

(1)

1.3.2 Vanishing Lyapunov exponents and the Lie group of continuous symmetries

(2)

1.3.3 Hamilton-Jacobi equation and the curvature of the wavefront

(3)

1.3.4 Elimination of the neutral directions

(1)

1.3.5 The local stretching rate in f = 2 hyperbolic Hamiltonian systems

(2)

1.4 Linear stability in billiards

(12)

1.4.1 Some definitions

(1)

1.4.2 The second fundamental form

(1)

1.4.3 The tangent space of the billiard

(1)

1.4.4 Free flight

(1)

1.4.5 Collision

(1)

1.4.6 Expanding and contracting horospheres

(3)

1.4.7 Two-dimensional hard-disk billiards (f = 2)

(4)

Chapter 2 Topological chaos

(24)

2.1 Topology of trajectories in phase space

(4)

2.1.1 Phase portrait and invariant set

(1)

2.1.2 Critical orbits: stationary points and periodic orbits

(1)

2.1.3 Nonwandering sets

(1)

2.1.4 Locally maximal invariant sets and global stability

(1)

2.1.5 Dense orbits and transitivity

(1)

2.1.6 Attractors, anti-attractors, and repellers

(1)

2.2 Hyperbolicity

(5)

2.2.1 Definition

(2)

2.2.2 Escape-time functions

(1)

2.2.3 Anosov and Axiom-A systems

(2)

2.3 Markov partition and symbolic dynamics in hyperbolic systems

(5)

2.3.1 Partitioning phase space with stable and unstable manifolds

(1)

2.3.2 Symbolic dynamics and shift

(2)

2.3.3 Markov topological shift

(2)

2.4 Topological entropy

(3)

2.4.1 Nets and separated subsets

(1)

2.4.2 Definition and properties

(2)

2.5 The spectrum of periodic orbits

(4)

2.5.1 Periodic orbits and the topological zeta function

(1)

2.5.2 Prime periodic orbits and fixed points

(2)

2.5.3 How to order a sum over periodic orbits?

(1)

2.5.4 The topological entropy of a hyperbolic invariant set

(1)

2.6 The topological zeta function of hyperbolic systems

(3)

2.6.1 The topological zeta function for a nonoverlapping partition

(1)

2.6.2 The topological zeta function for an overlapping partition

(2)

Chapter 3 Liouvillian dynamics

(59)

3.1 Statistical ensembles

(1)

3.2 Time evolution of statistical ensembles

(4)

3.2.1 Liouville equation

(1)

3.2.2 Frobenius-Perron and Koopman operators

(1)

3.2.3 Boundary conditions

(2)

3.3 Invariant measures

(3)

3.3.1 Definition

(1)

3.3.2 Selection of an invariant measure

(1)

3.3.3 The basic invariant measures of Hamiltonian systems

(1)

3.4 Correlation functions and spectral functions

(1)

3.5 Spectral theory on real frequencies

(7)

3.5.1 Spectral decomposition on a Hilbert space

(1)

3.5.2 Purely discrete real spectrum

(1)

3.5.3 Continuous real spectrum

(3)

3.5.4 Continuous real spectrum and the Wiener-Khinchin theorem

(1)

3.5.5 Continuous real spectrum and Gaussian fluctuations

(1)

3.6 Spectral theory on complex frequencies or resonance theory

(11)

3.6.1 Analytic continuation to complex frequencies

(4)

3.6.2 Singularities as a spectrum of generalized eigenvalues

(4)

3.6.3 Definition of a trace

(3)

3.7 Resonances of stationary points

(9)

3.7.1 Trace formula and eigenvalues

(2)

3.7.2 Eigenstates and spectral decompositions

(4)

3.7.3 Hamiltonian stationary point of saddle type

100

(3)

3.8 Resonances for hyperbolic sets with periodic orbits

103

(15)

3.8.1 Trace formula

103

(2)

3.8.2 The Selberg-Smale Zeta function

105

(5)

3.8.3 Formulation of the eigenvalue problem

110

(3)

3.8.4 Fredholm determinant

113

(1)

3.8.5 Fredholm theory for the eigenstates

114

(2)

3.8.6 Periodic-orbit averages of observables

116

(2)

3.9 Resonance spectrum at bifurcations

118

(5)

3.9.1 Pitchfork bifurcation

119

(1)

