Chaos, scattering and statistical mechanics:
Gespeichert in:
| Beteilige Person: | |
|---|---|
| Format: | Buch |
| Sprache: | Englisch |
| Veröffentlicht: |
Cambridge [u.a.]
Cambridge Univ. Press
2005
|
| Ausgabe: | 1. paperback version |
| Schriftenreihe: | Cambridge nonlinear science series
9 |
| Schlagwörter: | |
| Links: | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=017115916&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| Umfang: | XIX, 475 S. Ill., graph. Darst. |
| ISBN: | 0521018250 9780521018258 |
Internformat
MARC
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| 100 | 1 | |a Gaspard, Pierre |e Verfasser |4 aut | |
| 245 | 1 | 0 | |a Chaos, scattering and statistical mechanics |c Pierre Gaspard |
| 250 | |a 1. paperback version | ||
| 264 | 1 | |a Cambridge [u.a.] |b Cambridge Univ. Press |c 2005 | |
| 300 | |a XIX, 475 S. |b Ill., graph. Darst. | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
| 490 | 1 | |a Cambridge nonlinear science series |v 9 | |
| 650 | 0 | 7 | |a Streuung |0 (DE-588)4058056-8 |2 gnd |9 rswk-swf |
| 650 | 0 | 7 | |a Chaostheorie |0 (DE-588)4009754-7 |2 gnd |9 rswk-swf |
| 650 | 0 | 7 | |a Transporttheorie |0 (DE-588)4185936-4 |2 gnd |9 rswk-swf |
| 650 | 0 | 7 | |a Chaotisches System |0 (DE-588)4316104-2 |2 gnd |9 rswk-swf |
| 650 | 0 | 7 | |a Nichtgleichgewichtsstatistik |0 (DE-588)4136220-2 |2 gnd |9 rswk-swf |
| 655 | 7 | |8 1\p |0 (DE-588)1071861417 |a Konferenzschrift |2 gnd-content | |
| 689 | 0 | 0 | |a Chaotisches System |0 (DE-588)4316104-2 |D s |
| 689 | 0 | 1 | |a Streuung |0 (DE-588)4058056-8 |D s |
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| 689 | 1 | 1 | |a Chaostheorie |0 (DE-588)4009754-7 |D s |
| 689 | 1 | |5 DE-604 | |
| 689 | 2 | 0 | |a Transporttheorie |0 (DE-588)4185936-4 |D s |
| 689 | 2 | 1 | |a Chaostheorie |0 (DE-588)4009754-7 |D s |
| 689 | 2 | |5 DE-604 | |
| 830 | 0 | |a Cambridge nonlinear science series |v 9 |w (DE-604)BV004573757 |9 9 | |
| 856 | 4 | 2 | |m Digitalisierung UB Regensburg |q application/pdf |u http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=017115916&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |3 Inhaltsverzeichnis |
| 883 | 1 | |8 1\p |a cgwrk |d 20201028 |q DE-101 |u https://d-nb.info/provenance/plan#cgwrk | |
| 943 | 1 | |a oai:aleph.bib-bvb.de:BVB01-017115916 | |
Datensatz im Suchindex
| _version_ | 1819268701848862720 |
|---|---|
| adam_text | Contents
Preface
xvii
Introduction
1
Chapter
1
Dynamical systems and their linear stability
12
1.1
Dynamics in phase space
12
1.1.1
The group of time evolutions
12
1.1.2
The
Poincaré
map
13
1.1.3
Hamiltonian systems
14
1.1.4
Billiards
15
1.2
Linear stability and the tangent space
15
1.2.1
The fundamental matrix
15
1.2.2
Lyapunov exponents
16
1.2.3
Decomposition of the tangent space into orthogonal
directions
17
1.2.4
Homological decomposition of the multiplicative cocycle
19
1.2.5
The local stretching rates
20
1.2.6
Stable and unstable manifolds
22
1.3
Linear stability of Hamiltonian systems
23
1.3.1
Symplectic dynamics and the pairing rule of the Lyapunov
exponents
23
1.3.2
Vanishing Lyapunov exponents and the Lie group of continuous
symmetries
23
1.3.3
Hamilton-Jacobi equation and the curvature of the
v/avefront
25
vn
viii Contents
1.3.4 Elimination
of the neutral directions
28
1.3.5
The local stretching rate in
ƒ = 2
hyperbolic Hamiltonian
systems
29
1.4
Linear stability in billiards
31
1.4.1
Some definitions
32
1.4.2
The second fundamental form
33
1.4.3
The tangent space of the billiard
33
1.4.4
Free flight
34
1.4.5
Collision
35
1.4.6
Expanding and contracting horospheres
36
1.4.7
Two-dimensional hard-disk billiards
(ƒ = 2) 59
Chapter
2
Topological chaos
43
2.1
Topology of trajectories in phase space
43
2.1.1
