Preface ......................................................... v
Classical Statistical Dynamics .................................. 1
1 Introduction ................................................. 3
2 Probability Theory .......................................... 13
2.1 Sample Spaces and States ............................... 13
2.2 Random Variables, Algebras ............................. 24
2.3 Entropy ................................................ 34
2.4 Exercises .............................................. 39
3 Linear Dynamics ............................................. 43
3.1 Reversible Dynamics .................................... 43
3.2 Random Dynamics ........................................ 48
3.3 Convergence to Equilibrium ............................. 60
3.4 Markov Chains .......................................... 66
3.5 Exercises .............................................. 69
4 Isolated Dynamics ........................................... 73
4.1 The Boltzmann Map ...................................... 73
4.2 The Heat-Particle ...................................... 87
4.3 The Hard-Core Model of Chemical Kinetics ............... 94
4.3.1 Isomers and Diffusion in a Force-Field .......... 95
4.3.2 Markov Dynamics ................................ 100
4.3.3 Entropy Production ............................. 102
4.3.4 Osmosis ........................................ 103
4.3.5 Exchange Diffusion ............................. 104
4.3.6 General Diffusions ............................. 105
4.4 Chemical Reactions .................................... 106
4.4.1 Unimolecular Reactions ......................... 106
4.4.2 Balanced Reactions ............................. 107
4.5 Energy of Solvation ................................... 111
4.6 Activity-led Reactions ................................ 111
4.7 Exercises ............................................. 119
5 Isothermal Dynamics ........................................ 123
5.1 Legendre Transforms ................................... 124
5.2 The Free-energy Theorem ............................... 126
5.3 Chemical Kinetics ..................................... 130
5.4 Convergence in Norm ................................... 137
5.5 Dilation of Markov Chains ............................. 146
5.6 Exercises ............................................. 149
6 Driven Systems ............................................. 151
6.1 Sources and Sinks ..................................... 151
6.2 A Poor Conductor ...................................... 152
6.3 A Driven Chemical System .............................. 155
6.4 How to Add Noise ...................................... 162
6.5 Exercises ............................................. 165
7 Fluid Dynamics ............................................. 167
7.1 Hydrostatics of a Gas of Hard Spheres ................. 168
7.2 The Fundamental Equation .............................. 171
7.3 The Euler Equations ................................... 177
7.4 Entropy Production .................................... 178
7.5 A Correct Navier-Stokes System ........................ 181
Quantum Statistical Dynamics .................................. 187
8 Introduction to Quantum Theory ............................. 189
9 Quantum Probability ........................................ 197
9.1 Algebras of Observables ............................... 197
9.2 States ................................................ 204
9.3 Quantum Entropy ....................................... 213
9.4 Exercises ............................................. 217
10 Linear Quantum Dynamics .................................... 221
10.1 Reversible Dynamics ................................... 221
10.2 Random Quantum Dynamics ............................... 224
10.3 Quantum Dynamical Maps ................................ 228
10.4 Exercises ............................................. 236
11 Isolated Quantum Dynamics .................................. 237
11.1 The Quantum Boltzmann Map ............................. 237
11.2 The Quantum Heat-Particle ............................. 240
11.3 Fermions and Ions with a Hard Core .................... 256
11.4 The Quantum Boltzmann Equation ........................ 272
11.5 Exercises ............................................. 281
12 Isothermal and Driven Systems .............................. 283
12.1 Isothermal Quantum Dynamics ........................... 283
12.2 Convergence to Equilibrium ............................ 289
12.3 Driven Quantum Systems ................................ 292
12.4 Exercises ............................................. 296
13 Infinite Systems ........................................... 297
13.1 The Algebra of an Infinite System ..................... 299
13.2 The Reversible Dynamics ............................... 300
13.3 Return to Equilibrium ................................. 302
13.4 Irreversible Linear Dynamics .......................... 306
13.5 Exercises ............................................. 309
14 Proof of the Second Law .................................... 311
14.1 von Neumann Entropy ................................... 311
14.2 Entropy Increase in Quantum Mechanics ................. 312
14.3 The Quantum Kac Model ................................. 314
14.4 Equilibrium ........................................... 315
14.5 Thee-Limit ............................................ 316
14.6 The Marginals and Entropy ............................. 316
14.7 The Results ........................................... 317
15 Information Geometry ....................................... 319
15.1 The Jaynes-Ingarden Theory ............................ 319
15.2 Non-Linear Ising Dynamics ............................. 322
15.3 Ising Model Close to Equilibrium ...................... 327
15.4 Non-linear Heisenberg Model ........................... 329
15.5 Estimation; the Cramer-Rao Inequality ................. 333
15.6 Efron, Dawid and Amari ................................ 337
15.7 Entropy Methods, Exponential Families ................. 340
15.8 The Work of Pistone and Sempi ......................... 341
15.9 The Finite-Dimensional Quantum Info-Manifold .......... 346
15.10 Araki's Expansionals and the Analytic Manifold ....... 352
15.11 The Quantum Young Function ........................... 354
15.12 The Quantum Cramér Class ............................. 359
15.13 The Parameter-Free Quantum Manifold .................. 360
15.14 Exercises ............................................ 364
Bibliography .................................................. 367
Index ......................................................... 377
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