Chaos, Scattering and Statistical Mechanics (Cambridge Nonlinear Science Series)

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Chaos, Scattering and Statistical Mechanics (Cambridge Nonlinear Science Series)

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  • 製本 Hardcover:ハードカバー版/ページ数 475 p.
  • 言語 ENG
  • 商品コード 9780521395113
  • DDC分類 530.13011857

Full Description


This book describes 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. Chaotic Scattering is illustrated with disk scatterers and with examples of unimolecular chemical reactions and then generalized to transport in spatially extended systems. 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.

Table of Contents

Preface                                            xvii
Introduction 1 (11)
Chapter 1 Dynamical systems and their linear 12 (31)
stability
1.1 Dynamics in phase space 12 (3)
1.1.1 The group of time evolutions 12 (1)
1.1.2 The Poincare map 13 (1)
1.1.3 Hamiltonian systems 14 (1)
1.1.4 Billiards 15 (1)
1.2 Linear stability and the tangent space 15 (8)
1.2.1 The fundamental matrix 15 (1)
1.2.2 Lyapunov exponents 16 (1)
1.2.3 Decomposition of the tangent space 17 (2)
into orthogonal directions
1.2.4 Homological decomposition of the 19 (1)
multiplicative cocycle
1.2.5 The local stretching rates 20 (2)
1.2.6 Stable and unstable manifolds 22 (1)
1.3 Linear stability of Hamiltonian systems 23 (8)
1.3.1 Symplectic dynamics and the 23 (1)
pairing rule of the Lyapunov exponents
1.3.2 Vanishing Lyapunov exponents and 23 (2)
the Lie group of continuous symmetries
1.3.3 Hamilton-Jacobi equation and the 25 (3)
curvature of the wavefront
1.3.4 Elimination of the neutral 28 (1)
directions
1.3.5 The local stretching rate in f = 2 29 (2)
hyperbolic Hamiltonian systems
1.4 Linear stability in billiards 31 (12)
1.4.1 Some definitions 32 (1)
1.4.2 The second fundamental form 33 (1)
1.4.3 The tangent space of the billiard 33 (1)
1.4.4 Free flight 34 (1)
1.4.5 Collision 35 (1)
1.4.6 Expanding and contracting 36 (3)
horospheres
1.4.7 Two-dimensional hard-disk 39 (4)
billiards (f = 2)
Chapter 2 Topological chaos 43 (24)
2.1 Topology of trajectories in phase space 43 (4)
2.1.1 Phase portrait and invariant set 43 (1)
2.1.2 Critical orbits: stationary points 44 (1)
and periodic orbits
2.1.3 Nonwandering sets 44 (1)
2.1.4 Locally maximal invariant sets and 45 (1)
global stability
2.1.5 Dense orbits and transitivity 46 (1)
2.1.6 Attractors, anti-attractors, and 46 (1)
repellers
2.2 Hyperbolicity 47 (5)
2.2.1 Definition 47 (2)
2.2.2 Escape-time functions 49 (1)
2.2.3 Anosov and Axiom-A systems 50 (2)
2.3 Markov partition and symbolic dynamics 52 (5)
in hyperbolic systems
2.3.1 Partitioning phase space with 52 (1)
stable and unstable manifolds
2.3.2 Symbolic dynamics and shift 53 (2)
2.3.3 Markov topological shift 55 (2)
2.4 Topological entropy 57 (3)
2.4.1 Nets and separated subsets 58 (1)
2.4.2 Definition and properties 58 (2)
2.5 The spectrum of periodic orbits 60 (4)
2.5.1 Periodic orbits and the 60 (1)
topological zeta function
2.5.2 Prime periodic orbits and fixed 61 (2)
points
2.5.3 How to order a sum over periodic 63 (1)
orbits?
