Algorithm for the Cross Roll State 6.4.1 Hydrodynamic Equations and Conditions 6.4.2 Fourier Expansion Algorithm for Convective Transitions Convection Theory and Experiment
157 157
159 161 163
Equilibrium Statistical Mechanics 7.1 Hamilton's Equations of Motion 7.1.1 Classical versus Quantum Statistical Mechanics 7.2 Probability Density of an Isolated System 7.2.1 Ergodic Hypothesis 7.2.2 Time, Volume, and Surface Averages 7.2.3 Energy Uniformity 7.2.4 Trajectory Uniformity 7.2.5 Partition Function and Entropy 7.2.6 Internal Entropy of Phase Space Points 7.3
7.6
176 176 177 177 180 182 184 186
186
7.3.1
Maxwell-Boltzmann Distribution
186
7.3.2
Helmholtz Free
188
Energy
Probability Distribution for Other Systems 7.3.4 Equipartition Theorem Transition Probability 7.4.1 Stochastic Equations of Motion 7.4.2 Second Entropy 7.4.3 Mixed Parity Derivation of the Second Entropy and the Equations of Motion
7.5
174
Canonical Equilibrium System
7.3.3
7.4
173
7.4.4
Irreversibility
and
7.4.5
The Fokker-Planck
194
195 195
198 203
Dissipation
Equation Equilibrium Probability
192
and
205
Stationarity
of the 207
Evolution in Phase Space
210
7.5.1
Various Phase Functions
210
7.5.2
Compressibility
214
7.5.3
Liouville's Theorem
216
Reversibility 7.6.1
7.6.2
Isolated
System Canonical Equilibrium System
218 219 220
Detailed Contents
xiv
7.7
8
Trajectory Probability and Time Correlation 7.7.1 Trajectory Probability
226
226
7.7.2
Equilibrium Averages
227
7.7.3
Time Correlation Functions
227
7.7.4
Reversibility
229
Statistical Mechanics
Non-Equilibrium
233
8.1
General Considerations
233
8.2
Reservoir
235
8.2.1
Entropy Trajectory Entropy
235
8.2.2
Reduction
8.2.3
Fluctuation Form for the Reservoir
8.3
8.4
the Point
to
240
Space
240
8.3.2
Dependent Weight Fluctuation Form of the Second Entropy
8.3.3
Time Correlation Function
247
8.3.4
Stochastic, Dissipative Equations of Motion
249
8.3.5
Transition
8.3.6
Most
Changes
in
Probability
and Fokker-Planck
244
Equation
Likely
258 260 262
in
262
8.4.2
Entropy Irreversibility and Dissipation
8.4.3
Various Time Derivatives
268
Steady
State
273
Odd
8.6
Path
Entropy
8.6.1
Path
8.6.2
Fluctuation and Work Theorem
Projection
266
System
8.5
of the
Dynamic
Reservoir
275
Entropy
and Transitions
280 280
Entropy
287
Entropy for Mechanical Work 8.7.1 Evolution of the Reservoir Entropy and Transitions Path
8.7.2
Transition Theorems
Statistical Mechanics of
289 .
.
.
.
289 292
9.1.1
Steady Flow: Heat and Steady Heat Flow Canonical Equilibrium System
9.1.2
Fourier's Law of Heat Conduction
296
9.1.3
Second
299
Thermodynamics
Shear
of
Entropy
for Heat Flow
9.3
Space Probability Density Explicit Hamiltonian and First Energy Moment 9.2.2 Reservoir Entropy and Probability Density Most Likely Trajectory
9.4
Equipartition
9.5
Green-Kubo Expressions for the Thermal
9.2
....
Force with Constraints
Entropy and Time Derivatives
Change
8.4.4
9.1
239
Entropy
Foundations for Time
8.4.1
8.7
237
Entropy
Transitions and Motion in Phase 8.3.1
9
Functions
295 295 295
Phase
303
9.2.1
303
Theorem for Heat Flow
306 308
310
Conductivity
313
9.5.1
Isolated
System
313
9.5.2
Heat Reservoirs
315
9.5.3
Relation with Odd
Projection
318
Detailed Contents
9.6
Shear Flow
9.6.3 9.6.4
Equipartition
9.6.2
Second
10 Generalised
10.2
320
Entropy for Shear Flow Phase Space Probability Density Most Likely Trajectory
9.6.1
10.1
xv
322
323 326
Theorem
327
Langevin Equation
329
Free Brownian Particle
331
10.1.1
Time Correlation Functions
332
10.1.2
Mixed Parity Digression
335
10.1.3
Diffusion Constant
10.1.4
Trajectory Entropy
Langevin
337 and
and Smoluchowski
10.3 Perturbation
Correlation
338 342
Equations
Theory
343
10.3.1
Most
10.3.2
Alternative Derivation
347
10.3.3
Most
348
10.3.4
Stochastic
10.3.5 10.3.6
343
Likely Velocity
Likely
Position
Dissipative Equations of Motion Generalised Langevin Equation for Velocity Fluctuation Dissipation Theorem
348 351
353
10.3.7 Weiner-Khintchine Theorem 10.3.8
354
Exponentially Decaying Memory
10.4 Adiabatic Linear
Function
Response Theory
10.5 Numerical Results for
a
356
Brownian Particle in
a
Moving Trap
358
Langevin Theory
359
Smoluchowski Theory
360
10.5.3 Computer Simulations
360
10.5.6
Perturbation
Algorithm
361
10.6.1
361
Lag
Stochastic Trajectory
10.6 Generalised Langevin Equation in the Case of Mixed Parity
Equilibrium System
10.6.2
Regression
10.6.3
Time
10.6.4
Generalised Langevin Equation
364 .
.
.
366 366
of Fluctuation
371
Dependent Perturbation
373
Projector Operator Formalism
10.8 Harmonic Oscillator Model for the Memory Function
