Preface ........................................................ xv
Acknowledgments ............................................... xix
Part One Methods and Concepts .................................. 1
1 Introduction to Nonequilibrium Systems and Transport
Phenomena .................................................... 3
1.1 Introduction ............................................ 3
1.2 Classification of Nonequilibrium Phenomena .............. 4
1.3 Hierarchy of Description at Different Levels ............ 6
1.4 Individual-Based Models ................................. 7
1.4.1 Newton's Equations, Hamilton's Equations, and
Individual Trajectories .......................... 7
1.4.2 Langevin Equation ................................ 9
1.5 Population-Based Models ................................ 10
1.5.1 Master Equation and Irreversibility ............. 10
1.5.2 Fokker-Planck Equation .......................... 13
1.6 Fluid Flow: Theoretical Descriptions at Different
Levels ................................................. 14
1.6.1 Liouville Equation and Flow in Phase Space ...... 15
1.6.2 From Liouville Equation to Boltzmann Equation:
BBGKY Hierarchy ................................. 16
1.6.3 From Kinetic Theory to Navier-Stokes Equation ... 17
1.7 Back to Discrete Models: Mimicking Hydrodynamics with
Fictitious Particles ................................... 19
1.7.1 Driven-Diffusive Lattice Gas Models ............. 20
1.8 Phase Transitions, Critical Dynamics, and Kinetics of
Phase Ordering ......................................... 22
1.8.1 Critical Dynamics: Role of Symmetry and
Conservation Laws ............................... 22
1.8.2 Kinetics of Phase Ordering: Metastable and
Unstable Initial States ......................... 23
1.8.3 Phase Transitions in Driven Systems ............. 24
2 Methods for the Description of Stochastic Models ............ 27
2.1 Quantum Formalism ...................................... 28
2.1.1 Master Equation and Stochastic Hamiltonian ...... 28
2.1.2 Spectrum and Expectation Values ................. 31
2.1.3 Discrete Time Dynamics .......................... 33
2.2 Mean-Field and Cluster Methods ......................... 37
2.2.1 Mean-Field Approximations ....................... 37
2.3 Bethe Ansatz ........................................... 41
2.4 Matrix-Product Ansatz .................................. 43
2.4.1 MPA in Quantum Formalism ........................ 45
2.4.2 MPA for Discrete Time Updates ................... 48
2.4.3 Dynamical MPA ................................... 51
2.4.4 Relation with Bethe Ansatz ...................... 52
2.5 Other Methods .......................................... 52
2.5.1 Hydrodynamic Limit .............................. 52
2.5.2 Field-Theoretic Methods and Renormalization
Groups .......................................... 53
2.5.3 Similarity Transformations ...................... 54
2.5.4 Ultradiscrete Method ............................ 54
2.6 Numerical Methods ...................................... 57
2.6.1 Computer Simulation (MC Methods) ................ 58
2.6.2 Exact Diagonalization ........................... 60
2.6.3 Density-Matrix Renormalization Group ............ 61
2.6.4 Transfer-Matrix DMRG ............................ 63
2.7 Appendices ............................................. 66
2.7.1 Some Mathematics ................................ 66
2.7.2 MPA and Optimum Ground States for Quantum Spin
Chains .......................................... 66
2.7.3 Krebs-Sandow Theorem and Extensions ............. 68
3 Particle-Hopping Models of Transport Far from Equilibrium ... 71
3.1 Elements of Random Walk Theory ......................... 72
3.2 Asymmetric Simple Exclusion Process .................... 74
3.3 Zero-Range Process and Exact Results ................... 75
3.3.1 Exact Solution .................................. 76
3.3.2 Bethe Ansatz Solution ........................... 79
3.4 Extensions and Generalizations ......................... 81
3.4.1 Parallel Dynamics ............................... 81
3.4.2 Other Lattice Structures ........................ 82
3.4.3 ZRP with Disorder ............................... 83
3.4.4 ZRP with Fluctuating Particle Number ............ 83
