Microscopic chaos, fractals and transport in nonequilibrium statistical mechanics:
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| Main Author: | |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
New Jersey [u.a.]
World scientific
2007
|
| Series: | Advanced series in nonlinear dynamics
24 |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | XV, 441 S. Ill., graph. Darst. |
| ISBN: | 9789812565075 9812565078 |
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MARC
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| 100 | 1 | |a Klages, Rainer |d 1966- |e Verfasser |0 (DE-588)114197547 |4 aut | |
| 245 | 1 | 0 | |a Microscopic chaos, fractals and transport in nonequilibrium statistical mechanics |c Rainer Klages |
| 264 | 1 | |a New Jersey [u.a.] |b World scientific |c 2007 | |
| 300 | |a XV, 441 S. |b Ill., graph. Darst. | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
| 490 | 1 | |a Advanced series in nonlinear dynamics |v 24 | |
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Record in the Search Index
| _version_ | 1804137044483506176 |
|---|---|
| adam_text | Contents
Preface
vii
1.
Introduction and outline
1
1.1
Hamiltonian dynamical systems approach to nonequilib-
rium statistical mechanics
.................. 2
1.2
Thermostated dynamical systems approach to nonequilib-
rium statistical mechanics
.................. 7
1.3
The red thread through this book
.............. 11
Part
1:
Fractal transport coefficients
15
2.
*Deterministic diffusion
17
2.1
A simple model for deterministic diffusion
......... 17
2.2
A parameter-dependent fractal diffusion coefficient
.... 22
2.3
Summary
........................... 28
3.
Deterministic drift-diffusion
29
3.1
Drift-diffusion model: mathematical definition
...... 29
3.2
Calculating deterministic drift and diffusion coefficients
32
3.2.1
Twisted eigenstate method
............. 33
3.2.2
Transition matrix methods
............. 37
3.2.3
Numerical comparison of the different methods
. . 39
3.3
The phase diagram
...................... 40
3.4
Simple maps as deterministic ratchets
........... 49
3.5
Summary
........................... 54
xii
Contents
4.
Deterministic reaction-diffusion
55
4.1
A reactive-diffusive multibaker map
............ 55
4.1.1
Deterministic models of reaction-diffusion
..... 56
4.1.2
The Probenius-Perron operator
........... 60
4.2
Diffusive dynamics
...................... 62
4.2.1
+ Diffusive modes of the dyadic multibaker
.... 62
4.2.2
The parameter-dependent diffusion coefficient
. . 64
4.3
Reactive dynamics
...................... 70
4.3.1
Reactive modes of the dyadic multibaker
.... 70
4.3.2
The parameter-dependent reaction rate
...... 75
4.4
*Summary
........................... 81
5.
Deterministic diffusion and random perturbations
83
5.1
Disordered dynamical systems
................ 83
5.2
Noisy dynamical systems
.................. 89
5.3
*Summary
........................... 98
6.
Prom normal to anomalous diffusion
99
6.1
Deterministic diffusion and bifurcations
.......... 99
6.2
Anomalous diffusion in intermittent maps
......... 107
6.3
*Summary
........................... 119
7.
EVom
diffusive maps to Hamiltonian particle billiards
121
7.1
Correlated random walks in maps
.............. 121
7.2
Correlated random walks in billiards
............ 128
7.3
*Summary
........................... 134
8.
Designing billiards with irregular transport coefficients
137
8.1
Diffusion in the flower-shaped billiard
........... 137
8.2
+Random and correlated random walks
.......... 141
8.3
Diffusion in porous solids
.................. 148
8.4
*Summary
........................... 150
9.
Deterministic diffusion of granular particles
153
9.1
Resonances and diffusion in the bouncing ball billiard
. . 153
9.2
+Diffusion
by correlated random walks
........... 157
9.3
Vibratory conveyors
..................... 160
9.4
*Summary
........................... 161
Contents xiii
Part
2: Thermostated
dynamical systems
163
10.
Motivation: coupling a system to a thermal reservoir
165
10.1
*Why thermostats?
...................... 165
10.2
*Modeling thermal reservoirs: the Langevin equation
. . . 167
10.3
Equilibrium velocity distributions for thermostated systems
173
10.4
Applying thermostats: the periodic
Lorentz
gas
...... 179
10.5
*Summary
........................... 183
11.