3.9.2 Hopf bifurcation

120

(2)

3.9.3 Homoclinic bifurcation in a two-dimensional flow

122

(1)

3.10 Liouvillian dynamics of one-dimensional maps

123

(3)

Chapter 4 Probabilistic chaos

126

(45)

4.1 Dynamical randomness and the entropy per unit time

126

(5)

4.1.1 A model of observation

127

(1)

4.1.2 Information redundancy and algorithmic complexity

128

(2)

4.1.3 Kolmogorov-Sinai entropy per unit time

130

(1)

4.2 The large-deviation formalism

131

(7)

4.2.1 Separated subsets

132

(1)

4.2.2 Topological pressure and the dynamical invariant measures

132

(3)

4.2.3 Pressure functions based on the Lyapunov exponents

135

(1)

4.2.4 Entropy function and Legendre transform

136

(2)

4.3 Closed hyperbolic systems

138

(4)

4.3.1 The microcanonical measure as a Sinai-Ruelle-Bowen dynamical measure

138

(2)

4.3.2 The pressure function for closed systems

140

(1)

4.3.3 Generating functions of observables and transport coefficients

141

(1)

4.4 Open hyperbolic systems

142

(10)

4.4.1 The nonequilibrium invariant measure of the repeller

143

(2)

4.4.2 Connection with the dynamical invariant measure and the pressure function

145

(3)

4.4.3 Fractal repellers in two-degrees-of-freedom systems and their dimensions

148

(4)

4.5 Generalized zeta functions

152

(4)

4.5.1 Frobenius-Perron operators in the large-deviation formalism

152

(2)

4.5.2 The topological pressure as an eigenvalue

154

(2)

4.6 Probabilistic Markov chains and lattice gas automata

156

(7)

4.6.1 Markov chain models

156

(1)

4.6.2 Isomorphism between Markov chains and area-preserving maps

157

(2)

4.6.3 Repellers of Markov chains

159

(2)

4.6.4 Large-deviation formalism of Markov chains

161

(2)

4.7 Special fractals generated by uniformly hyperbolic maps

163

(6)

4.7.1 Condition on the mean sojourn time in a domain

163

(4)

4.7.2 Fractal repeller generated in trajectory reconstruction

167

(2)

4.8 Nonhyperbolic systems

169

(2)

Chapter 5 Chaotic scattering

171

(53)

5.1 Classical scattering theory

171

(5)

5.1.1 Motivations

171

(1)

5.1.2 Classical scattering function and time delay

172

(2)

5.1.3 The different types of trajectories

174

(2)

5.1.4 Scattering operator for statistical ensembles

176

(1)

5.2 Hard-disk scatterers

176

(34)

5.2.1 Generalities

176

(1)

5.2.2 The dynamics

177

(2)

5.2.3 The Birkhoff mapping

179

(1)

5.2.4 The one-disk scatterer

180

(1)

5.2.5 The two-disk scatterer

181

(2)

5.2.6 The three-disk scatterer

183

(18)

5.2.7 The four-disk scatterer

201

(9)

5.3 Hamiltonian mapping of scattering type

210

(6)

5.3.1 Definition of the model

210

(1)

5.3.2 Metamorphoses of the phase portraits

211

(4)

5.3.3 Characteristic quantities of the chaotic repeller

215

(1)

5.4 Application to the molecular transition state

216

(5)

5.4.1 Model of photodissociation of HgI(2)

216

(2)

5.4.2 Transition from a periodic to a chaotic repeller

218

(1)

5.4.3 Three-branched Smale repeller and its characterization

219

(2)

5.5 Further applications of chaotic scattering

221

(3)

Chapter 6 Scattering theory of transport

224

(51)

6.1 Scattering and transport

224

(1)

6.2 Diffusion and chaotic scattering

225

(9)

6.2.1 Large scatterers and the diffusion equation

225

(1)

6.2.2 Escape-time function and escape rate

226

(3)

6.2.3 Phenomenology of the escape process

229

(2)

6.2.4 The escape-rate formula for diffusion

231

(3)

6.3 The periodic Lorentz gas

234

(16)

6.3.1 Definition

234

(1)

6.3.2 The Liouville invariant measure

235

(1)

6.3.3 Finite and infinite horizons

236

(1)

6.3.4 Chaotic properties of the infinite Lorentz gas

237

(7)