2.5.4 The topological entropy of a 64 (1)
hyperbolic invariant set
2.6 The topological zeta function of 64 (3)
hyperbolic systems
2.6.1 The topological zeta function for 65 (1)
a nonoverlapping partition
2.6.2 The topological zeta function for 65 (2)
an overlapping partition
Chapter 3 Liouvillian dynamics 67 (59)
3.1 Statistical ensembles 67 (1)
3.2 Time evolution of statistical ensembles 68 (4)
3.2.1 Liouville equation 68 (1)
3.2.2 Frobenius-Perron and Koopman 69 (1)
operators
3.2.3 Boundary conditions 70 (2)
3.3 Invariant measures 72 (3)
3.3.1 Definition 72 (1)
3.3.2 Selection of an invariant measure 73 (1)
3.3.3 The basic invariant measures of 74 (1)
Hamiltonian systems
3.4 Correlation functions and spectral 75 (1)
functions
3.5 Spectral theory on real frequencies 76 (7)
3.5.1 Spectral decomposition on a 76 (1)
Hilbert space
3.5.2 Purely discrete real spectrum 77 (1)
3.5.3 Continuous real spectrum 78 (3)
3.5.4 Continuous real spectrum and the 81 (1)
Wiener-Khinchin theorem
3.5.5 Continuous real spectrum and 82 (1)
Gaussian fluctuations
3.6 Spectral theory on complex frequencies 83 (11)
or resonance theory
3.6.1 Analytic continuation to complex 83 (4)
frequencies
3.6.2 Singularities as a spectrum of 87 (4)
generalized eigenvalues
3.6.3 Definition of a trace 91 (3)
3.7 Resonances of stationary points 94 (9)
3.7.1 Trace formula and eigenvalues 94 (2)
3.7.2 Eigenstates and spectral 96 (4)
decompositions
3.7.3 Hamiltonian stationary point of 100 (3)
saddle type
3.8 Resonances for hyperbolic sets with 103 (15)
periodic orbits
3.8.1 Trace formula 103 (2)
3.8.2 The Selberg-Smale Zeta function 105 (5)
3.8.3 Formulation of the eigenvalue 110 (3)
problem
3.8.4 Fredholm determinant 113 (1)
3.8.5 Fredholm theory for the eigenstates 114 (2)
3.8.6 Periodic-orbit averages of 116 (2)
observables
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 122 (1)
two-dimensional flow
3.10 Liouvillian dynamics of 123 (3)
one-dimensional maps
Chapter 4 Probabilistic chaos 126 (45)
4.1 Dynamical randomness and the entropy 126 (5)
per unit time
4.1.1 A model of observation 127 (1)
4.1.2 Information redundancy and 128 (2)
algorithmic complexity
4.1.3 Kolmogorov-Sinai entropy per unit 130 (1)
time
4.2 The large-deviation formalism 131 (7)
4.2.1 Separated subsets 132 (1)
4.2.2 Topological pressure and the 132 (3)
dynamical invariant measures
4.2.3 Pressure functions based on the 135 (1)
Lyapunov exponents
4.2.4 Entropy function and Legendre 136 (2)
transform
4.3 Closed hyperbolic systems 138 (4)
4.3.1 The microcanonical measure as a 138 (2)
Sinai-Ruelle-Bowen dynamical measure
4.3.2 The pressure function for closed 140 (1)
systems
4.3.3 Generating functions of 141 (1)
observables and transport coefficients
4.4 Open hyperbolic systems 142 (10)
4.4.1 The nonequilibrium invariant 143 (2)
measure of the repeller
4.4.2 Connection with the dynamical 145 (3)
invariant measure and the pressure
function
4.4.3 Fractal repellers in 148 (4)
two-degrees-of-freedom systems and their
dimensions
4.5 Generalized zeta functions 152 (4)
4.5.1 Frobenius-Perron operators in the 152 (2)
large-deviation formalism
4.5.2 The topological pressure as an 154 (2)
eigenvalue
4.6 Probabilistic Markov chains and 156 (7)
lattice gas automata
4.6.1 Markov chain models 156 (1)
4.6.2 Isomorphism between Markov chains 157 (2)
and area-preserving maps
4.6.3 Repellers of Markov chains 159 (2)
4.6.4 Large-deviation formalism of 161 (2)
Markov chains
4.7 Special fractals generated by 163 (6)
uniformly hyperbolic maps
4.7.1 Condition on the mean sojourn time 163 (4)