3.4.5 Generalizations ................................. 86
3.4.6 Dynamical Urn Models ............................ 86
3.4.7 Misanthrope Process ............................. 87
3.4.8 Relation of ZRP to Other Models and Some
Applications .................................... 88
3.5 Physics of the ZRP ..................................... 90
3.5.1 Condensation Transition ......................... 90
3.5.2 Dynamics and Coarsening ......................... 94
3.5.3 Criterion for Phase Separation .................. 95
3.6 Particle-Hopping Models with Factorized Stationary
States ................................................. 97
3.6.1 Models with Pair-Factorized Steady States ...... 100
3.7 Generalized Mass Transport Models ..................... 102
3.7.1 Models with Continuous States .................. 102
3.7.2 Asymmetric Random Average Process .............. 103
3.7.3 Chipping Model ................................. 105
3.8 Appendix .............................................. 106
3.8.1 Derivation of the Factorization Criterion ...... 106
4 Asymmetric Simple Exclusion Process - Exact Results ........ 109
4.1 ASEP with Periodic Boundary Conditions ................ 111
4.1.1 Random-Sequential Dynamics ..................... 111
4.1.2 Bethe Ansatz for Translationally Invariant
Systems ........................................ 113
4.1.3 Mean-Field Theories for Parallel Dynamics ...... 116
4.1.4 Mapping to ZRP ................................. 123
4.1.5 Paradisical Mean-Field Theory .................. 123
4.1.6 Combinatorial Solution for Parallel Dynamics ... 124
4.1.7 Ordered-Sequential and Sublattice-Parallel
Updates ........................................ 126
4.1.8 Shuffled Dynamics .............................. 128
4.2 ASEP with Open Boundary Conditions .................... 131
4.2.1 Mean-Field Theory .............................. 132
4.2.2 Recursion Relations ............................ 134
4.2.3 Matrix-Product Ansatz .......................... 135
4.2.4 Exact Phase Diagram ............................ 136
4.2.5 Phase Transitions .............................. 141
4.2.6 Relation with Combinatorics .................... 142
4.2.7 Bethe Ansatz ................................... 142
4.2.8 Dynamical MPA .................................. 143
4.2.9 Hydrodynamic Limit ............................. 143
4.3 Partially Asymmetric Version .......................... 145
4.3.1 MPA Solution ................................... 146
4.3.2 Bethe-Ansatz Solution .......................... 148
4.3.3 Phase Diagram of the PASEP ..................... 148
4.4 Extension of the ASEP to Other Update Types ........... 151
4.4.1 Ordered-Sequential Updates ..................... 151
4.4.2 Sublattice-Parallel Update ..................... 154
4.4.3 Parallel Update ................................ 154
4.5 Boundary-Induced Phase Transitions .................... 158
4.5.1 Domain Wall Picture ............................ 158
4.5.2 Extremal Principle and Steady-State
Selection ...................................... 161
4.5.3 More on Shock Dynamics ......................... 161
4.5.4 Fluctuations and Large Deviation Functions ..... 162
4.6 Extensions of ASEP .................................... 163
4.6.1 Quenched Disorder .............................. 164
4.6.2 Disorder in Open Systems ....................... 173
4.6.3 Langmuir Kinetics .............................. 174
4.6.4 Extended Particles ............................. 177
4.6.5 Other Boundary Conditions ...................... 177
4.6.6 Long-Range Hopping ............................. 179
4.6.7 ASEP Beyond One Dimension ...................... 180
4.7 Multispecies Models ................................... 181
4.7.1 Models with Second-Class Particles ............. 181
4.7.2 ABC Model ...................................... 183
4.7.3 AHR Model ...................................... 184
4.8 Other Related Models .................................. 185
4.8.1 Staggered Hopping Rates ........................ 185
4.8.2 Two-Parameter Model ............................ 186
4.8.3 Restricted ASEP ................................ 188
4.8.4 KLS Model ...................................... 190
4.8.5 Asymmetric Avalanche Process ................... 191
4.8.6 Higher Velocities .............................. 193