Phase portrait and invariant set
43
2.1.2
Critical orbits: stationary points and periodic orbits
44
2.1.3
Nonwandering sets
44
2.1.4
Locally maximal invariant sets and global stability
45
2.1.5
Dense orbits and transitivity
46
2.1.6
Attractors, anti-attractors, and repellers
46
2.2
Hyperbolicky
47
2.2.1
Definition
47
2.2.2
Escape-time functions
49
2.2.3
Anosov and
Axiom
-А
systems
50
2.3
Markov partition and symbolic dynamics in hyperbolic
systems
52
2.3.1
Partitioning phase space with stable and unstable
manifolds
52
2.3.2
Symbolic dynamics and shift
53
2.3.3
Markov topological shift
55
2.4
Topological entropy
57
2.4.1
Nets and separated subsets
58
2.4.2
Definition and properties
58
2.5
The spectrum of periodic orbits
60
25.1
Periodic orbits and the topological
zeta
function
60
2.5.2
Prime periodic orbits and fixed points
61
2.5.3
How to order a sum over periodic orbits?
63
2.5.4
The topological entropy of a hyperbolic invariant set
64
2.6
The topological
zeta
function of hyperbolic systems
64
Contente
ІХ
2.6.1
The
topologicul
zeta
function
for
a
nonoverlapping
partition
65
2.6.2
The topological
zeta
function for an overlapping partition
65
Chapter
3
Liouvillian dynamics
67
3.1
Statistical ensembles
67
3.2
Time evolution of statistical ensembles
68
3.2.1
Liouville equation
68
3.2.2
Frobenius-Perron and
Koopman
operators
69
3.2.3
Boundary conditions
70
3.3
Invariant measures
72
3.3.1
Definition
72
3.3.2
Selection of an invariant measure
73
3.3.3
The basic invariant measures of Hamiltonian systems
74
3.4
Correlation functions and spectral functions
75
3.5
Spectral theory on real frequencies
76
3.5.1
Spectral decomposition on
a Hubert
space
76
3.5.2
Purely discrete real spectrum
77
3.5.3
Continuous real spectrum
78
3.5.4
Continuous real spectrum and the Wiener-Khinchin
theorem
81
3.5.5
Continuous real spectrum and Gaussian fluctuations
82
3.6
Spectral theory on complex frequencies or resonance
theory
83
3.6.1
Analytic continuation to complex frequencies
83
3.6.2
Singularities as a spectrum of generalized eigenvalues
87
3.6.3
Definition of a trace
91
3.7
Resonances of stationary points
94
3.7.1
Trace formula and eigenvalues
94
3.7.2
Eigenstates and spectral decompositions
96
3.7.3
Hamiltonian stationary point of saddle type
100
3.8
Resonances for hyperbolic sets with periodic orbits
103
3.8.1
Trace formula
103
3.8.2
The Seiberg-Smale
Zeta
function
105
3.8.3
Formulation of the eigenvalue problem
110
3.8.4
Fredholm
determinant
113
3.8.5
Fredholm
theory for the eigenstates
114
3.8.6
Periodic-orbit averages of
observables
116
3.9
Resonance spectrum at bifurcations
118
Contents
3.9.1
Pitchfork bifurcation
119
3.9.2 Hopf
bifurcation
120
3.9.3
Homoclinic bifurcation in a two-dimensional flow
122
3.10 Liouvülian
dynamics of one-dimensional maps
123
Chapter
4
Probabilistic chaos
126
4.1
Dynamical randomness and the entropy per unit time
126
4.1.1
A model of observation
127
4.1.2
Information redundancy and algorithmic complexity
128
4.1.3
Kolmogorov-Sinai entropy per unit time
130
4.2
The large-deviation formalism
131
4.2.1
Separated subsets
132
4.2.2
Topologicał
pressure and the dynamical invariant
measures
132
4.2.3
Pressure functions based on the Lyapunov exponents
135
4.2.4
Entropy function and Legendre transform
136
4.3
Closed hyperbolic systems
138
4.3.1
The microcanonical measure as a Sinai-Ruelle-Bowen
dynamical measure
138
4.3.2
The pressure function for closed systems
140
4.3.3
Generating functions of
observables
and transport
coefficients
141
4.4
Open hyperbolic systems
142
4.4.1
The nonequilibrium invariant measure of the
repeller
143
4.4.2
Connection with the dynamical invariant measure and the
pressure function
145
4.4.3
Fractal repellers in two-degrees-of-freedom systems and their
dimensions