The Gaussian thermostat
185
11.1
Construction of the Gaussian thermostat
.......... 185
11.2
Chaos and transport in Gaussian thermostated systems
. 189
11.2.1
Phase space contraction and entropy production
. 189
11.2.2
Lyapunov exponents and transport coefficients
. . 190
11.2.3
Nonequilibrium fractal attractors
......... 193
11.2.4
Electrical conductivity
............... 198
11.3
Summary
........................... 202
12.
The
Nosé-Hoover
thermostat
205
12.1
The dissipative Liouville equation
.............. 205
12.2
Construction of the
Nosé-Hoover
thermostat
........ 208
12.2.1
Heuristic derivation
................. 208
12.2.2
Physics of this thermostat
............. 210
12.3
Properties of the
Nosé-Hoover
thermostat
......... 213
12.3.1
Chaos and transport
................ 213
12.3.2
Generalized Hamiltonian formalism
....... 215
12.3.3
Fractals and transport
............... 218
12.4 +
Subtleties of
Nosé-Hoover
dynamics
............ 222
12.4.1
Necessary conditions and generalizations
..... 222
12.4.2
Thermal reservoirs in nonequilibrium
....... 226
12.5
*Summary
........................... 227
13.
Universalities in Gaussian and
Nosé-Hoover
dynamics?
231
13.1
Non-Hamiltonian nonequilibrium steady states
...... 231
13.2
Phase space contraction and entropy production
..... 235
13.3
Transport coefficients and dynamical systems quantities
. 240
13.4
Fractal attractors for nonequilibrium steady states
.... 247
13.5
Nonlinear response in the driven periodic
Lorentz
gas
. . 251
xiv Contents
13.6 *Sunimary........................... 253
14.
Gaussian
and Nosé-Hoover
thermostats revisited
257
14.1
Non-
ideal Gaussian thermostat
............... 257
14.2
Non-ideal Nose-Hoover thermostat
............. 261
14.3
Further alternative thermostats
.............. 264
14.4
*Summary
........................... 266
15.
Stochastic and deterministic boundary thermostats
269
15.1
Stochastic boundary thermostats
.............. 270
15.2
Deterministic boundary thermostats
............ 271
15.3
+Boundary thermostats from first principles
........ 273
15.4
Deterministic boundary thermostats for the driven peri¬
odic
Lorentz
gas
....................... 279
15.4.1
Phase space contraction and entropy production
. 280
15.4.2
Attractors, bifurcations and conductivity
..... 283
15.4.3
Lyapunov exponents
................. 286
15.5
Hard disk fluid under shear and heat flow
......... 287
15.5.1
Homogeneously and inhomogeneously driven shear
and heat flows
.................... 288
15.5.2
Shear and heat flows thermostated by determinis¬
tic scattering
..................... 291
15.6
»Summary
........................... 300
16.
Active Brownian particles and
Nosé-Hoover
dynamics
303
16.1
Brownian motion of migrating cells?
............ 304
16.2
+Moving biological entities as active Brownian particles
. 306
16.3
Bimodal velocity distributions and
Nosé-Hoover
dynamics
308
16.4
»Summary
........................... 314
Part
3:
Outlook and conclusions
317
17.
Further topics in chaotic transport theory
319
17.1
Fluctuation relations
..................... 320
17.1.1
Entropy fluctuation in nonequilibrium steady states
320
17.1.2
The Gallavotti-Cohen fluctuation theorem
.... 321
17.1.3
The Evans-Searles fluctuation theorem
...... 327
17.1.4
Jarzynski work relation and Crooks relation
. . . 328
Contents xv
17.2
Lyapunov modes
....................... 331
17.3
Fourier s law
......................... 337
17.3.1
The basic problem
.................. 338
17.3.2
Heat conduction in anharmonic chaotic chains
. . 340
17.3.3
Heat conduction in chaotic particle billiards
. . . 344
17.4
Pseudochaotic diffusion
................... 347
17.4.1
Microscopic chaos and diffusion?
.......... 348
17.4.2
Polygonal billiard channels
............. 352
17.5
*Summary
........................... 364
18. *
Conclusions
367
18.1
Microscopic chaos and nonequilibrium statistical mechan¬
ics: the big picture
...................... 367
18.2
Assessment of the main results
............... 371
18.2.1
Existence of fractal transport coefficients
..... 371
18.2.2
Universalities in thermostated dynamical systems?
