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Dorfman J.R. Introduction to Chaos in Nonequilibrium Statistical Mechanics
Dorfman J.R.  Introduction to Chaos in Nonequilibrium Statistical Mechanics









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: Introduction to Chaos in Nonequilibrium Statistical Mechanics

: Dorfman J.R.

:

This book is an introduction to the applications in nonequilibrium statistical mechanics of chaotic dynamics, and also to the use of techniques in statistical mechanics important for an understanding of the chaotic behaviour of fluid systems. The fundamental concepts of dynamical systems theory are reviewed and simple examples are given. Advanced topics including SRB and Gibbs measures, unstable periodic orbit expansions, and applications to billiard-ball systems, are then explained. The text emphasises the connections between transport coefficients, needed to describe macroscopic properties of fluid flows, and quantities, such as Lyapunov exponents and Kolmogorov-Sinai entropies, which describe the microscopic, chaotic behaviour of the fluid. Later chapters consider the roles of the expanding and contracting manifolds of hyperbolic dynamical systems and the large number of particles in macroscopic systems. Exercises, detailed references and suggestions for further reading are included.


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ed2k: ed2k stats

: 1999

: 287

: 06.12.2004

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Absolutely continuous measure      132
Anosov system      17 122 123 125 126 148 183 186 228 232 234
Arnold cat map      114
Arnold cat map and irreversibility      228
Arnold cat map, ergodicity and mixing property      115
Arnold cat map, H-function      231
Arnold cat map, H-theorem      115 228
Arnold cat map, KS entropy      127
Arnold cat map, Lyapunov exponents      114
Arnold cat map, measure-preserving property      112
Arnold cat map, periodic points      204
Arnold cat map, stable and unstable manifolds      114 115 228
Arnold cat map, topological entropy      207
Arnold cat map, topological zeta-function      207 (see also Toral automorphism)
Arnold cat mapas Anosov diffeomorphism      124
Arrow of time      4 104
Attractor      158 160 163
Attractor, Axiom-A      124
Attractor, hyperbolic      176 177
Axiom-A system      17 124
Axiom-A system, non-wandering set of points      124
Bakers transformation      89
Bakers transformation and irreversibility      94 108
Bakers transformation as Bernoulli shift      96
Bakers transformation, area-preserving property      104
Bakers transformation, Bernoulli sequences      94 96 203
Bakers transformation, Bogoliubovs arguments      94 110
Bakers transformation, Boltzmann equation      9093
Bakers transformation, cylinder set      97
Bakers transformation, dyadic (bi-infinite sequence) representation      95 96
Bakers transformation, dynamical partition function      190 215
Bakers transformation, ergodicity and mixing property      90 93 104106 108
Bakers transformation, Gibbs measure      190
Bakers transformation, H-theorem      9092
Bakers transformation, invariant measure      105
Bakers transformation, KS entropy      121 122 127
Bakers transformation, Lyapunov exponents      103 104 122
Bakers transformation, Markov partition      124 125
Bakers transformation, measure-preserving property      89
Bakers transformation, partition of the unit square      119121
Bakers transformation, periodic points      98 109 130 204
Bakers transformation, Probenius Perron equation      90
Bakers transformation, separated set      190
Bakers transformation, stable and unstable manifolds      105 106 124
Bakers transformation, symbolic dynamics      96 120
Bakers transformation, topological entropy      190 207
Bakers transformation, topological pressure      192 215
Bakers transformation, topological zeta-function      207
BBGKY hierarchy equations      51 53 232
Bernoulli shift/process      104 109 118 143
Bernoulli system      122
Bijection      130
Billiard systems      240242 253
Billiard systems, dispersing/semi-dispersing      251
Billiard systems, ergodicity and mixing property      13 73 242
Billiard systems, stable and unstable manifolds      241
Birkhoffs individual ergodic theorem      11 62 63 177 246
Birkhoffs individual ergodic theorem, Hopfs proof      11
Boltzmann (transport) equation      30 40 227
Boltzmann (transport) equation, assumption of molecular chaos      3 25
Boltzmann (transport) equation, Chapman Enskog method      39
Boltzmann (transport) equation, collision cylinder      24 25 27
Boltzmann (transport) equation, direct collision      22 2628 32
Boltzmann (transport) equation, free-streaming term      22
Boltzmann (transport) equation, gain term      22 29
Boltzmann (transport) equation, impact parameter      2629 242
Boltzmann (transport) equation, loss term      22 24
Boltzmann (transport) equation, restituting collision      22 27 28 32 248
Boltzmann (transport) equation, Sonine polynomials      43
Boltzmann (transport) equation, wall (collision) term      22 31
Box-Counting Dimension      144
Bunimovich stadium      253 254
Cellular automata lattice gases (CALG)      217
