Microscopic Chaos, Fractals And Transport in Nonequilibrium Statistical Mechanics (Advanced Series in Nonlinear Dynamics)

Contents......Page 12
Preface......Page 8
1. Introduction and outline......Page 18
1.1 Hamiltonian dynamical systems approach to nonequilibrium statistical mechanics......Page 19
1.2 Thermostated dynamical systems approach to nonequilibrium statistical mechanics......Page 24
1.3 The red thread through this book......Page 28
Part 1: Fractal transport coefficients......Page 32
2.1 A simple model for deterministic diffusion......Page 34
2.2 A parameter-dependent fractal diffusion coefficient......Page 39
2.3 Summary......Page 45
3.1 Drift-diffusion model: mathematical definition......Page 46
3.2 +Calculating deterministic drift and diffusion coefficients......Page 49
3.2.1 Twisted eigenstate method......Page 50
3.2.2 Transition matrix methods......Page 54
3.2.3 Numerical comparison of the different methods......Page 56
3.3 The phase diagram......Page 57
3.4 Simple maps as deterministic ratchets......Page 66
3.5 Summary......Page 71
4.1 A reactive-diffusive multibaker map......Page 72
4.1.1 Deterministic models of reaction-diffusion......Page 73
4.1.2 The Frobenius-Perron operator......Page 77
4.2.1 +Diffusive modes of the dyadic multibaker......Page 79
4.2.2.1 Fractal forms in the Taylor-Green-Kubo formula for dif- fusion......Page 81
4.2.2.2 An area-preserving multibaker with a fractal diffusion co- efficient......Page 84
4.3.1 +Reactive modes of the dyadic multibaker......Page 87
4.3.2 The parameter-dependent reaction rate......Page 92
4.4 Summary......Page 98
5.1 Disordered dynamical systems......Page 100
5.2 Noisy dynamical systems......Page 106
5.3 Summary......Page 115
6.1 Deterministic diffusion and bifurcations......Page 116
6.2 Anomalous diffusion in intermittent maps......Page 124
6.3 Summary......Page 136
7.1 Correlated random walks in maps......Page 138
7.2 Correlated random walks in billiards......Page 145
7.3 Summary......Page 151
8.1 Diffusion in the ower-shaped billiard......Page 154
8.2 +Random and correlated random walks......Page 158
8.3 Diffusion in porous solids......Page 165
8.4 Summary......Page 167
9.1 Resonances and diffusion in the bouncing ball billiard......Page 170
9.2 +Diffusion by correlated random walks......Page 174
9.3 Vibratory conveyors......Page 177
9.4 Summary......Page 178
Part 2: Thermostated dynamical systems......Page 180
10.1 Why thermostats?......Page 182
10.2 Modeling thermal reservoirs: the Langevin equation......Page 184
10.3 Equilibrium velocity distributions for thermostated systems......Page 190
10.4 Applying thermostats: the periodic Lorentz gas......Page 196
10.5 Summary......Page 200
11.1 Construction of the Gaussian thermostat......Page 202
11.2.1 Phase space contraction and entropy production......Page 206
11.2.2 Lyapunov exponents and transport coefficients......Page 207
11.2.3 Nonequilibrium fractal attractors......Page 210
11.2.4 Electrical conductivity......Page 215
11.3 Summary......Page 219
12.1 The dissipative Liouville equation......Page 222
12.2.1 Heuristic derivation......Page 225
12.2.2 Physics of this thermostat......Page 227
12.3.1 Chaos and transport......Page 230
12.3.2 +Generalized Hamiltonian formalism......Page 232
12.3.3 Fractals and transport......Page 235
12.4.1 Necessary conditions and generalizations......Page 239
12.4.2 Thermal reservoirs in nonequilibrium......Page 243
12.5 Summary......Page 244
13.1 Non-Hamiltonian nonequilibrium steady states......Page 248
13.2 Phase space contraction and entropy production......Page 252
13.3 Transport coefficients and dynamical systems quantities......Page 257
13.4 Fractal attractors for nonequilibrium steady states......Page 264
13.5 Nonlinear response in the driven periodic Lorentz gas......Page 268
13.6 Summary......Page 270
14.1 Non-ideal Gaussian thermostat......Page 274
14.2 Non-ideal Nose-Hoover thermostat......Page 278
14.3 +Further alternative thermostats......Page 281
14.4 Summary......Page 283
15. Stochastic and deterministic boundary thermostats......Page 286
15.1 Stochastic boundary thermostats......Page 287
15.2 Deterministic boundary thermostats......Page 288
15.3 +Boundary thermostats from first principles......Page 290
15.4 Deterministic boundary thermostats for the driven periodic Lorentz gas......Page 296
15.4.1 Phase space contraction and entropy production......Page 297
15.4.2 Attractors, bifurcations and conductivity......Page 300
15.4.3 Lyapunov exponents......Page 303
15.5 Hard disk fluid under shear and heat flow......Page 304
15.5.1 Homogeneously and inhomogeneously driven shear and heat flows......Page 305
15.5.2 Shear and heat flows thermostated by deterministic scattering......Page 308
15.6 Summary......Page 317
16. Active Brownian particles and Nos e-Hoover dynamics......Page 320
16.1 Brownian motion of migrating cells?......Page 321
16.2 +Moving biological entities as active Brownian particles......Page 323
16.3 +Bimodal velocity distributions and Nose-Hoover dynamics......Page 325
16.4 Summary......Page 331
Part 3: Outlook and conclusions......Page 334
17. Further topics in chaotic transport theory......Page 336
17.1.1 Entropy fluctuation in nonequilibrium steady states......Page 337
17.1.2 The Gallavotti-Cohen fluctuation theorem......Page 338
17.1.3 The Evans-Searles fluctuation theorem......Page 344
17.1.4 Jarzynski work relation and Crooks relation......Page 345
17.2 Lyapunov modes......Page 348
17.3 Fourier's law......Page 354
17.3.1 The basic problem......Page 355
17.3.2 Heat conduction in anharmonic chaotic chains......Page 357
17.3.3 Heat conduction in chaotic particle billiards......Page 361
17.4 Pseudochaotic diffusion......Page 364
17.4.1 Microscopic chaos and diffusion?......Page 365
17.4.2 Polygonal billiard channels......Page 369
17.5 Summary......Page 381
18.1 Microscopic chaos and nonequilibrium statistical mechanics: the big picture......Page 384
18.2.1 Existence of fractal transport coefficients......Page 388
18.2.2 Universalities in thermostated dynamical systems?......Page 391
18.3 Important open questions......Page 393
18.3.1 Fractal transport coefficients......Page 394
18.3.2 Thermostated dynamical systems......Page 396
Note added in proof......Page 397
Bibliography......Page 398
Index......Page 452