374
18.3
Important open questions
.................. 376
18.3.1
Fractal transport coefficients
............ 377
18.3.2
Thermostated dynamical systems
......... 379
Note added in proof
......................... 380
Bibliography
381
Index
435
|
| adam_txt |
Contents
Preface
vii
1.
"Introduction and outline
1
1.1
Hamiltonian dynamical systems approach to nonequilib-
rium statistical mechanics
. 2
1.2
Thermostated dynamical systems approach to nonequilib-
rium statistical mechanics
. 7
1.3
The red thread through this book
. 11
Part
1:
Fractal transport coefficients
15
2.
*Deterministic diffusion
17
2.1
A simple model for deterministic diffusion
. 17
2.2
A parameter-dependent fractal diffusion coefficient
. 22
2.3
Summary
. 28
3.
Deterministic drift-diffusion
29
3.1
Drift-diffusion model: mathematical definition
. 29
3.2
"'"Calculating deterministic drift and diffusion coefficients
32
3.2.1
Twisted eigenstate method
. 33
3.2.2
Transition matrix methods
. 37
3.2.3
Numerical comparison of the different methods
. . 39
3.3
The phase diagram
. 40
3.4
Simple maps as deterministic ratchets
. 49
3.5
'Summary
. 54
xii
Contents
4.
Deterministic reaction-diffusion
55
4.1
A reactive-diffusive multibaker map
. 55
4.1.1
Deterministic models of reaction-diffusion
. 56
4.1.2
The Probenius-Perron operator
. 60
4.2
Diffusive dynamics
. 62
4.2.1
+ Diffusive modes of the dyadic multibaker
. 62
4.2.2
The parameter-dependent diffusion coefficient
. . 64
4.3
Reactive dynamics
. 70
4.3.1
"'"Reactive modes of the dyadic multibaker
. 70
4.3.2
The parameter-dependent reaction rate
. 75
4.4
*Summary
. 81
5.
Deterministic diffusion and random perturbations
83
5.1
Disordered dynamical systems
. 83
5.2
Noisy dynamical systems
. 89
5.3
*Summary
. 98
6.
Prom normal to anomalous diffusion
99
6.1
Deterministic diffusion and bifurcations
. 99
6.2
Anomalous diffusion in intermittent maps
. 107
6.3
*Summary
. 119
7.
EVom
diffusive maps to Hamiltonian particle billiards
121
7.1
Correlated random walks in maps
. 121
7.2
Correlated random walks in billiards
. 128
7.3
*Summary
. 134
8.
Designing billiards with irregular transport coefficients
137
8.1
Diffusion in the flower-shaped billiard
. 137
8.2
+Random and correlated random walks
. 141
8.3
Diffusion in porous solids
. 148
8.4
*Summary
. 150
9.
Deterministic diffusion of granular particles
153
9.1
Resonances and diffusion in the bouncing ball billiard
. . 153
9.2
+Diffusion
by correlated random walks
. 157
9.3
Vibratory conveyors
. 160
9.4
*Summary
. 161
Contents xiii
Part
2: Thermostated
dynamical systems
163
10.
Motivation: coupling a system to a thermal reservoir
165
10.1
*Why thermostats?
. 165
10.2
*Modeling thermal reservoirs: the Langevin equation
. . . 167
10.3
Equilibrium velocity distributions for thermostated systems
173
10.4
Applying thermostats: the periodic
Lorentz
gas
. 179
10.5
*Summary
. 183
11.
The Gaussian thermostat
185
11.1
Construction of the Gaussian thermostat
. 185
11.2
Chaos and transport in Gaussian thermostated systems
. 189
11.2.1
Phase space contraction and entropy production
. 189
11.2.2
Lyapunov exponents and transport coefficients
. . 190
11.2.3
Nonequilibrium fractal attractors
. 193
11.2.4
Electrical conductivity
. 198
11.3
Summary
. 202
12.
The
Nosé-Hoover
thermostat
205
12.1
The dissipative Liouville equation
. 205
12.2
Construction of the
Nosé-Hoover
thermostat
. 208
12.2.1
Heuristic derivation
. 208
12.2.2
Physics of this thermostat
. 210
12.3
Properties of the
Nosé-Hoover
thermostat
. 213
12.3.1
Chaos and transport
. 213
12.3.2
"'"Generalized Hamiltonian formalism
. 215
12.3.3
Fractals and transport
. 218
12.4 +
Subtleties of
Nosé-Hoover
dynamics
. 222
12.4.1
Necessary conditions and generalizations
. 222
12.4.2
Thermal reservoirs in nonequilibrium
. 226
12.5
*Summary
. 227
13.
