Microscopic Chaos, Fractals And Transport in Nonequilibrium Statistical Mechanics

by Klages, Rainer

• ISBN13: 9789812565075
• ISBN10: 9812565078
• Format: Hardcover
• Copyright: 2007-09-30
• Publisher: World Scientific Pub Co Inc

Summary

A valuable introduction for newcomers as well as an important reference and source of inspiration for established researchers, this book provides an up-to-date summary of central topics in the field of nonequilibrium statistical mechanics and dynamical systems theory. Understanding macroscopic properties of matter starting from microscopic chaos in the equations of motion of single atoms or molecules is a key problem in nonequilibrium statistical mechanics. Of particular interest both for theory and applications are transport processes such as diffusion, reaction, conduction and viscosity. Recent advances towards a deterministic theory of nonequilibrium statistical physics are summarized: both Hamiltonian dynamical systems under nonequilibrium boundary conditions and non-Hamiltonian modeling of nonequilibrium steady states by using thermal reservoirs. The surprising new results include transport coefficients that are fractal functions of control parameters, fundamental relations between transport coefficients and chaos quantities, and an understanding of nonequilibrium entropy production in terms of fractal measures and attractors. The theory is particularly useful for the description of many-particle systems with properties in-between conventional thermodynamics and nonlinear science, as they are frequently encountered on nanoscales.

Table of Contents

Preface	p. vii
Introduction and outline	p. 1
Hamiltonian dynamical systems approach to nonequilibrium statistical mechanics	p. 2
Thermostated dynamical systems approach to nonequilibrium statistical mechanics	p. 7
The red thread through this book	p. 11
Fractal transport coefficients	p. 15
Deterministic diffusion	p. 17
A simple model for deterministic diffusion	p. 17
A parameter-dependent fractal diffusion coefficient	p. 22
Summary	p. 28
Deterministic drift-diffusion	p. 29
Drift-diffusion model: mathematical definition	p. 29
Calculating deterministic drift and diffusion coefficients	p. 32
Twisted eigenstate method	p. 33
Transition matrix methods	p. 37
Numerical comparison of the different methods	p. 39
The phase diagram	p. 40
Simple maps as deterministic ratchets	p. 49
Summary	p. 54
Deterministic reaction-diffusion	p. 55
A reactive-diffusive multibaker map	p. 55
Deterministic models of reaction-diffusion	p. 56
The Frobenius-Perron operator	p. 60
Diffusive dynamics	p. 62
Diffusive modes of the dyadic multibaker	p. 62
The parameter-dependent diffusion coefficient	p. 64
Reactive dynamics	p. 70
Reactive modes of the dyadic multibaker	p. 70
The parameter-dependent reaction rate	p. 75
Summary	p. 81
Deterministic diffusion and random perturbations	p. 83
Disordered dynamical systems	p. 83
Noisy dynamical systems	p. 89
Summary	p. 98
From normal to anomalous diffusion	p. 99
Deterministic diffusion and bifurcations	p. 99
Anomalous diffusion in intermittent maps	p. 107
Summary	p. 119
From diffusive maps to Hamiltonian particle billiards	p. 121
Correlated random walks in maps	p. 121
Correlated random walks in billiards	p. 128
Summary	p. 134
Designing billiards with irregular transport coefficients	p. 137
Diffusion in the flower-shaped billiard	p. 137
Random and correlated random walks	p. 141
Diffusion in porous solids	p. 148
Summary	p. 150
Deterministic diffusion of granular particles	p. 153
Resonances and diffusion in the bouncing ball billiard	p. 153
Diffusion by correlated random walks	p. 157
Vibratory conveyors	p. 160
Summary	p. 161
Thermostated dynamical systems	p. 163
Motivation: coupling a system to a thermal reservoir	p. 165
Why thermostats?	p. 165
Modeling thermal reservoirs: the Langevin equation	p. 167
Equilibrium velocity distributions for thermostated systems	p. 173
Applying thermostats: the periodic Lorentz gas	p. 179
Summary	p. 183
The Gaussian thermostat	p. 185
Construction of the Gaussian thermostat	p. 185
Chaos and transport in Gaussian thermostated systems	p. 189
Phase space contraction and entropy production	p. 189
Lyapunov exponents and transport coefficients	p. 190
Nonequilibrium fractal attractors	p. 193
Electrical conductivity	p. 198
Summary	p. 202
The Nose-Hoover thermostat	p. 205
The dissipative Liouville equation	p. 205
Construction of the Nose-Hoover thermostat	p. 208
Heuristic derivation	p. 208
Physics of this thermostat	p. 210
Properties of the Nose-Hoover thermostat	p. 213
Chaos and transport	p. 213
Generalized Hamiltonian formalism	p. 215
Fractals and transport	p. 218
Subtleties of Nose-Hoover dynamics	p. 222
Necessary conditions and generalizations	p. 222
Thermal reservoirs in nonequilibrium	p. 226
Summary	p. 227
Universalities in Gaussian and Nose-Hoover dynamics?	p. 231
Non-Hamiltonian nonequilibrium steady states	p. 231
Phase space contraction and entropy production	p. 235
Transport coefficients and dynamical systems quantities	p. 240
Fractal attractors for nonequilibrium steady states	p. 247
Nonlinear response in the driven periodic Lorentz gas	p. 251
Summary	p. 253
Gaussian and Nose-Hoover thermostats revisited	p. 257
Non-ideal Gaussian thermostat	p. 257
Non-ideal Nose-Hoover thermostat	p. 261
Further alternative thermostats	p. 264
Summary	p. 266
Stochastic and deterministic boundary thermostats	p. 269
Stochastic boundary thermostats	p. 270
Deterministic boundary thermostats	p. 271
Boundary thermostats from first principles	p. 273
Deterministic boundary thermostats for the driven periodic Lorentz gas	p. 279
Phase space contraction and entropy production	p. 280
Attractors, bifurcations and conductivity	p. 283
Lyapunov exponents	p. 286
Hard disk fluid under shear and heat flow	p. 287
Homogeneously and inhomogeneously driven shear and heat flows	p. 288
Shear and heat flows thermostated by deterministic scattering	p. 291
Summary	p. 300
Active Brownian particles and Nose-Hoover dynamics	p. 303
Brownian motion of migrating cells?	p. 304
Moving biological entities as active Brownian particles	p. 306
Bimodal velocity distributions and Nose-Hoover dynamics	p. 308
Summary	p. 314
Outlook and conclusions	p. 317
Further topics in chaotic transport theory	p. 319
Fluctuation relations	p. 320
Entropy fluctuation in nonequilibrium steady states	p. 320
The Gallavotti-Cohen fluctuation theorem	p. 321
The Evans-Searles fluctuation theorem	p. 327
Jarzynski work relation and Crooks relation	p. 328
Lyapunov modes	p. 331
Fourier's law	p. 337
The basic problem	p. 338
Heat conduction in anharmonic chaotic chains	p. 340
Heat conduction in chaotic particle billiards	p. 344
Pseudochaotic diffusion	p. 347
Microscopic chaos and diffusion?	p. 348
Polygonal billiard channels	p. 352
Summary	p. 364
Conclusions	p. 367
Microscopic chaos and nonequilibrium statistical mechanics: the big picture	p. 367
Assessment of the main results	p. 371
Existence of fractal transport coefficients	p. 371
Universalities in thermostated dynamical systems?	p. 374
Important open questions	p. 376
Fractal transport coefficients	p. 377
Thermostated dynamical systems	p. 379
Note added in proof	p. 380
Bibliography	p. 381
Index	p. 435
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