in a domain
4.7.2 Fractal repeller generated in 167 (2)
trajectory reconstruction
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 172 (2)
time delay
5.1.3 The different types of trajectories 174 (2)
5.1.4 Scattering operator for 176 (1)
statistical ensembles
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 211 (4)
portraits
5.3.3 Characteristic quantities of the 215 (1)
chaotic repeller
5.4 Application to the molecular 216 (5)
transition state
5.4.1 Model of photodissociation of 216 (2)
HgI(2)
5.4.2 Transition from a periodic to a 218 (1)
chaotic repeller
5.4.3 Three-branched Smale repeller and 219 (2)
its characterization
5.5 Further applications of chaotic 221 (3)
scattering
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 225 (1)
equation
6.2.2 Escape-time function and escape 226 (3)
rate
6.2.3 Phenomenology of the escape process 229 (2)
6.2.4 The escape-rate formula for 231 (3)
diffusion
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 237 (7)
Lorentz gas
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 254 (3)
and the escape-rate formula
6.5 Escape-rate formalism for general 257 (7)
transport coefficients
6.5.1 General context 257 (1)
6.5.2 Transport coefficients and their 258 (3)
Helfand moments
6.5.3 Generalization of the escape-rate 261 (3)
formula
6.6 Escape-rate formalism for chemical 264 (4)
reaction rates
6.6.1 Nonequilibrium thermodynamics of 265 (1)
chemical reactions
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 269 (1)
escape-rate formalism
6.7.3 Relation to the 270 (2)
thermostatted-system approach
6.7.4 The escape-rate formalism in the 272 (3)
presence of external forces
Chapter 7 Hydrodynamic modes of diffusion 275 (68)
7.1 Hydrodynamics from Liouvillian dynamics 275 (3)
7.1.1 Historical background and 275 (1)
motivation
7.1.2 Pollicott-Ruelle resonances and 276 (2)
quasiperiodic boundary conditions
7.2 Liouvillian dynamics for systems 278 (15)
symmetric under a group of spatial
translations
7.2.1 Introduction 278 (1)
7.2.2 Suspended flows of infinite 279 (3)
spatial extension
7.2.3 Assumptions on the properties of 282 (1)
the mapping
7.2.4 Invariant measures 283 (1)
7.2.5 Time-reversal symmetry 284 (1)
7.2.6 The Frobenius-Perron operator on 284 (1)
the infinite lattice
7.2.7 Spatial Fourier transforms 285 (1)
7.2.8 The Frobenius-Perron operators in 286 (2)
the wavenumber subspaces
7.2.9 Time-reversal symmetry for the 288 (1)
k-components
7.2.10 Reduction to the Frobenius-Perron 289 (1)
operator of the mapping
7.2.11 Eigenvalue problem and zeta 289 (2)
function
7.2.12 Consequences of time-reversal 291 (1)
symmetry on the resonances
7.2.13 Relation to the eigenvalue 292 (1)
problem for the flow
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 295 (1)
eigenstate with respect to the wavenumber
7.3.4 Diffusion matrix 296 (1)
7.3.5 Higher-order diffusion coefficients 297 (2)
7.3.6 Eigenvalues and the Van Hove 299 (3)
function
7.3.7 Periodic-orbit formula for the 302 (1)
diffusion coefficient
7.3.8 Consequences of the lattice 303 (2)
symmetry under a point group
7.4 Deterministic diffusion in the 305 (7)
periodic Lorentz gas
7.4.1 Properties of the infinite Lorentz 305 (4)
gas
7.4.2 Diffusion and its dispersion 309 (1)
relation
7.4.3 Cumulative functions of the 310 (2)
eigenstates
7.5 Deterministic diffusion in the 312 (22)
periodic multibaker
7.5.1 Properties of the periodic 312 (1)
multibaker
7.5.2 The Frobenius-Perron operator and 313 (3)
its Pollicott-Ruelle resonances
7.5.3 Generalized spectral decomposition 316 (2)
