A kinetic view of statistical physics

Preface . . xi
Conventions . . xiv

1 Aperitifs . . 1
1.1 Diffusion . . 1
1.2 Single-species annihilation/coalescence . . 4
1.3 Two-species annihilation . . 9
1.4 Notes . . 10

2 Diffusion . . 12
2.1 The probability distribution . . 12
2.2 Central limit theorem . . 15
2.3 Walks with broad distributions . . 17
2.4 Application to gravity: the Holtsmark distribution . . 22
2.5 First-passage properties . . 26
2.6 Exit probabilities and exit times . . 30
2.7 Reaction rate theory . . 37
2.8 The Langevin approach . . 40
2.9 Application to surface growth . . 44
2.10 Notes . . 51
2.11 Problems . . 51

3 Collisions . . 59
3.1 Kinetic theory . . 51
3.2 The Lorentz gas . . 63
3.3 Lorentz gas in an external field . . 70
3.4 Collisional impact . . 75
3.5 Maxwell molecules and very hard particles . . 77
3.6 Inelastic gases . . 81
3.7 Ballistic agglomeration . . 89
3.8 Single-lane traffic . . 92
3.9 Notes . . 96
3.10 Problems . . 97

4 Exclusion . . 103
4.1 Symmetric exclusion process . . 103
4.2 Asymmetric exclusion process . . 108
4.3 Hydrodynamic approach . . 112
4.4 Microscopic approach . . 118
4.5 Open systems . . 123
4.6 Notes . . 130
4.7 Problems . . 131

5 Aggregation . . 134
5.1 The master equations . . 134
5.2 Exact solution methods . . 137
5.3 Gelation . . 145
5.4 Scaling . . 153
5.5 Aggregation with input . . 156
5.6 Exchange-driven growth . . 163
5.7 Notes . . 167
5.8 Problems . . 168

6 Fragmentation . . 172
6.1 Binary fragmentation . . 172
6.2 Planar fragmentation . . 180
6.3 Reversible polymerization . . 185
6.4 Collisional fragmentation . . 191
6.5 Notes . . 195
6.6 Problems . . 195

7 Adsorption . . 199
7.1 Random sequential adsorption in one dimension . . 199
7.2 Phase space structure . . 206
7.3 Adsorption in higher dimensions . . 213
7.4 Reversible adsorption . . 220
7.5 Polymer translocation . . 226
7.6 Notes . . 229
7.7 Problems . . 230

8 Spin dynamics . . 233
8.1 Phenomenology of coarsening . . 233
8.2 The voter model . . 235
8.3 Ising-Glauber model . . 244
8.4 Mean-field approximation . . 247
8.5 Glauber dynamics in one dimension . . 249
8.6 Glauber dynamics in higher dimensions . . 258
8.7 Spin-excnange dynamics . . 264
8.8 Cluster dynamics . . 269
8.9 Notes . . 273
8.10 Problems . . 274

9 Coarsening . . 277
9.1 Models . . 277
9.2 Free evolution . . 280
9.3 Case studies in non-conservative dynamics . . 283
9.4 Final states . . 292
9.5 Defects . . 294
9.6 Conservative dynamics . . 302
9.7 Extremal dynamics . . 307
9.8 Nucleation and growth . . 312
9.9 Notes . . 317
9.10 Problems . . 318

10 Disorder . . 322
10.1 Disordered spin chain . . 322
10.2 Random walk in a random potential . . 331
10.3 Random walk in random velocity fields . . 339
10.4 Notes . . 343
10.5 Problems . . 343

11 Hysteresis . . 346
11.1 Homogeneous f erromagnets . . 346
11.2 Perturbation analysis . . 349
11.3 Disordered ferromagnets . . 357
11.4 Mean-field model . . 361
11.5 Hysteresis in the random-field Ising chain . . 366
11.6 Notes . . 370
11.7 Problems . . 370

12 Population dynamics . . 373
12.1 Continuum formulation . . 373
12.2 Discrete reactions . . 382
12.3 Small-fluctuatHi expansion . . 391
12.4 Large fluctuations . . 394
12.5 Notes . . 399
12.6 Problems . . 400

13 Diffusive reactions . . 404
13.1 Role of the spatial dimension . . 404
13.2 The trapping reaction . . 409
13.3 Two-species annihilation . . 414
13.4 Single-species reactions in one dimension . . 417
13.5 Reactions in spatial gradients . . 428
13.6 Notes . . 436
13.7 Problems . . 437

14 Complex networks . . 441
14.1 Non-lattice networks . . 441
14.2 Evolving random graphs . . 443
14.3 Random recursive trees . . 451
14.4 Preferential attachment . . 456
14.5 Fluctuations in networks . . 460
14.6 Notes . . 465
14.7 Problems . . 466

References . . 471
Index . . 483