Statistical Mechanics of Nonequilibrium Liquids

✍ Scribed by G. P. Morriss, D. J. Evans

• Publisher: Australian National University E Press
• Year: 2007
• Tongue: English
• Leaves: 318
• Category: Library

✦ Synopsis

In recent years the interaction between dynamical systems theory and non-equilibrium statistical mechanics has been enormous. The discovery of fluctuation theorems as a fundamental structure common to almost all non-equilibrium systems, and the connections with the free energy calculation methods of Jarzynski and Crooks, have excited both theorists and experimentalists. This graduate level book charts the development and theoretical analysis of molecular dynamics as applied to equilibrium and non-equilibrium systems. Designed for both researchers in the field and graduate students of physics, it connects molecular dynamics simulation with the mathematical theory to understand non-equilibrium steady states. It also provides a link between the atomic, nano, and macro worlds. The book ends with an introduction to the use of non-equilibrium statistical mechanics to justify a thermodynamic treatment of non-equilibrium steady states, and gives a direction to further avenues of exploration.

✦ Table of Contents

Statistical Mechanics of Nonequilibrium Liquids......Page 1
Contents......Page 5
Preface......Page 9
Biographies......Page 11
List of Symbols......Page 13
1. Introduction......Page 17
References......Page 26
2.1 The Conservation Equations......Page 27
2.2 Entropy Production......Page 33
2.3 Curie’s Theorem......Page 36
2.4 Non-Markovian Constitutive Relations: Viscoelasticity......Page 43
References......Page 48
Lagrange's equations......Page 49
Hamiltonian mechanics......Page 50
Gauss' Principle of Least Constraint......Page 52
Gauss' Principle for Holonomic Constraints......Page 55
Gauss' Principle for Nonholonomic Constraints......Page 57
3.2 Phase Space......Page 58
3.3 Distribution Functions and the Liouville Equation......Page 59
Time Evolution of the distribution function......Page 62
Properties of Liouville Operators......Page 63
Schrödinger and Heisenberg Representations......Page 64
3.4 Ergodicity, Mixing and Lyapunov Exponents......Page 67
Lyapunov Exponents......Page 69
3.5 Equilibrium Time Correlation Functions......Page 72
3.6 Operator Identities......Page 75
The Dyson Decomposition of Propagators......Page 76
3.7 The Irving-Kirkwood Procedure......Page 79
3.8 Instantaneous Microscopic Representation of Fluxes......Page 85
3.9 The Kinetic Temperature......Page 89
References......Page 90
4.1 The Langevin Equation......Page 93
4.2 Mori-Zwanzig Theory......Page 96
4.3 Shear Viscosity......Page 100
4.4 Green-Kubo Relations for Navier-Stokes Transport Coefficients......Page 105
References......Page 108
5.1 Adiabatic Linear Response Theory......Page 109
The Gaussian Isokinetic Thermostat......Page 114
Nosé-Hoover thermostat - canonical ensemble......Page 118
5.3 Isothermal Linear Response Theory......Page 124
5.4 The Equivalence of Thermostatted Linear Responses......Page 128
References......Page 131
6.1 Introduction......Page 133
BookmarkTitle:......Page 135
BookmarkTitle:......Page 136
6.2 Self Diffusion......Page 140
6.3 Couette Flow and Shear Viscosity......Page 144
Lees Edwards Shearing Periodic Boundaries......Page 145
The SLLOD Algorithm......Page 149
6.4 Thermostatting Shear Flows......Page 158
Thermostats for streaming or convecting flows - PUT......Page 159
6.5 Thermal Conductivity......Page 161
6.6 Norton Ensemble Methods......Page 164
Gaussian Constant Colour Current Algorithm......Page 165
6.7 Constant-Pressure Ensembles......Page 168
Isothermal-Isobaric molecular dynamics......Page 169
6.8 Constant Stress Ensemble......Page 171
References......Page 179
7.1 Kubo’s Form for the Nonlinear Response......Page 183
7.2 Kawasaki Distribution Function......Page 184
7.3 The Transient Time Correlation Function Formalism......Page 188
7.4 Trajectory Mappings......Page 192
7.5. Numerical Results for the Transient Time-Correlation Function......Page 199
7.6. Differential Response Functions......Page 204
7.7 Numerical Results for the Kawasaki Representation......Page 210
7.8 The Van Kampen Objection to Linear Response Theory......Page 214
References......Page 223
8.2 Time Evolution of Phase Variables......Page 225
8.3 The Inverse Theorem......Page 227
8.4 The Associative Law and Composition Theorem......Page 230
8.5 Time Evolution of the Distribution Function......Page 232
8.6 Time Ordered Exponentials......Page 233
8.7 Schrödinger and Heisenberg Representations......Page 234
8.8 The Dyson Equation......Page 236
8.9 Relation Between p- and f- Propagators......Page 237
8.10 Time Dependent Response Theory......Page 239
8.11 Renormalisation......Page 242
8.12 Discussion......Page 244
References......Page 245
9.1 Introduction......Page 247
Transient Time Correlation Function Approach......Page 248
Kawasaki representation......Page 249
9.3 The Compressibility and Isobaric Specific Heat......Page 253
9.4 Differential Susceptibility......Page 255
Kawasaki Representation......Page 257
9.5 The Inverse Burnett Coefficients......Page 258
References......Page 260
10.1 Introduction......Page 261
10.2 Chaotic Dynamical Systems......Page 263
The Quadratic Map......Page 264
The Lorenz Model......Page 272
10.3 The Characterization of Chaos......Page 274
The Fractal and Information Dimensions......Page 275
Generalized Dimensions......Page 276
The Probability Distribution on the Attractor......Page 278
Lyapunov Exponents......Page 280
Lyapunov Dimension......Page 281
10.4 Chaos in Planar Couette Flow......Page 283
Information Dimension......Page 284
Generalized Dimensions......Page 285
Lyapunov Exponents......Page 288
10.5 Green's Expansion for the Entropy......Page 297
References......Page 307
Index......Page 311