Statistical Mechanics of Nonequilibrium Liquids (Cambridge 2008)

Denis J. Evans, Gary Morriss0521857910, 9780521857918, 9780511397097

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 :
Cover……Page 1
Half-title……Page 3
Title……Page 5
Copyright……Page 6
Contents……Page 7
Preface to the second edition……Page 11
Preface to the first edition……Page 13
1 Introduction……Page 15
2.1 The conservation equations……Page 25
2.2 Entropy production……Page 31
2.3 Curie’s theorem……Page 34
2.4 Non-Markovian constitutive relations: viscoelasticity……Page 41
Newtonian mechanics……Page 47
Lagrangian mechanics……Page 48
Hamiltonian mechanics……Page 49
The fundamental principles of mechanics……Page 51
Gauss’ principle of least constraint……Page 53
Gauss’ principle for holonomic constraints……Page 56
Gauss’ principle for nonholonomic constraints……Page 57
3.2 Phase space……Page 58
3.3 Distribution functions and the Liouville equation……Page 60
Time evolution of the distribution function……Page 62
Time evolution of phase variables……Page 63
Schrodinger and Heisenberg representations……Page 64
3.4 Ergodicity, mixing, and Lyapunov exponents……Page 67
Lyapunov exponents……Page 70
3.5 Equilibrium time-correlation functions……Page 73
3.6 Operator identities……Page 76
The Dyson decomposition of propagators……Page 77
3.7 The Irving–Kirkwood procedure……Page 80
3.8 Instantaneous microscopic representation of fluxes……Page 82
k-Space representations……Page 89
3.9 Microscopic representation of the temperature……Page 91
4.1 The Langevin equation……Page 93
4.2 Mori–Zwanzig theory……Page 97
4.3 Shear viscosity……Page 101
4.4 Green–Kubo relations for Navier–Stokes transport coefficients……Page 106
5.1 Adiabatic linear response theory……Page 109
The Gaussian isokinetic thermostat……Page 114
Hamiltonian formulation of the GIK thermostat……Page 119
Nosé–Hoover thermostat – canonical ensemble……Page 121
5.3 Isothermal linear response theory……Page 125
5.4 The equivalence of thermostatted linear responses……Page 130
6.1 Introduction……Page 133
6.2 Self diffusion……Page 139
Lees–Edwards shearing periodic boundaries……Page 144
The SLLOD algorithm……Page 147
Thermostats for secondary or convecting flows……Page 156
6.5 Elongational flows……Page 160
6.6 Thermal conductivity……Page 164
6.7 Norton ensemble methods……Page 166
Gaussian constant color current algorithm……Page 167
6.8 Constant-pressure ensembles……Page 170
Isothermal-isobaric molecular dynamics……Page 171
Isobaric–isoenthalpic molecular dynamics……Page 173
6.9 Constant stress ensembles……Page 174
7.1 Kubo’s form for the nonlinear response……Page 181
7.2 Kawasaki distribution function……Page 183
7.3 The transient time-correlation function formalism……Page 187
7.4 Trajectory mappings……Page 191
Dynamics……Page 193
Numerical results for the transient time-correlation function……Page 195
7.5 Differential response functions……Page 199
Numerical results for the Kawasaki representation……Page 205
7.6 The van Kampen objection to linear response theory……Page 207
7.7 Time-dependent response theory……Page 214
The inverse theorem……Page 215
The associative law and composition theorem……Page 216
The distribution function……Page 217
Response theory……Page 219
8.1 Introduction……Page 223
8.2 Chaotic dynamical systems……Page 225
The quadratic map……Page 226
The Lorenz model……Page 233
Simple properties……Page 234
8.3 The characterization of chaos……Page 235
The capacity and information dimensions……Page 236
Correlation dimension……Page 237
Generalized dimensions……Page 238
The probability distribution on the attractor……Page 239
Lyapunov exponents……Page 242
Lyapunov dimension……Page 243
8.4 Chaos in planar Couette flow……Page 244
Generalized dimensions……Page 246
Lyapunov exponents……Page 247
Numerical conjugate pairing……Page 249
8.5 Conjugate pairing of Lyapunov exponents……Page 252
8.6 Periodic orbit measures……Page 258
Time evolution of densities……Page 262
Evolution operators……Page 265
Spectral determinants……Page 266
8.7 Positivity of transport coefficients……Page 269
9.1 Introduction……Page 273
9.2 The specific heat……Page 274
Transient time-correlation function approach……Page 275
Kawasaki representation……Page 276
9.3 The compressibility and isobaric specific heat……Page 279
9.4 The fluctuation theorem……Page 281
The transient fluctuation theorem……Page 282
9.5 Gallavotti and Cohen fluctuation theorem……Page 288
9.6 The Jarzynski equality……Page 291
9.7 The Crooks relation……Page 294
9.8 Experimental verification……Page 296
10.1 The thermodynamic temperature……Page 297
Nonequilibrium systems……Page 302
10.2 Green’s expansion for the entropy……Page 306
10.3 Prospects……Page 313
References……Page 315
index……Page 323