Chaos, Scattering and Statistical Mechanics

by Pierre Gaspard

• ISBN13: 9780521395113
• ISBN10: 0521395119
• Format: Hardcover
• Copyright: 1998-06-28
• Publisher: Cambridge University Press

Summary

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.

Table of Contents

Preface	xvii
Introduction	1	(11)
Chapter 1 Dynamical systems and their linear stability	12	(31)
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 into orthogonal directions	17	(2)
1.2.4 Homological decomposition of the multiplicative cocycle	19	(1)
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 pairing rule of the Lyapunov exponents	23	(1)
1.3.2 Vanishing Lyapunov exponents and the Lie group of continuous symmetries	23	(2)
1.3.3 Hamilton-Jacobi equation and the curvature of the wavefront	25	(3)
1.3.4 Elimination of the neutral directions	28	(1)
1.3.5 The local stretching rate in f = 2 hyperbolic Hamiltonian systems	29	(2)
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 horospheres	36	(3)
1.4.7 Two-dimensional hard-disk billiards (f = 2)	39	(4)
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 and periodic orbits	44	(1)
2.1.3 Nonwandering sets	44	(1)
2.1.4 Locally maximal invariant sets and global stability	45	(1)
2.1.5 Dense orbits and transitivity	46	(1)
2.1.6 Attractors, anti-attractors, and repellers	46	(1)
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 in hyperbolic systems	52	(5)
2.3.1 Partitioning phase space with stable and unstable manifolds	52	(1)
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 topological zeta function	60	(1)
2.5.2 Prime periodic orbits and fixed points	61	(2)
2.5.3 How to order a sum over periodic orbits?	63	(1)
2.5.4 The topological entropy of a hyperbolic invariant set	64	(1)
2.6 The topological zeta function of hyperbolic systems	64	(3)
2.6.1 The topological zeta function for a nonoverlapping partition	65	(1)
2.6.2 The topological zeta function for an overlapping partition	65	(2)
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 operators	69	(1)
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 Hamiltonian systems	74	(1)
3.4 Correlation functions and spectral functions	75	(1)
3.5 Spectral theory on real frequencies	76	(7)
3.5.1 Spectral decomposition on a Hilbert space	76	(1)
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 Wiener-Khinchin theorem	81	(1)
3.5.5 Continuous real spectrum and Gaussian fluctuations	82	(1)
3.6 Spectral theory on complex frequencies or resonance theory	83	(11)
3.6.1 Analytic continuation to complex frequencies	83	(4)
3.6.2 Singularities as a spectrum of generalized eigenvalues	87	(4)
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 decompositions	96	(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