7.5.4 Analysis of the associated 318 (8)
one-dimensional map
7.5.5 The root states of the 326 (8)
two-dimensional map
7.6 Extensions to the other transport 334 (6)
processes
7.6.1 Gaussian fluctuations of the 334 (2)
Helfand moments
7.6.2 The minimal models of transport 336 (1)
7.6.3 Viscosity and self-diffusion in 337 (2)
two-particle fluids
7.6.4 Spontaneous symmetry breaking and 339 (1)
Goldstone hydrodynamic modes
7.7 Chemio-hydrodynamic modes 340 (3)
Chapter 8 Systems maintained out of 343 (44)
equilibrium
8.1 Nonequilibrium systems in Liouvillian 343 (2)
dynamics
8.2 Nonequilibrium steady states of 345 (5)
diffusion
8.2.1 Phenomenological description of 345 (1)
the steady states
8.2.2 Deterministic description of the 346 (4)
steady states
8.3 From the hydrodynamic modes to the 350 (11)
nonequilibrium steady states
8.3.1 From the eigenstates to the 350 (2)
nonequilibrium steady states
8.3.2 Microscopic current and Fick's law 352 (1)
8.3.3 Nonequilibrium steady states of 353 (2)
the periodic Lorentz gas
8.3.4 Nonequilibrium steady states of 355 (4)
the periodic multibaker
8.3.5 Nonequilibrium steady states of 359 (2)
the Langevin process
8.4 From the finite to the infinite 361 (5)
multibaker
8.5 Generalization to the other transport 366 (2)
processes
8.6 Entropy production 368 (17)
8.6.1 Irreversible thermodynamics and 370 (1)
the problem of entropy production
8.6.2 Comparison with deterministic 371 (2)
schemes
8.6.3 Open systems and their Poisson 373 (3)
suspension
8.6.4 The XXX-entropy 376 (2)
8.6.5 Entropy production in the 378 (5)
multibaker map
8.6.6 Summary 383 (2)
8.7 Comments on far-from-equilibrium 385 (2)
systems
Chapter 9 Noises as microscopic chaos 387 (46)
9.1 Differences and similarities between 387 (3)
noises and chaos
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 391 (1)
interval T and a partition XXX
9.2.3 Partition (XXX, XXX)-entropy per 392 (1)
unit time
9.2.4 Cohen-Procaccia (XXX, XXX)-entropy 393 (1)
per unit time
9.2.5 Shannon-Kolmogorov (XXX, 394 (3)
XXX)-entropy per unit time
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, 402 (4)
amplitude-continuous random processes
9.3.5 Time- and amplitude-continuous 406 (5)
random processes
9.3.6 White noise 411 (1)
9.3.7 Levy flights 411 (1)
9.3.8 Classification of the time random 412 (3)
processes
9.4 Spacetime random processes 415 (4)
9.4.1 (XXX, XXX)-entropy per unit time 415 (1)
and volume
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 417 (1)
equations
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 419 (1)
processes
9.5 Random processes of statistical 419 (10)
mechanics
9.5.1 Ideal gases 420 (4)
9.5.2 The Lorentz gases and the 424 (1)
hard-sphere gases
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 429 (1)
Brownian motion
9.6.2 A lower bound on the positive 430 (1)
Lyapunov exponents
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 433 (5)
statistical ensembles
10.1.2 Dynamical chaos 438 (3)
10.1.3 Fractal repellers and chaotic 441 (1)
scattering
10.1.4 Scattering theory of transport 442 (1)
10.1.5 Relaxation to equilibrium 442 (5)
10.1.6 Nonequilibrium steady states and 447 (1)
entropy production
10.1.7 Irreversibility 448 (3)
10.1.8 Possible experimental support for 451 (1)
the hypothesis of microscopic chaos
10.2 Perspectives and open questions 452 (6)
10.2.1 Extensions to general and 453 (1)
dissipative dynamical systems
10.2.2 Extensions in nonequilibrium 454 (2)
statistical mechanics
10.2.3 Extensions to quantum-mechanical 456 (2)
systems
References 458 (13)
Index 471