148
4.5
Generalized
zeta
functions
152
4.5.1
Frobenius-Perron operators in the large-deviation
formalism
152
4.5.2
The topological pressure as an eigenvalue
154
4.6
Probabilistic Markov chains and lattice gas automata
156
4.6.1
Markov chain models
156
4.6.2
Isomorphism between Markov chains and area-preserving
maps
157
4.6.3
Repellers of Markov chains
159
4.6.4
Large-deviation formalism of Markov chains
161
Contents
ХІ
4.7 Special
fractals generated by uniformly hyperbolic
maps
163
4.7.1
Condition on the mean sojourn time in a domain
163
4.7.2
Fractal
repeller
generated in trajectory reconstruction
167
4.8
Nonhyperbolic systems
169
Chapters Chaotic scattering
171
5.1
Classical scattering theory
171
5.1.1
Motivations
171
5.1.2
Classical scattering function and time delay
172
5.1.3
The different types of trajectories
174
5.1.4
Scattering operator for statistical ensembles
176
5.2
Hard-disk scatterers
176
5.2.1
Generalities
176
5.2.2
The dynamics
177
5.2.3
The Birkhoff mapping
179
5.2.4
The one-disk scatterer
180
5.2.5
The two-disk scatterer
181
5.2.6
The three-disk scatterer
183
5.2.7
The four-disk scatterer
201
5.3
Hamiltonian mapping of scattering type
210
5.3.1
Definition of the model
210
5.3.2
Metamorphoses of the phase portraits
211
5.3.3
Characteristic quantities of
tne
chaotic
repeller
215
5.4
Application to the molecular transition state
216
5.4.1
Model of
photodissociation ofHgli
216
5.4.2
Transition from a periodic to a chaotic
repeller
218
5.4.3
Three-branched
Smale repeller
and its characterization
219
5.5
Further applications of chaotic scattering
221
Chapter
6
Scattering theory of transport
224
6.1
Scattering and transport
224
6.2
Diffusion and chaotic scattering
225
6.2.1
Large scatterers and the diffusion equation
225
6.2.2
Escape-time function and escape rate
226
6.2.3
Phenomenology of the escape process
229
6.2.4
The escape-rate formula for diffusion
231
6.3
The periodic
Lorentz gas
234
xii Contents
6.3.1 Definition 234
6.3.2 The Liouville invariant
measure
235
6.3.3
Finite and
infinite
horizons
236
6.3.4
Chaotic properties of the infinite
Lorentz gas
237
6.3.5
The open
Lorentz gas
244
6.4
The multibaker mapping
250
6.4.1
Definition
250
6.4.2
The Pollicott-Ruelle resonances and the escape-rate
formula
254
6.5
Escape-rate formalism for general transport coefficients
257
6.5.1
General context
257
6.5.2
Transport coefficients and their Helfand moments
258
6.5.3
Generalization of the escape-rate formula
261
6.6
Escape-rate formalism for chemical reaction rates
264
6.6.1
Noneąuilibrium
thermodynamics of chemical reactions
265
6.6.2
The master equation approach
266
6.7
Discussion
268
6.7.1
Summary
268
6.7.2
Further applications of the escape-rate formalism
269
6.7.3
Relation to the thermostatted-system approach
270
6.7.4
The escape-rate formalism in the presence of external
forces
272
Chapter
7
Hydrodynamic modes of diffusion
275
7.1
Hydrodynamics from Liouvillian dynamics
275
7.1.1
Historical background and motivation
275
7.1.2
Pollicott-Ruelle resonances and quasiperiodic boundary
conditions
276
7.2
Liouviffian dynamics for systems symmetric under a group
of spatial translations
278
7.2.1
Introduction
278
7.2.2
Suspended flows of infinite spatial extension
279
7.2.3
Assumptions on the properties of the mapping
282
7.2.4
Invariant measures
283
7.2.5
Ћте
-reversal
symmetry
284
7.2.6
The Frobenius-Perron operator on the infinite lattice
284
7.2.7
Spatial Fourier transforms
285
7.2.8
The Frobenius-Perron operators in the wavenumber
subspaces
286
Contents xiii
7.2.9
Time-reversal symmetry for the k-components
288
7.2.10
Reduction to the Frobenius-Perron operator of the
mapping
289
7.2.11
Eigenvalue problem and
zeta
function
289
7.2.12
Consequences of time-reversal symmetry on the
resonances