Universalities in Gaussian and
Nosé-Hoover
dynamics?
231
13.1
Non-Hamiltonian nonequilibrium steady states
. 231
13.2
Phase space contraction and entropy production
. 235
13.3
Transport coefficients and dynamical systems quantities
. 240
13.4
Fractal attractors for nonequilibrium steady states
. 247
13.5
Nonlinear response in the driven periodic
Lorentz
gas
. . 251
xiv Contents
13.6 *Sunimary. 253
14.
Gaussian
and Nosé-Hoover
thermostats revisited
257
14.1
Non-
ideal Gaussian thermostat
. 257
14.2
Non-ideal Nose-Hoover thermostat
. 261
14.3
"'"Further alternative thermostats
. 264
14.4
*Summary
. 266
15.
Stochastic and deterministic boundary thermostats
269
15.1
Stochastic boundary thermostats
. 270
15.2
Deterministic boundary thermostats
. 271
15.3
+Boundary thermostats from first principles
. 273
15.4
Deterministic boundary thermostats for the driven peri¬
odic
Lorentz
gas
. 279
15.4.1
Phase space contraction and entropy production
. 280
15.4.2
Attractors, bifurcations and conductivity
. 283
15.4.3
Lyapunov exponents
. 286
15.5
Hard disk fluid under shear and heat flow
. 287
15.5.1
Homogeneously and inhomogeneously driven shear
and heat flows
. 288
15.5.2
Shear and heat flows thermostated by determinis¬
tic scattering
. 291
15.6
»Summary
. 300
16.
Active Brownian particles and
Nosé-Hoover
dynamics
303
16.1
Brownian motion of migrating cells?
. 304
16.2
+Moving biological entities as active Brownian particles
. 306
16.3
"'"Bimodal velocity distributions and
Nosé-Hoover
dynamics
308
16.4
»Summary
. 314
Part
3:
Outlook and conclusions
317
17.
Further topics in chaotic transport theory
319
17.1
Fluctuation relations
. 320
17.1.1
Entropy fluctuation in nonequilibrium steady states
320
17.1.2
The Gallavotti-Cohen fluctuation theorem
. 321
17.1.3
The Evans-Searles fluctuation theorem
. 327
17.1.4
Jarzynski work relation and Crooks relation
. . . 328
Contents xv
17.2
Lyapunov modes
. 331
17.3
Fourier's law
. 337
17.3.1
The basic problem
. 338
17.3.2
Heat conduction in anharmonic chaotic chains
. . 340
17.3.3
Heat conduction in chaotic particle billiards
. . . 344
17.4
Pseudochaotic diffusion
. 347
17.4.1
Microscopic chaos and diffusion?
. 348
17.4.2
Polygonal billiard channels
. 352
17.5
*Summary
. 364
18. *
Conclusions
367
18.1
Microscopic chaos and nonequilibrium statistical mechan¬
ics: the big picture
. 367
18.2
Assessment of the main results
. 371
18.2.1
Existence of fractal transport coefficients
. 371
18.2.2
Universalities in thermostated dynamical systems?