Cellular automata lattice gases (CALG), exclusion principle      218
Cellular automata lattice gases (CALG), Frisch Hasslacher Pomeau (FHP) model      218
Cellular automata lattice gases (CALG), stochastic/deterministic      218
Center subspace/manifold      123
Chaotic hypothesis      234
Chapman Enskog method      39 41
Chapman Enskog method for Boltzmann equation      39 83
Chapman Enskog method for Liouville equation      79
Chapman Kolmogorov equation      220
Characteristic function      50 64 68 70
Cumulative (distribution) function      168 172
Diffeomorphism      90 183
Diffusion coefficient      40 45 46 152
Diffusion coefficient and fractal structure      195 201
Diffusion coefficient and mean square displacement      46 86
Diffusion coefficient and Takagi function      200
Diffusion coefficient for one-dimensional maps      196
Diffusion coefficient, Einstein relation      46
Diffusion coefficient, generating function method      211 214
Diffusion coefficient, Green Kubo expression      83 86
Diffusion coefficient, time-dependent      86
Diffusion coefficient, using periodic orbit expansion      207
Diffusion coefficientand velocity autocorrelation-function      86
Diffusion equation      40 46 152
Dynamical chaos      12 240
Dynamical partition function      149 150 183187 192 215
Dynamical phase transition      192 224
Ehrenfest urn model      56
Ehrenfest urn model, Loschmidts paradox      56
Ehrenfest urn model, Zermelos paradox      56
Ehrenfest wind-tree model      47 48 247
Ergodic hypothesis      13
Ergodicity and Gibbs mixing      68 69
Ergodicity and metric indecomposability      66 6870
Ergodicity and random perturbations      61 62
Ergodicity for hyperbolic systems      148
Ergodicity for periodic orbits      208
Ergodicity of sequences of numbers      65 66
Ergodicity, Birkhoffs theorem      11 62 63 177 246
Ergodicity, equal-times-in-equal-areas rule      5 7 8 62 64 65
Ergodicity, ergodic hypothesis      5 7 8 58 60 62
Ergodicity, escape-rate formalism      15 16 118 136 138 141 149 153 176 186
Ergodicity, Hopfs method      108
Ergodicity, infinite/long-time limit/average      6 7 5860 62 63 105 131
Extended Lorentz Boltzmann equation      246 249
Fermi Pasta Ulam model      12
Fokker Planck equation      11
Fokker Planck equation and Green Kubo formulae      82
Fractal set      144
Fractal set, box-counting dimension      144 145
Fractal set, Cantor set      137 139 142144 146
Fractal set, information dimension      160
Fractal set, multifractal      166 168
Fractal set, self-similarity      146
Fredholm determinant method      210 211
Frobenius Perron equation      129 134 168 175 180 259
Frobenius Perron operator      134 198 201 203 208 213 259
Gallavotti Cohen fluctuation formula      14 181 182
Generating partition      121
Gibbs measure      16 160 164 183 185 188 210 214 215
Gibbs mixing      9 67
Gibbs mixing and ergodicity      68 69
Green Kubo formulae      12
Green Kubo formulae for maps      195 196
Green Kubo formulae, diffusion coefficient      83 86
Green Kubo formulae, electrical conductivity      79
Green Kubo formulae, fractal forms      197
Green Kubo formulae, intermediate scattering function      85
Green Kubo formulae, linear response derivation      75
Green Kubo formulae, Liouville operator      76
Green Kubo formulae, long-time tails/algebraic decay      218 233 237
Green Kubo formulae, time-correlation function      75 79
Green Kubo formulae, time-displacement operator      77
Green Kubo formulae, using Fokker Planck equation      82
Green Kubo formulae, van Kampens objections      79
H-Theorem      2 3 3133 115 227 228
H-theorem, detailed balance      32
H-theorem, H-function      31
Hausdorff dimension      146 147
Homoclinic and heteroclinic tangles      10 142
Hyperbolic system      13 17 122124 148 152 177 187189 191 192 203 205 227 228 232 234 236 237 241 242
Invariant measure      51 59 62 63 123 132 163 164 170 176
Invariant tori      10
Jacobian derivative of a map      129 165 180 209 214
Jump function      198
Kac ring model      34 39 217 247 254
Kac ring model and irreversibility      36
Kac ring model, Lifshitz tails      38
Kac ring model, Loschmidts paradox      36
Kac ring model, Poincare recurrence      35
Kac ring model, Zermelos paradox      36 38
Kac ring model, anti-Boltzmann behavior      38
Kadanoff Tang formula      209
Kolmogorov Arnold Moser (KAM) theorem      12
Kolmogorov Sinai (KS) entropy      118 119 132 137 150 234
Langevin equation      11 233
Law of Universal Return      54
Lebesgue singular function      170173
Lifshitz tails      38
Liouville equation      49 70
Liouville equation, volume-preserving property      51 54
Liouville measure      51 123
Lorentz gas      154 161 181 211 215 241 247
Lorentz gas as a billiard system      240
Lorentz gas, dilute, random      246 249
Lorentz gas, ergodicity and mixing property      240 246
Lorentz gas, Grad limit      237
Lorentz gas, Lyapunov exponents      240 244 246 251
Lorentz gas, periodic orbit      212
Lorentz gas, separation of nearby trajectories      72 73
Lorentz lattice gases      218
Lorentz lattice gases, BBGKY hierarchy equations      221
