This book provides an introduction to localised excitations in spatially discrete systems, from the experimental, numerical and mathematical points of view. Also known as discrete breathers, nonlinear lattice excitations and intrinsic localised modes; these are spatially localised time periodic motions in networks of dynamical units. Examples of such networks are molecular crystals, biomolecules, and arrays of Josephson superconducting junctions. The book also addresses the formation of discrete breathers and their potential role in energy transfer in such systems.
Author(s): Thierry Dauxois, Anna Litvak-Hinenzon, Robert Mackay, Anna Spanoudaki
Year: 2004
Language: English
Pages: 428
CONTENTS ......Page 12
Preface ......Page 6
1 Introduction ......Page 19
2 A bit on numerics of solving ODEs ......Page 24
3.1 Targeted initial conditions ......Page 28
3.2 Breathers in transient processes ......Page 35
3.3 Breathers in thermal equilibrium ......Page 41
4 Obtaining breathers up to machine precision: Part I ......Page 43
4.1 Method No.1 - designing a map ......Page 44
4.2 Method No.2 - saddles on the rim with space-time separation ......Page 48
4.3 Method No.3 - homoclinic orbits with time-space separation ......Page 49
5 Obtaining breathers up to machine precision: Part II ......Page 51
5.1 Method No.4 - Newton in phase space ......Page 53
5.2 Method No.5 - steepest descent in phase space ......Page 55
5.3 Symmetries ......Page 56
6 Perturbing breathers ......Page 57
6.1 Linear stability analysis ......Page 58
6.2 Plane wave scattering ......Page 61
7 Breathers in dissipative systems ......Page 65
7.1 Obtaining dissipative breathers ......Page 66
7.2 Perturbing dissipative breathers ......Page 67
8 Computing quantum breathers ......Page 69
8.1 The dimer ......Page 72
8.2 The trimer ......Page 76
8.3 Quantum roto-breathers ......Page 83
9 Some applications instead of conclusions ......Page 84
References ......Page 86
CHAPTER 2 VIBRATIONAL SPECTROSCOPY AND QUANTUM LOCALIZATION ......Page 91
1.1 Nonlinear dynamics and energy localization ......Page 92
1.2 Nonlinear dynamics and vibrational spectroscopy ......Page 94
2.1 Some definitions ......Page 96
2.2 Optical techniques ......Page 100
2.3 Neutron scattering techniques ......Page 102
2.4 A (not so) simple example ......Page 107
3.1 The harmonic approximation: Normal modes ......Page 110
3.3 Local modes ......Page 111
3.4 Local versus normal mode separability ......Page 114
3.5 The water molecule ......Page 118
3.6 The algebraic force-field Hamiltonian ......Page 122
3.7 Other molecules ......Page 125
3.8 Local modes and energy localization ......Page 127
4 Crystals ......Page 128
4.1 The harmonic approximation: Phonons ......Page 129
4.2 Phonon-phonon interaction ......Page 131
4.3 Phonon-electron interaction ......Page 134
4.4 Local modes ......Page 137
4.5 Nonlinear dynamics ......Page 141
5.2 Optical vibrational spectroscopy and energy localization ......Page 159
5.3 Inelastic neutron scattering spectroscopy of solitons ......Page 160
References ......Page 161
1 Introduction ......Page 167
2 Normally Hyperbolic versus General Case ......Page 170
3 Hamiltonian versus General Case ......Page 172
4 Improving a slow manifold ......Page 178
5 Symplectic slow manifolds ......Page 180
6 The Methods of Collective Coordinates ......Page 187
7 Velocity Splitting ......Page 189
8 Poisson slow manifolds ......Page 191
9 Slow manifolds with Internal Oscillation ......Page 192
10 Internal oscillation: U(1)-symmetric Hamiltonians ......Page 194
11 Internal oscillation: General Hamiltonians ......Page 199
12 Bounds on time evolution ......Page 203
Acknowledgements ......Page 205
References ......Page 206
1 Introduction ......Page 211
2.2 Superconducting tunnel junctions ......Page 212
2.3 Long Josephson junctions ......Page 217
