Chaos, Complexity and Transport: Theory and Applications

CONTENTS......Page 10
Preface......Page 6
THEORY......Page 14
Out-of-Equilibrium Phase Transitions in Mean-Field Hamiltonian Dynamics P.-H. Chavanis, G. De Ninno, D. Fanelli and S. Ruff0......Page 16
1. Introduction......Page 17
2. On the emergence of quasi-stationary states: Predictions from the Lynden-Bell theory within the Vlasov picture......Page 19
3. Properties of the homogeneous Lynden-Bell distribution......Page 24
4. Stability of the Lynden-Bell homogeneous phase......Page 28
5. The rectangular water-bag initial condition: phase diagram in the (Mo, 27) plane......Page 30
6. The general case: Phase diagram in the ( fo , U ) plane......Page 33
7. Conclusions......Page 37
References......Page 38
1. Introduction......Page 40
2. Kicked Two Coupled Oscillators......Page 42
3. Symmetry of the Stochastic Web......Page 43
4. More Coupled Oscillators......Page 48
5 . Conclusion......Page 50
References......Page 51
1. Introduction......Page 53
2.1. Separating the Shallow Direction......Page 55
2.2. Substrate Coordinates......Page 56
3. Equations of Motion......Page 58
3.2. Solution in Terms of Characteristics......Page 59
4. Fluid Particle Trajectories......Page 61
5 . Lyapunov Exponents and Chaos......Page 64
6 . Discussion......Page 65
References......Page 66
1. Introduction......Page 68
2. The construction......Page 69
3. Mixing......Page 74
4. Discussion......Page 76
Appendix......Page 78
Acknowledgements......Page 79
References......Page 80
1. Introduction......Page 82
2. Entropy for collision map......Page 88
3. Hard disks......Page 91
4. Concluding remarks......Page 93
Appendix A. Collision Map......Page 94
References......Page 95
1. Introduction......Page 97
2. Wave equation......Page 101
3. Kinetics Theory and Bose-Einstein condensation......Page 102
4. Dynamics before collapse......Page 105
5.1. Early stage......Page 106
5.2. Late stage: The appearance of coherence and the Bogoluibov spectra......Page 109
6. Comments and remarks......Page 112
References......Page 117
1. Introduction......Page 119
2. Parrondo games and their deterministic version......Page 120
3. Periodic orbit theory of deterministic Parrondo games......Page 124
4. Periodic hopping framework......Page 128
References......Page 130
Separatrix Chaos: New Approach to the Theoretical lleatment S. M. Soskin, R. Mannella and 0. M. Yevtushenko......Page 132
1. Introduction......Page 133
2. Basic ideas......Page 134
3. Application to the double-separatrix chaos......Page 137
4. Single-separatrix layer: estimates of the largest width......Page 139
References......Page 141
1. Introduction......Page 142
References......Page 148
1. Introduction......Page 149
2. Derivation of the control term......Page 150
3.1. Application to the standard map......Page 153
3.2. Application to the tokamap......Page 155
References......Page 157
(1) PLASMA & FLUIDS......Page 158
1.1. General Background......Page 160
1.2. Steady-state Power Exhaust in Fusion Plasmas......Page 161
1.3. Transient Particle and Energy Bursts in Fusion Plasmas......Page 163
2.1. General Theoretical Approach......Page 164
2.2. Applications of Hamiltoniun Mapping Models to Circular Limited and Poloidully Diverted Tokamaks......Page 167
3.1. General Background......Page 168
3.2. Calculations of Homoclinic Tangles and Heteroclinic intersections in Circular Limited Tokamaks......Page 170
3.3. Calculations of Homoclinic Tangles in Poloidally Diverted Tokamaks......Page 174
4.2. Improved Confinement and Transport Barrier Physics in Circular Limited Tokamaks with Stochastic Magnetic Boundary Layers......Page 176
4.3. Signatures of Homoclinic Tangles in Poloidally Diverted Tokamaks......Page 181
6. Summary and Conclusions......Page 183
References......Page 185
1. Introduction......Page 190
2. Chaotic mixing in the alternating vortex chain......Page 192
3. Experimental methods......Page 193
4.1. Mode-locking in oscillating vortex chain......Page 194
4.2. Freezing of fronts in the presence of a uniform wind......Page 195
5. Collective oscillatory behavior and synchronization by chaotic mixing......Page 198
References......Page 200