6.3.5 The open Lorentz gas

244

(6)

6.4 The multibaker mapping

250

(7)

6.4.1 Definition

250

(4)

6.4.2 The Pollicott-Ruelle resonances and the escape-rate formula

254

(3)

6.5 Escape-rate formalism for general transport coefficients

257

(7)

6.5.1 General context

257

(1)

6.5.2 Transport coefficients and their Helfand moments

258

(3)

6.5.3 Generalization of the escape-rate formula

261

(3)

6.6 Escape-rate formalism for chemical reaction rates

264

(4)

6.6.1 Nonequilibrium thermodynamics of chemical reactions

265

(1)

6.6.2 The master equation approach

266

(2)

6.7 Discussion

268

(7)

6.7.1 Summary

268

(1)

6.7.2 Further applications of the escape-rate formalism

269

(1)

6.7.3 Relation to the thermostatted-system approach

270

(2)

6.7.4 The escape-rate formalism in the presence of external forces

272

(3)

Chapter 7 Hydrodynamic modes of diffusion

275

(68)

7.1 Hydrodynamics from Liouvillian dynamics

275

(3)

7.1.1 Historical background and motivation

275

(1)

7.1.2 Pollicott-Ruelle resonances and quasiperiodic boundary conditions

276

(2)

7.2 Liouvillian dynamics for systems symmetric under a group of spatial translations

278

(15)

7.2.1 Introduction

278

(1)

7.2.2 Suspended flows of infinite spatial extension

279

(3)

7.2.3 Assumptions on the properties of the mapping

282

(1)

7.2.4 Invariant measures

283

(1)

7.2.5 Time-reversal symmetry

284

(1)

7.2.6 The Frobenius-Perron operator on the infinite lattice

284

(1)

7.2.7 Spatial Fourier transforms

285

(1)

7.2.8 The Frobenius-Perron operators in the wavenumber subspaces

286

(2)

7.2.9 Time-reversal symmetry for the k-components

288

(1)

7.2.10 Reduction to the Frobenius-Perron operator of the mapping

289

(1)

7.2.11 Eigenvalue problem and zeta function

289

(2)

7.2.12 Consequences of time-reversal symmetry on the resonances

291

(1)

7.2.13 Relation to the eigenvalue problem for the flow

292

(1)

7.3 Deterministic diffusion

293

(12)

7.3.1 Introduction

293

(1)

7.3.2 Mean drift

294

(1)

7.3.3 The first derivative of the eigenstate with respect to the wavenumber

295

(1)

7.3.4 Diffusion matrix

296

(1)

7.3.5 Higher-order diffusion coefficients

297

(2)

7.3.6 Eigenvalues and the Van Hove function

299

(3)

7.3.7 Periodic-orbit formula for the diffusion coefficient

302

(1)

7.3.8 Consequences of the lattice symmetry under a point group

303

(2)

7.4 Deterministic diffusion in the periodic Lorentz gas

305

(7)

7.4.1 Properties of the infinite Lorentz gas

305

(4)

7.4.2 Diffusion and its dispersion relation

309

(1)

7.4.3 Cumulative functions of the eigenstates

310

(2)

7.5 Deterministic diffusion in the periodic multibaker

312

(22)

7.5.1 Properties of the periodic multibaker

312

(1)

7.5.2 The Frobenius-Perron operator and its Pollicott-Ruelle resonances

313

(3)

7.5.3 Generalized spectral decomposition

316

(2)

7.5.4 Analysis of the associated one-dimensional map

318

(8)

7.5.5 The root states of the two-dimensional map

326

(8)

7.6 Extensions to the other transport processes

334

(6)

7.6.1 Gaussian fluctuations of the Helfand moments

334

(2)

7.6.2 The minimal models of transport

336

(1)

7.6.3 Viscosity and self-diffusion in two-particle fluids

337

(2)

7.6.4 Spontaneous symmetry breaking and Goldstone hydrodynamic modes

339

(1)

7.7 Chemio-hydrodynamic modes

340

(3)

Chapter 8 Systems maintained out of equilibrium

343

(44)

8.1 Nonequilibrium systems in Liouvillian dynamics

343

(2)

8.2 Nonequilibrium steady states of diffusion

345

(5)

8.2.1 Phenomenological description of the steady states

345

(1)

8.2.2 Deterministic description of the steady states

346

(4)