2.4 Quantum effects in Josephson junctions ......Page 218
3 Modeling Josephson arrays ......Page 219
3.1 Series arrays ......Page 221
3.3 dc-SQUID ......Page 222
3.4 JJ parallel array ......Page 223
3.5 JJ ladder array ......Page 224
3.6 2D arrays ......Page 225
4.1 Vortices in 2D arrays ......Page 227
4.2 2D arrays with small junctions ......Page 229
4.3 Kinks in parallel arrays ......Page 230
5 Discrete breathers in Josephson arrays ......Page 235
5.1 Oscillobreather in an ac biased parallel array ......Page 237
5.3 The ladder array ......Page 238
5.4 Rotobreathers in a dc biased ladder ......Page 239
5.6 DBs in two-dimensional Josephson junction arrays ......Page 256
Acknowledgments ......Page 258
References ......Page 259
1 Introduction ......Page 265
2 Fabrication of Josephson arrays ......Page 266
2.1 Materials ......Page 267
2.2 Layout ......Page 270
2.3 Junction parameters ......Page 272
3 Measurement techniques ......Page 273
3.1 Generation of localized excitations ......Page 274
3.2 Hot probe imaging techniques ......Page 275
4.1 Fluxons in Josephson arrays ......Page 277
4.2 Rotobreathers in Josephson ladders ......Page 280
4.3 Meandered states in 2-D Josephson arrays ......Page 282
5.1 Single Josephson junction ......Page 283
5.2 Coupled Josephson junctions ......Page 286
Acknowledgments ......Page 287
References ......Page 288
2 Protein structure ......Page 291
3 Energetics of protein stabilisation ......Page 293
4 Protein folding ......Page 294
4.1 On-lattice models ......Page 295
4.2 Off-lattice models ......Page 301
4.3 More detailed models ......Page 302
5.2 Collective motions ......Page 303
5.3 Low-frequency normal modes ......Page 307
6 Dissipation of energy in proteins ......Page 315
Acknowledgments ......Page 316
References ......Page 317
1 Introduction: The Story of Davidov's Soliton ......Page 319
2.1 Harmonic and Anharmonic Potential Energy Surfaces ......Page 321
2.2 Linear and Nonlinear Spectroscopy ......Page 323
3.1 Theoretical Background ......Page 325
3.2 Experimental Observation ......Page 327
4.1 Theoretical Background ......Page 328
4.2 Experimental Observation ......Page 330
5.1 Theoretical Background ......Page 332
5.2 Experimental Observation ......Page 333
6 Conclusion and Outlook ......Page 337
Acknowledgments ......Page 338
Appendix: Feynman Diagram Description of Linear and Nonlinear Spectroscopy ......Page 339
References ......Page 341
1 Introduction ......Page 343
2.1 Local modes in small molecules ......Page 344
2.2 Local modes in large molecules ......Page 345
2.3 Local modes in crystals ......Page 347
2.4 Localisation of vibrations and chemical reaction rates ......Page 348
2.5 Fluctuational opening in DNA ......Page 349
3 Quantum self-trapping ......Page 351
4 Discussion ......Page 355
References ......Page 357
1 Introduction/Outlook ......Page 359
2 Thermal DNA denaturation: A domain-wall driven transition? ......Page 361
3 ILMs in DNA dynamics? ......Page 363
4.1 Definitions Notation ......Page 366
4.2 Thermodynamics ......Page 368
Acknowledgments ......Page 369
References ......Page 370
1 Introduction ......Page 373
2 Discrete Breathers ......Page 377
2.1 DBs in periodic lattices ......Page 378
2.2 DBs in random systems ......Page 389
3 Targeted energy transfer ......Page 394
3.1 Nonlinear resonance ......Page 396
3.2 Targeted energy transfer in a nonlinear dimer ......Page 398
3.3 Targeted energy transfer through discrete breathers ......Page 401
4 Ultrafast Electron Transfer ......Page 405
4.1 Nonlinear dynamical model for ET ......Page 408
4.2 ET in the Dimer ......Page 412
4.3 Catalytic ET in a trimer ......Page 413
4.4 The example of bacterial photosynthetic reaction center ......Page 414
5 Conclusions and perspectives ......Page 418
References ......Page 420
Index ......Page 423