4.8.7 Reconstituting Dimers .......................... 194
4.9 Appendices ............................................ 195
4.9.1 Mapping of ASEP to Surface Growth Model ........ 195
4.9.2 Mapping of the ASEP to an Ising Model .......... 196
4.9.3 Solution of the Mean-Field Recursion
Relations for the ASEP ......................... 197
4.9.4 Results Obtained from Normal-Ordering of
Matrices ....................................... 199
4.9.5 Dimension of Matrices in the MPA for the
ASEP ........................................... 200
4.9.6 Representations of the Matrix Algebra of the
ASEP ........................................... 201
4.9.7 Mean-Field Approximation of the DTASEP ......... 205
Part Two Applications ........................................ 207
5 Modeling of Traffic and Transport Processes ................ 209
5.1 Introduction .......................................... 209
5.1.1 Some Practical Questions ....................... 210
5.1.2 Some Fundamental Questions ..................... 211
5.2 Classification of Models .............................. 212
5.2.1 Model Characteristics .......................... 212
5.2.2 Model Classes .................................. 214
6 Vehicular Traffic I: Empirical Facts ....................... 215
6.1 Measurement Techniques and Detectors .................. 215
6.2 Observables and Data Analysis ......................... 216
6.3 Formation and Characterization of Traffic Jams ........ 221
6.3.1 Jams Induced by Bottlenecks .................... 222
6.3.2 Spontaneous Traffic Jams ....................... 223
6.3.3 Experiment on Spontaneous Jam Formation ........ 224
6.4 Fundamental Diagram ................................... 226
6.5 Metastability and Hysteresis .......................... 228
6.6 Phases of Traffic Flow ................................ 230
6.6.1 Level of Service Classification ................ 230
6.6.2 Traffic Phases and Phase Transitions ........... 231
6.6.3 Gas-Liquid Analogy ............................. 234
6.7 Ramps, Intersections, and Other Inhomogeneities ....... 235
6.8 Headway Distributions ................................. 236
6.9 Optimal-Velocity Function ............................. 238
6.10 Correlation Functions ................................. 239
6.11 Psychological Effects ................................. 240
7 Vehicular Traffic II: The Nagel-Schreckenberg Model ........ 243
7.1 Definition of the Model ............................... 244
7.1.1 Update Rules ................................... 244
7.1.2 Relation with Other Models ..................... 247
7.2 Fundamental Diagram and Limiting Cases of the NaSch
Model ................................................. 248
7.2.1 Fundamental Diagram ............................ 248
7.2.2 NaSch Model in the Deterministic Limit p = 0 ... 250
7.2.3 NaSch Model in the Deterministic Limit p = 1 ... 251
7.2.4 NaSch Model with νmax = 1 ...................... 251
7.2.5 NaSch Model in the Limit νmax = ∞ .............. 253
7.3 Analytical Theories for NaSch Model with νmax > 1 ..... 255
7.3.1 SOMF Theory for the NaSch Model ................ 255
7.3.2 Cluster-Approximations for the NaSch Model ..... 256
7.3.3 pMF Theory of the NaSch Model .................. 257
7.3.4 Car-Oriented Mean-Field Theory of the NaSch
Model .......................................... 259
7.4 Spatio-Temporal Organization of Vehicles .............. 260
7.4.1 Microscopic Structure of the Stationary
State .......................................... 260
7.4.2 Spatial Correlations ........................... 261
7.4.3 Headway Distributions .......................... 262
7.4.4 Distributions of Jam Sizes and Gaps between
Jams ........................................... 263
7.4.5 Distribution of Lifetimes of Jams .............. 265
7.4.6 Temporal Correlations and Relaxation Time ...... 266
7.4.7 Structure Factor ............................... 267
7.4.8 Phase Transition ............................... 268
7.4.9 Boundary-Induced Phase Transitions ............. 270
7.5 Appendices ............................................ 272
7.5.1 Details of SOMF for NaSch ...................... 272
7.5.2 Details of PMF for NaSch ....................... 276
7.5.3 Details of COMF for NaSch ...................... 277