8.3 From the hydrodynamic modes to the nonequilibrium steady states

350

(11)

8.3.1 From the eigenstates to the nonequilibrium steady states

350

(2)

8.3.2 Microscopic current and Fick's law

352

(1)

8.3.3 Nonequilibrium steady states of the periodic Lorentz gas

353

(2)

8.3.4 Nonequilibrium steady states of the periodic multibaker

355

(4)

8.3.5 Nonequilibrium steady states of the Langevin process

359

(2)

8.4 From the finite to the infinite multibaker

361

(5)

8.5 Generalization to the other transport processes

366

(2)

8.6 Entropy production

368

(17)

8.6.1 Irreversible thermodynamics and the problem of entropy production

370

(1)

8.6.2 Comparison with deterministic schemes

371

(2)

8.6.3 Open systems and their Poisson suspension

373

(3)

8.6.4 The XXX-entropy

376

(2)

8.6.5 Entropy production in the multibaker map

378

(5)

8.6.6 Summary

383

(2)

8.7 Comments on far-from-equilibrium systems

385

(2)

Chapter 9 Noises as microscopic chaos

387

(46)

9.1 Differences and similarities between noises and chaos

387

(3)

9.2 (XXX,XXX)-entropy per unit time

390

(7)

9.2.1 Dynamical processes

390

(1)

9.2.2 Entropy of a process over a time interval T and a partition XXX

391

(1)

9.2.3 Partition (XXX, XXX)-entropy per unit time

392

(1)

9.2.4 Cohen-Procaccia (XXX, XXX)-entropy per unit time

393

(1)

9.2.5 Shannon-Kolmogorov (XXX, XXX)-entropy per unit time

394

(3)

9.3 Time random processes

397

(18)

9.3.1 Deterministic processes

397

(1)

9.3.2 Bernoulli and Markov chains

398

(1)

9.3.3 Birth-and-death processes

399

(3)

9.3.4 Time-discrete, amplitude-continuous random processes

402

(4)

9.3.5 Time- and amplitude-continuous random processes

406

(5)

9.3.6 White noise

411

(1)

9.3.7 Levy flights

411

(1)

9.3.8 Classification of the time random processes

412

(3)

9.4 Spacetime random processes

415

(4)

9.4.1 (XXX, XXX)-entropy per unit time and volume

415

(1)

9.4.2 Deterministic cellular automata

415

(1)

9.4.3 Lattice gas automata

416

(1)

9.4.4 Coupled map lattices

416

(1)

9.4.5 Nonlinear partial differential equations

417

(1)

9.4.6 Stochastic spin dynamics

417

(1)

9.4.7 Spacetime Gaussian fields

417

(1)

9.4.8 Sporadic spacetime random processes

419

(1)

9.4.9 Classification of spacetime random processes

419

(1)

9.5 Random processes of statistical mechanics

419

(10)

9.5.1 Ideal gases

420

(4)

9.5.2 The Lorentz gases and the hard-sphere gases

424

(1)

9.5.3 The Boltzmann-Lorentz process

425

(4)

9.6 Brownian motion and microscopic chaos

429

(4)

9.6.1 Hamiltonian and Langevin models of Brownian motion

429

(1)

9.6.2 A lower bound on the positive Lyapunov exponents

430

(1)

9.6.3 Some conclusions

431

(2)

Chapter 10 Conclusions and perspectives

433

(25)

10.1 Overview of the results

433

(19)

10.1.1 From dynamical instability to statistical ensembles

433

(5)

10.1.2 Dynamical chaos

438

(3)

10.1.3 Fractal repellers and chaotic scattering

441

(1)

10.1.4 Scattering theory of transport

442

(1)

10.1.5 Relaxation to equilibrium

442

(5)

10.1.6 Nonequilibrium steady states and entropy production

447

(1)

10.1.7 Irreversibility

448

(3)

10.1.8 Possible experimental support for the hypothesis of microscopic chaos

451

(1)

10.2 Perspectives and open questions

452

(6)

10.2.1 Extensions to general and dissipative dynamical systems

453

(1)

10.2.2 Extensions in nonequilibrium statistical mechanics

454

(2)

10.2.3 Extensions to quantum-mechanical systems

456

(2)

References

458

(13)

Index

471

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Chaos, Scattering and Statistical Mechanics

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