374
18.3
Important open questions
. 376
18.3.1
Fractal transport coefficients
. 377
18.3.2
Thermostated dynamical systems
. 379
Note added in proof
. 380
Bibliography
381
Index
435 |
| any_adam_object | 1 |
| any_adam_object_boolean | 1 |
| author | Klages, Rainer 1966- |
| author_GND | (DE-588)114197547 |
| author_facet | Klages, Rainer 1966- |
| author_role | aut |
| author_sort | Klages, Rainer 1966- |
| author_variant | r k rk |
| building | Verbundindex |
| bvnumber | BV022782352 |
| classification_rvk | UG 3500 UG 3900 |
| ctrlnum | (OCoLC)255793947 (DE-599)BVBBV021756639 |
| dewey-full | 530.13 |
| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 530 - Physics |
| dewey-raw | 530.13 |
| dewey-search | 530.13 |
| dewey-sort | 3530.13 |
| dewey-tens | 530 - Physics |
| discipline | Physik |
| discipline_str_mv | Physik |
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| id | DE-604.BV022782352 |
| illustrated | Illustrated |
| index_date | 2024-07-02T18:36:43Z |
| indexdate | 2024-07-09T21:06:02Z |
| institution | BVB |
| isbn | 9789812565075 9812565078 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-015987835 |
| oclc_num | 255793947 |
| open_access_boolean | |
| owner | DE-703 DE-20 DE-355 DE-BY-UBR DE-83 DE-29T |
| owner_facet | DE-703 DE-20 DE-355 DE-BY-UBR DE-83 DE-29T |
| physical | XV, 441 S. Ill., graph. Darst. |
| publishDate | 2007 |
| publishDateSearch | 2007 |
| publishDateSort | 2007 |
| publisher | World scientific |
| record_format | marc |
| series | Advanced series in nonlinear dynamics |
| series2 | Advanced series in nonlinear dynamics |
| spelling | Klages, Rainer 1966- Verfasser (DE-588)114197547 aut Microscopic chaos, fractals and transport in nonequilibrium statistical mechanics Rainer Klages New Jersey [u.a.] World scientific 2007 XV, 441 S. Ill., graph. Darst. txt rdacontent n rdamedia nc rdacarrier Advanced series in nonlinear dynamics 24 Statistische Mechanik (DE-588)4056999-8 gnd rswk-swf Chaotisches System (DE-588)4316104-2 gnd rswk-swf Fraktal (DE-588)4123220-3 gnd rswk-swf Transportprozess (DE-588)4185932-7 gnd rswk-swf Nichtgleichgewichtsthermodynamik (DE-588)4130850-5 gnd rswk-swf Transportprozess (DE-588)4185932-7 s Statistische Mechanik (DE-588)4056999-8 s Nichtgleichgewichtsthermodynamik (DE-588)4130850-5 s Chaotisches System (DE-588)4316104-2 s Fraktal (DE-588)4123220-3 s DE-604 Advanced series in nonlinear dynamics 24 (DE-604)BV004464593 24 Digitalisierung UB Regensburg application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=015987835&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Klages, Rainer 1966- Microscopic chaos, fractals and transport in nonequilibrium statistical mechanics Advanced series in nonlinear dynamics Statistische Mechanik (DE-588)4056999-8 gnd Chaotisches System (DE-588)4316104-2 gnd Fraktal (DE-588)4123220-3 gnd Transportprozess (DE-588)4185932-7 gnd Nichtgleichgewichtsthermodynamik (DE-588)4130850-5 gnd |
| subject_GND | (DE-588)4056999-8 (DE-588)4316104-2 (DE-588)4123220-3 (DE-588)4185932-7 (DE-588)4130850-5 |
| title | Microscopic chaos, fractals and transport in nonequilibrium statistical mechanics |
| title_auth | Microscopic chaos, fractals and transport in nonequilibrium statistical mechanics |
| title_exact_search | Microscopic chaos, fractals and transport in nonequilibrium statistical mechanics |
| title_exact_search_txtP | Microscopic chaos, fractals and transport in nonequilibrium statistical mechanics |
| title_full | Microscopic chaos, fractals and transport in nonequilibrium statistical mechanics Rainer Klages |
| title_fullStr | Microscopic chaos, fractals and transport in nonequilibrium statistical mechanics Rainer Klages |
| title_full_unstemmed | Microscopic chaos, fractals and transport in nonequilibrium statistical mechanics Rainer Klages |
| title_short | Microscopic chaos, fractals and transport in nonequilibrium statistical mechanics |
| title_sort | microscopic chaos fractals and transport in nonequilibrium statistical mechanics |
| topic | Statistische Mechanik (DE-588)4056999-8 gnd Chaotisches System (DE-588)4316104-2 gnd Fraktal (DE-588)4123220-3 gnd Transportprozess (DE-588)4185932-7 gnd Nichtgleichgewichtsthermodynamik (DE-588)4130850-5 gnd |
| topic_facet | Statistische Mechanik Chaotisches System Fraktal Transportprozess Nichtgleichgewichtsthermodynamik |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=015987835&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| volume_link | (DE-604)BV004464593 |
| work_keys_str_mv | AT klagesrainer microscopicchaosfractalsandtransportinnonequilibriumstatisticalmechanics |