8 Vehicular Traffic III: Other CA Models ..................... 281
8.1 Slow-to-Start Rules, Metastability, and Hysteresis .... 282
8.1.1 General Remarks ................................ 282
8.1.2 The Velocity-Dependent-Randomization Model ..... 283
8.1.3 Takayasu-Takayasu Slow-to-Start Rule ........... 288
8.1.4 The BJH Model of Slow-to-Start Rule ............ 289
8.1.5 Other Slow-to-Start Rules ...................... 289
8.1.6 Flow Optimization and Metastable States ........ 290
8.2 Cruise-Control Limit .................................. 291
8.3 CA Models of Synchronized Traffic ..................... 294
8.3.1 Brake-Light or Comfortable Driving Model ....... 295
8.3.2 Kerner-Klenov-Wolf Model ....................... 299
8.3.3 Mechanical Restrictions Model of Lee et al ..... 301
8.4 Other CA Models ....................................... 304
8.4.1 Fukui-lshibashi Model .......................... 304
8.4.2 Velocity-Dependent Braking Model ............... 305
8.4.3 Time-Oriented CA Model ......................... 306
8.4.4 Models with Anticipation ....................... 307
8.4.5 Galilei-Invariant Model ........................ 309
8.4.6 Car-Following CA ............................... 311
8.5 CA from Ultradiscrete Method .......................... 313
8.5.1 Generalizations of BCA ......................... 314
8.5.2 Euler-Lagrange Transformation .................. 315
8.5.3 Traffic Models in Lagrange Form ................ 316
8.6 CA Models of Multilane Traffic ........................ 319
8.6.1 Classification of Lane Changing Rules .......... 319
8.6.2 CA Models of Bidirectional Traffic ............. 322
8.7 Effects of Quenched Disorder .......................... 324
8.7.1 Randomness in the Braking Probability .......... 324
8.7.2 Random νmax .................................... 326
8.7.3 Randomly Placed Bottlenecks .................... 326
8.7.4 Ramps .......................................... 328
8.8 Bus-Route Model ....................................... 329
8.9 Accidents ............................................. 332
9 Vehicular Traffic IV: Non-CA Approaches .................... 335
9.1 Fluid Dynamical Theories .............................. 336
9.1.1 Lighthill-Whitham-Richards Theory and
Kinematic Waves ................................ 337
9.1.2 Diffusion Term in LWR Theory and Its Effects ... 340
9.1.3 Greenshields Model and Burgers Equation ........ 341
9.2 Second-Order Fluid Dynamical Theories ................. 342
9.2.1 Special Models ................................. 344
9.2.2 Instabilities and Jam Formation ................ 345
9.2.3 Problems with Second-Order Models .............. 348
9.2.4 Aw-Rascle Model ................................ 348
9.2.5 Fluid-Dynamical Models and Synchronized
Traffic ........................................ 349
9.2.6 Fluid-Dynamical Theories of Traffic on
Multilane Highways and in Cities ............... 350
9.3 Gas-Kinetic Models .................................... 351
9.3.1 Prigogine Model ................................ 351
9.3.2 Paveri-Fontana Model ........................... 353
9.3.3 Derivation of Fluid-Dynamical Equations from
Gas-Kinetic Equations .......................... 356
9.4 Car-Following Models .................................. 357
9.4.1 Follow-the-Leader Model ........................ 358
9.4.2 Optimal Velocity Model and Its Extensions ...... 360
9.4.3 Generalized Force Models ....................... 364
9.4.4 Intelligent Driver Model ....................... 366
9.4.5 Kerner-Klenov Model ............................ 368
9.4.6 Inertial Car-Following Model ................... 369
9.5 Coupled-Map Models .................................... 371
9.5.1 Gipps Model .................................... 372
9.5.2 Krauss Model (SK Model) ........................ 373
9.5.3 Yukawa-Kikuchi Model ........................... 375
9.5.4 Nagel-Herrmann Model ........................... 376
9.6 Other Approaches ...................................... 377
9.6.1 Probabilistic Traffic Flow Theory .............. 377
9.6.2 Cell Transmission Model ........................ 379
9.6.3 Queueing Models ................................ 381
10 Transport on Networks ...................................... 383