291
7.2.13
Relation to the eigenvalue problem for the flow
292
7.3
Deterministic diffusion
293
7.3.1
Introduction
293
7.3.2
Mean drift
294
7.3.3
The first derivative of the eigenstate with respect to the
wavenumber
295
7.3.4
Diffusion matrix
296
7.3.5
Higher-order diffusion coefficients
297
7.3.6
Eigenvalues and the Van Hove function
299
7.3.7
Periodic-orbit formula for the diffusion coefficient
302
7.3.8
Consequences of the lattice symmetry under a
point group
303
7.4
Deterministic diffusion in the periodic
Lorentz gas
305
7.4.1
Properties of the infinite
Lorentz gas
305
7.4.2
Diffusion and its dispersion
геШіоп
309
7.4.3
Cumulative functions of the eigenstates
310
7.5
Deterministic diffusion in the periodic multibaker
312
7.5.1
Properties of the periodic multibaker
312
7.5.2
The Frobenius-Perron operator and its Pollicott-Ruelle
resonances
313
7.5.3
Generalized spectral decomposition
316
7.5.4
Analysis of the associated one-dimensional map
318
7.5.5
The root states of the two-dimensional map
326
7.6
Extensions to the other transport processes
334
7.6.1
Gaussian fluctuations of the Helfand moments
334
7.6.2
The minimal models of transport
336
7.6.3
Viscosity and self-diffusion in two-particle fluids
337
7.6.4
Spontaneous symmetry breaking and Goldstone hydrodynamic
modes
339
7.7
Chemio-hydrodynamic modes
340
XIV
Contents
Chapeei
8 Systems
maintained out of equilibrium
343
8.1
Nonequilibrium systems in Liouvillian dynamics
343
8.2
Nonequilibrium steady states of diffusion
345
8.2.1
Phenomenological description of the steady states
345
8.2.2
Deterministic description of the steady states
346
8.3
From the hydrodynamic modes to the nonequilibrium steady
states
350
8.3.1
From the eigenstates to the nonequilibrium steady states
350
8.3.2
Microscopic current and Fick s law
352
8.3.3
Nonequilibrium steady states of the periodic
Lorentz gas
353
8.3.4
Nonequilibrium steady states of the periodic multibaker
355
8.3.5
Nonequilibrium steady states of the
Langevin
process
359
8.4
From the finite to the infinite multibaker
361
8.5
Generalization to the other transport processes
366
8.6
Entropy production
368
8.6.1
Irreversible thermodynamics and the problem of entropy
production
370
8.6.2
Comparison with deterministic schemes
371
8.6.3
Open systems and their
Poisson
suspension
373
8.6.4
The
ε
-entropy
376
8.6.5
Entropy production in the multibaker map
378
8.6.6
Summary
383
8.7
Comments on far-from-equilibrium systems
385
Chapter
9
Noises as microscopic chaos
387
9.1
Differences and similarities between noises and chaos
387
9.2
(£,т)-епігору
per unit time
390
9.2.1
Dynamical processes
390
9.2.2
Entropy of a process over a time interval
T
and a
partition
<$ 391
9.2.3
Partition (e,r)-entropy per unit time
392
9.2.4
Cohen-Procaccia (e,x)-entropy
per unit time
393
9.2.5
Shannon-Kolmogorov
(ε,
r)-entropy per unit time
394
9.3
Time random processes
397
9.3.1
Determimstic processes
397
9.3.2
Bernoulli and Markov chains
398
9.3.3
Birth-and-death processes
399
9.3.4
Ћте
-discrete,
amplitude-continuous random processes
402
Contents
XV
9.3.5
Time- and amplitude-continuous random processes
406
9.3.6
White noise
411
9.3.7
Levy flights
411
9.3.8
Classification of the time random processes
412
9.4
Spacetime random processes
415
9.4.1
(ε,
x)-entropy per unit time and volume
415
9.4.2
Deterministic cellular automata
415
9.4.3
Lattice gas automata
416
9.4.4
Coupled map lattices
416
9.4.5
Nonlinear partial differential equations
417
9.4.6
Stochastic spin dynamics
417
9.4.7
Spacetime Gaussian fields
417
9.4.8
Sporadic spacetime random processes
418
9.4.9
Classification of spacetime random processes
419
9.5
Random processes of statistical mechanics
419
9.5.1
Ideal gases
420
9.5.2
The
Lorentz
gases and the hard-sphere gases
424