Lorentz lattice gases, Boltzmann equation      221
Lorentz lattice gases, Chapman Kolmogorov equation for      220
Lorentz lattice gases, dynamical partition function      222 223
Lorentz lattice gases, thermodynamic formalism      222
Lorentz lattice gases, topological pressure      223226
Loschmidts paradox      2 36 56
Lyapunov exponent      9 100 101
Lyapunov exponent and information production      119
Lyapunov exponent and separation of nearby initial states      101
Lyapunov exponent for a periodic orbit      209
Lyapunov exponent, symplectic condition      104 114 159 186 241
Lyapunov exponent, using Boltzmann equation      240
Lyapunov exponent, using Probenius Perron equation      130
Macroscopic and microscopic chaos      238
Manifolds/directions, stable and unstable      105 106 108110 114 115 119 123 124 142 160 165 177 203 227 228 232 241
Markov partition      122 124 125 176
Markov partition and KS entropy      124 125
Markov partition and Markov process      125 126
Markov partition and Markov transition matrix      126
Markov partitionand Pesins theorem      126
Maxwell Boltzmann distribution function      2 33 40 41 65
Measure/volume-preserving transformation      134
Mechanical problem of statistical mechanics      4
Metric indecomposability      6870
Molecular chaos      240
Molecular chaos, assumption of      3 25
Multiplicative ergodic theorem of Oseledets      102
Navier Stokes equations      233 238
Non-exponential separation of phase points      240
Nonequilibrium stationary states      14 157 201
Open systems/absorbing boundaries      129 136 153 186 188
Partition of a set      119
Periodic orbit expansion      16 152 178 203
Periodic orbit/point      69 124 126 205
Periodic orbit/point, prime      205
Periodic orbit/point, reducible      205
Pesins theorem      126
Poincare recurrence      65
Poincare recurrence, quantum systems      55 57
Poincare recurrence, recurrence theorem      2 6 35 54 55 124 241
Poincare recurrence, recurrence time      3 36 55 235 236
Quantum chaos      11 16 203 262264
Random matrices      262
Random perturbations      61 236 262
Random perturbations and ergodicity      61 62
Repeller      16 138 139 142 146 154 163 164 251
Repeller, forward      142
Repeller, invariant      143 145
Repeller, KS entropy      139
Repeller, Lyapunov exponents      139 147
Repeller, time-reversed      143
Rotation map      64
Rotation map, ergodicity      64 65
Rotation map, KS entropy      122
Rotation map, partition of the unit circle      122
Ruelle zeta-function      214
Ruelle Pollicott resonances      16 201 202 259
Ruelles inequality      126
Second law of thermodynamics      1 5 33 55 74
Second law of thermodynamics, Clausius form      30
Separated set      122 189 203 215
Shear flow      181 260
Sinai Ruelle Bowen (SRB) measure      15 71 126 160 163 164 170 175
Sinai Ruelle Bowen (SRB) measure, Hausdorff dimension      175
Sinai Ruelle Bowen (SRB) measure, Hausdorff dimension, Kaplan Yorke formula      174
Sinai Ruelle Bowen (SRB) measure, Hausdorff dimension, Youngs formula      173 175
Sinai Ruelle Bowen (SRB) theorem      176 177
Sinais formula      242
Smale horseshoe      142145
Smale horseshoe, escape rate      142
Sonine polynomials      43
Spatiotemporal chaos      18
Stosszahlansatz      25 29 31 3335 37 47 232 235 237
Tagged particle diffusion      39 75
Tagged particle diffusion, Boltzmann equation      40
Tagged particle diffusion, Chapman Enskog method      39 41
Tagged particle diffusion, cumulant expansion      85
Tagged particle diffusion, diffusion coefficient      40 45 46
Tagged particle diffusion, diffusion equation      40 46
Tagged particle diffusion, generating function for moments      85
Tagged particle diffusion, Green Kubo formula      83
Tagged particle diffusion, probability current      41 42
Tagged particle diffusion, probability density      40 84
Tagged particle diffusion, Sonine polynomials      43
Tagged particle diffusion, hydrodynamic stage      41
Tagged particle diffusion, kinetic stage      41
Takagi function      200202
Thermodynamic formalism for chaos      13 150 183 215 218
Thermodynamic/large-system limit      86 234238
Thermostatted system      163 164 167 172 174 251
Thermostatted system, coarse-grained entropy      178
Thermostatted system, diffusion coefficient      154
Thermostatted system, entropy production      158 178 179
Thermostatted system, friction coefficient      155 158
Thermostatted system, Gaussian thermostat method      15 154 156 216
Thermostatted system, Liouville equation      156 157
Thermostatted system, macroscopic escape rate      153
Thermostatted system, microscopic escape rate      154
Time-reversal operator      167
Time-reversal operator for a map      167
Toral automorphism      111 112 114 203205 207 228
Tpological entropy      185 190 207
Tpological pressure      149 150 183 185 187 190 191
Tpological zeta-function      205207
Tpological zeta-function and Riemann zeta-function      206
Transitive system      123
Virial/cluster expansions      11 12
Viscous flow      154 261
Weak limit      71
Zermelos paradox      2 36 38 56
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