10.1 Networks and Transport ................................ 383
10.2 BML Model of City Traffic ............................. 384
10.2.1 Phase Transition ............................... 385
10.2.2 Generalizations and Extensions of the BML
Model .......................................... 386
10.2.3 More Realistic Description of Streets and
Junctions ...................................... 388
10.3 Chowdhury-Schadschneider Model ........................ 390
10.3.1 Crossroads with Signals ........................ 390
10.3.2 ChSch Model .................................... 391
10.3.3 Traffic Signal Optimization .................... 395
10.4 Highway and City Networks ............................. 398
10.4.1 Online Simulation of Traffic Networks .......... 398
10.4.2 Network Analysis ............................... 400
10.4.3 Braess Paradox ................................. 401
10.5 Computer Networks and Internet Traffic ................ 402
11 Pedestrian Dynamics ........................................ 407
11.1 Introduction .......................................... 408
11.2 Empirical Observations and Collective Phenomena ....... 409
11.2.1 Individual Properties .......................... 409
11.2.2 Observables .................................... 410
11.2.3 Fundamental Diagram ............................ 413
11.2.4 Flows at Bottlenecks ........................... 415
11.2.5 Collective Phenomena ........................... 418
11.3 Cellular Automata Models .............................. 423
11.3.1 Fukui-lshibashi Model .......................... 424
11.3.2 Blue-Adler Model ............................... 428
11.3.3 Gipps-Marksjös Model ........................... 428
11.4 Floor Field CA ........................................ 430
11.4.1 General Principle .............................. 430
11.4.2 Update Rules ................................... 432
11.4.3 Construction of the Static Floor Field ......... 435
11.4.4 Conflicts and Friction ......................... 436
11.4.5 Other Generalizations and Interactions ......... 437
11.4.6 Moving Beyond Nearest Neighbors: νmax > 1 ...... 440
11.4.7 Collective Effects ............................. 441
11.4.8 Evacuation Simulations ......................... 444
11.5 Other Models .......................................... 447
11.5.1 Fluid-Dynamic and Gas-Kinetic Models ........... 447
11.5.2 Social-Force Models ............................ 449
11.5.3 Lattice Gas Models ............................. 454
11.5.4 Optimal Velocity Model ......................... 456
11.5.5 Active Walker Models ........................... 458
12 Traffic Phenomena In Biology ............................... 461
12.1 Introduction .......................................... 461
12.1.1 Different Types of Traffic in Biology .......... 462
12.2 TASEP for Hard Rods: Minimal Model of Transcription
and Translation ....................................... 462
12.2.1 TASEP for Hard Rods: Minimal Models of Traffic
of Ribosomes and RNAPs ......................... 464
12.2.2 TASEP for Hard Rods with Internal States:
Effects of Individual Mechano-Chemistry ........ 466
12.3 TASEP for Particles with Langmuir Kinetics: Minimal
Model of Kinesin Traffic .............................. 466
12.3.1 TASEP-Like Generic Models of Molecular Motor
Traffic ........................................ 467
12.3.2 Traffic of Interacting Particles with
"Internal States" and Langmuir Kinetics:
Effects of Individual Mechano-Chemistry of
KIF1A .......................................... 468
12.4 Traffic in Social Insect Colonies: Ant-Trails ......... 472
12.4.1 Model of Single-Lane Unidirectional
Ant-Traffic .................................... 473
12.4.2 Model of Single-Lane Bidirectional Ant-
Traffic ........................................ 478
12.4.3 Model of Two-Lane Bidirectional Ant-Traffic .... 483
12.4.4 Experimental Investigations of Ant-Traffic ..... 485
12.4.5 Empirical Results for Fundamental Diagrams of
Ant-Trails ..................................... 486
Guide to the Literature ....................................... 489
Bibliography .................................................. 491
Index ......................................................... 549
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