9.5.3
The Boltzmann-Lorentz process
425
9.6
Brownian motion and microscopic chaos
429
9.6.1
Hamiltonian and
Langevin
models of Brownian motion
429
9.6.2
A lower bound on the positive Lyapunov exponents
430
9.6.3
Some conclusions
431
Chapter
10
Conclusions and perspectives
433
10.1
Overview of the results
433
10.1.1
From dynamical instability to statistical ensembles
433
10.1.2
Dynamical chaos
438
10.1.3
Fractal repellers and chaotic scattering
441
10.1.4
Scattering theory of transport
442
10.1.5
Relaxation to equilibrium
442
10.1.6
Nonequilibrium steady states and entropy production
447
10.1.7
¡ireversibility
448
10.1.8
Possible experimental support for the hypothesis of microscopic
chaos
451
10.2
Perspectives and open questions
452
10.2.1
Extensions to general and
dissipatine
dynamical systems
453
xvi Contents
10.2.2
Extensions
in nonequilibrium
statistical mechanics
454
10.2.3
Extensions to quantum-mechanical systems
456
References
458
Index
471
|
| any_adam_object | 1 |
| author | Gaspard, Pierre |
| author_facet | Gaspard, Pierre |
| author_role | aut |
| author_sort | Gaspard, Pierre |
| author_variant | p g pg |
| building | Verbundindex |
| bvnumber | BV035311171 |
| classification_rvk | UG 3100 UG 3900 |
| classification_tum | PHY 063f MAT 587f PHY 057f |
| ctrlnum | (OCoLC)173162499 (DE-599)BVBBV035311171 |
| discipline | Physik Mathematik |
| edition | 1. paperback version |
| format | Book |
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| genre | 1\p (DE-588)1071861417 Konferenzschrift gnd-content |
| genre_facet | Konferenzschrift |
| id | DE-604.BV035311171 |
| illustrated | Illustrated |
| indexdate | 2024-12-20T13:28:16Z |
| institution | BVB |
| isbn | 0521018250 9780521018258 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-017115916 |
| oclc_num | 173162499 |
| open_access_boolean | |
| owner | DE-355 DE-BY-UBR DE-11 |
| owner_facet | DE-355 DE-BY-UBR DE-11 |
| physical | XIX, 475 S. Ill., graph. Darst. |
| publishDate | 2005 |
| publishDateSearch | 2005 |
| publishDateSort | 2005 |
| publisher | Cambridge Univ. Press |
| record_format | marc |
| series | Cambridge nonlinear science series |
| series2 | Cambridge nonlinear science series |
| spellingShingle | Gaspard, Pierre Chaos, scattering and statistical mechanics Cambridge nonlinear science series Streuung (DE-588)4058056-8 gnd Chaostheorie (DE-588)4009754-7 gnd Transporttheorie (DE-588)4185936-4 gnd Chaotisches System (DE-588)4316104-2 gnd Nichtgleichgewichtsstatistik (DE-588)4136220-2 gnd |
| subject_GND | (DE-588)4058056-8 (DE-588)4009754-7 (DE-588)4185936-4 (DE-588)4316104-2 (DE-588)4136220-2 (DE-588)1071861417 |
| title | Chaos, scattering and statistical mechanics |
| title_auth | Chaos, scattering and statistical mechanics |
| title_exact_search | Chaos, scattering and statistical mechanics |
| title_full | Chaos, scattering and statistical mechanics Pierre Gaspard |
| title_fullStr | Chaos, scattering and statistical mechanics Pierre Gaspard |
| title_full_unstemmed | Chaos, scattering and statistical mechanics Pierre Gaspard |
| title_short | Chaos, scattering and statistical mechanics |
| title_sort | chaos scattering and statistical mechanics |
| topic | Streuung (DE-588)4058056-8 gnd Chaostheorie (DE-588)4009754-7 gnd Transporttheorie (DE-588)4185936-4 gnd Chaotisches System (DE-588)4316104-2 gnd Nichtgleichgewichtsstatistik (DE-588)4136220-2 gnd |
| topic_facet | Streuung Chaostheorie Transporttheorie Chaotisches System Nichtgleichgewichtsstatistik Konferenzschrift |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=017115916&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| volume_link | (DE-604)BV004573757 |
| work_keys_str_mv | AT gaspardpierre chaosscatteringandstatisticalmechanics |