1. Introduction......Page 202
2. Continuous-time random walks......Page 206
3. LQvy distributions......Page 208
4.1. Crash course on fractional diffeerential operators......Page 210
4.2. Finding the fluid l i m i t of CTRWs......Page 212
5. Applications to numerical simulations of turbulence in magnetically-confined plasmas......Page 214
References......Page 217
Rotating Rayleigh-Birnard Convection in Cylinders J. J. Sa'nchea-Alvarez, E. Serre, E. Crespo Del Arc0 and F. H. Busse......Page 220
1. Introduction......Page 221
2. Numerical integration of the Boussinesq equations......Page 222
3. Results for circular boxes......Page 223
4. Results in an annular cavity......Page 225
5 . Concluding remarks......Page 228
References......Page 229
1. Introduction......Page 230
2.1. Dust particles formed by sputtering......Page 232
2.2. Dust particles formed by reactive gases......Page 234
3. Instabilities of the grown dust cloud......Page 236
References......Page 238
1. Introduction......Page 240
2. Power spectra of approximate signals......Page 242
3. Clustering information in the approximate signal......Page 244
References......Page 248
1. Introduction......Page 250
2. Experimental results and discussion......Page 251
3. Conclusion......Page 255
References......Page 256
1. Introduction......Page 258
2. The model......Page 260
3. Hamiltonian structure......Page 261
4. Numerical simulations......Page 263
References......Page 266
1. Introduction......Page 268
2. Formulation of the problem......Page 269
3.3. Weak stability......Page 271
4. Conclusions......Page 273
References......Page 274
(2) OTHERS......Page 276
1. Introduction......Page 278
2.1. The dynamical equations......Page 280
2.2. Generalized free energy and H-theorem......Page 282
2.3. Stationary solution......Page 283
2.4. Minimum of free energy......Page 284
2.5. Particular cases......Page 285
2.6. Generalized Smoluchowski equation......Page 286
2.7. Kinetic derivation of the generalized Keller-Segel model......Page 288
3.1. The standard Keller-Segel model: Boltzmann entropy......Page 290
3.2. Generalized Keller-Segel model with power law digusion: Tsallis entropy......Page 291
3.3. Generalized Keller-Segel model with logarithmic dinusion: logotropes......Page 292
3.4. Generalized Keller-Segel models with power law diffusion and power law drift: Tsallis entropy......Page 293
3.5. Generalized Keller-Segel models with a filling factor: Fermi-Dirac entropy......Page 294
3.6. Generalized Keller-Segel models incorporating anomalous diffusion and filling factor......Page 295
References......Page 298
1. Introduction......Page 300
3. Analytical conditions......Page 302
4. Illustrations......Page 306
5. Conclusions......Page 309
Appendix......Page 310
References......Page 311
1. Introduction......Page 313
2. Description of the model......Page 314
3. Equations of motion and dynamics around L1 and L2......Page 316
4. Results......Page 318
References......Page 320
1. Introduction......Page 322
2. Deriving energetic particle profiles by the propagator formalism......Page 323
3. Data analysis......Page 326
References......Page 329
1. Introduction......Page 331
2. Classical ray dynamics......Page 333
3. Wave dynamics......Page 336
4. Conclusion......Page 338
References......Page 339
Displacement Effects on Fermi Acceleration in Randomized Driven Billiards A. K. Karlis, P. K. Papachristou, F. K. Diakonos, V. Constantoudis and P. Schmelcher......Page 340
1. Introduction......Page 341
2. Fermi-Ulam model with a sawtooth wall driving function......Page 342
3. Time-dependent Lorentz gas......Page 345
4. Concluding remarks......Page 348
References......Page 349
1. Introduction......Page 350
2. Diffusion model of the relaxation......Page 351
3. Numerical calculations......Page 353
4. Where come long waiting times from?......Page 355
5. Summary......Page 357
References......Page 358
1. Introduction......Page 359
2. Simulation method......Page 361
3. Nodal pattern analysis......Page 362
4. Discussions......Page 366
References......Page 367
1. Introduction......Page 369
2. Definition and behavior of GALI......Page 370
3. Dynamical study of a 6D standard map......Page 371
References......Page 376