The Frenkel-Kontorova Model: Concepts, Methods, and Applications

Oleg M. Braun, Yuri S. Kivshar

Springer, Feb 20, 2004 - Science - 472 pages

This book presents an overview of the basic concepts, methods and applications of nonlinear low-dimensional solid-state physics based on the Frenkel--Kontorova model and its generalizations. It allows a nonspecialist to acquire foundations in the interdisciplinary concepts and methods of both solid-state physics and nonlinear science. It covers many important topics such as the nonlinear dynamics of discrete systems, the dynamics of solitons and their interaction, commensurate and incommensurate systems, statistical mechanics of nonlinear systems, ratchets, and nonequilibrium dynamics of interacting many-body systems. Also treated are the principal nonlinear equations along with the properties of their localized solutions and methods for their analysis.

Contents

• 1 Introduction: 1
• 12 The SineGordon Equation: 5
• 2 Physical Models: 9
• 22 A Mechanical Model: 10
• 23 Dislocation Dynamics: 12
• 24 Surfaces and Adsorbed Atomic Layers: 14
• 25 Incommensurate Phases in Dielectrics: 18
• 26 Crowdions and Lattice Defects: 20
• 27 Magnetic Chains: 21
• 28 Josephson Junctions: 23
• 29 Nonlinear Models of the DNA Dynamics: 25
• 210 HydrogenBonded Chains: 27
• 211 Models of Interfacial Slip: 29
• 3 Kinks: 31
• 32 Dynamics of Kinks: 38
• 322 Moving Kinks: 40
• 323 Trapped Kinks: 42
• 324 Multiple Kinks: 44
• 33 Generalized OnSite Potential: 47
• 331 Basic Properties: 48
• 332 Kink Internal Modes: 50
• 333 Nonsinusoidal OnSite Potential: 54
• 334 Multiple Well Potential: 58
• 335 MultiBarrier Potential: 63
• 34 Disordered Substrates: 66
• 341 Effective Equation of Motion: 68
• 342 Point Defects: 72
• 343 External Inhomogeneous Force: 73
• 35 Anharmonic Interatomic Interaction: 75
• 351 ShortRange Interaction: 77
• 352 Nonconvex Interatomic Potentials: 82
• 353 KacBaker Interaction: 89
• 354 LongRange Interaction: 92
• 355 Compacton Kinks: 96
• 4 Breathers: 99
• 412 SmallAmplitude Breathers: 102
• 42 Breather Collisions: 103
• 421 ManySoliton Effects: 105
• 422 Fractal Scattering: 107
• 423 Soliton Cold Gas: 109
• 43 Impurity Modes: 111
• 432 Soliton Interactions with Impurities: 116
• 44 Discrete Breathers: 121
• 442 Existence and Stability: 122
• 443 The Discrete NLS Equation: 125
• 444 Dark Breathers: 131
• 445 Rotobreathers: 134
• 45 TwoDimensional Breathers: 136
• 46 Physical Systems and Applications: 138
• 5 Ground State: 141
• 52 FixedDensity Chain: 149
• 522 Incommensurate Configurations: 159
• 53 FreeEnd Chain: 165
• 531 FrankvanderMerwe Transition: 167
• 532 Devils Staircase and Phase Diagram: 171
• 54 Generalizations of the FK Model: 174
• 542 Anharmonic Interatomic Potential: 177
• 543 Nonconvex Interaction: 184
• 6 Statistical Mechanics: 195
• 62 General Formalism: 197
• GlassLike Properties: 202
• 632 Configurational Excitations: 205
• 633 TwoLevel Systems and Specific Heat: 208
• Gas of Quasiparticles: 211
• 641 Sharing of the Phase Space and Breathers: 214
• 642 KinkPhonon Interaction: 215
• 643 KinkKink Interaction: 218
• 65 Statistical Mechanics of the FK Chain: 220
• 652 The PseudoSchrodinger Equation: 225
• 653 Susceptibility: 227
• 654 Hierarchy of Superkink Lattices: 233
• 655 EqualTime Correlation Functions: 234
• 656 Generalized FK Models: 239
• 7 Thermalized Dynamics: 243
• 711 Basic Formulas: 245
• 712 Mori Technique: 247
• 713 Diffusion Coefficients: 249
• 714 Noninteracting Atoms: 251
• 715 Interacting Atoms: 253
• 72 Diffusion of a Single Kink: 257
• 721 Langevin Equation: 258
• 722 Intrinsic Viscosity: 261
• 723 Anomalous Diffusion: 263
• 724 Kink Diffusion Coefficient: 265
• 73 Dynamic Correlation Functions: 268
• 74 Mass Transport Problem: 272
• 741 Diffusion in a Homogeneous Gas: 273
• 742 Approximate Methods: 276
• 743 Phenomenological Approach: 281
• 744 SelfDiffusion Coefficient: 284
• 745 Properties of the Diffusion Coefficients: 286
• 8 Driven Dynamics: 291
• 82 Nonlinear Response of Noninteracting Atoms: 292
• 821 Overdamped Case: 293
• 822 Underdamped Case: 294
• 83 Overdamped FK Model: 300
• 84 Driven Kink: 306
• 85 Instability of Fast Kinks: 308
• 86 Supersonic and Multiple Kinks: 316
• 87 LockedtoSliding Transition: 323
• 88 Hysteresis: 328
• 89 Traffic Jams: 330
• Dissipative Dynamics: 334
• 811 Periodic Driving of Underdamped Systems: 339
• 9 Ratchets: 343
• 92 Different Types of Ratchets: 345
• 922 Diffusional Ratchets: 346
• 923 Inertial Ratchets: 353
• 93 Solitonic Ratchets: 356
• 931 Symmetry Conditions: 357
• 933 Pulsating Ratchets: 361
• 94 Experimental Realizations: 363
• 10 FiniteLength Chain: 365
• 102 Ground State and Excitation Spectrum: 366
• 1022 Continuum Approximation: 369
• 1023 Discrete Chains: 370
• 1024 Vibrational Spectrum: 372
• 103 Dynamics of a Finite Chain: 374
• 1032 Adiabatic Trajectories: 375
• 1033 Diffusion of Short Chains: 379
• 1034 Stimulated Diffusion: 381
• 11 TwoDimensional Models: 383
• 112 Scalar Models: 385
• 1121 Statistical Mechanics: 389
• 1122 Dynamic Properties: 391
• 113 Zigzag Model: 392
• 1131 Ground State: 394
• 1132 Aubry Transitions: 397
• 1133 Classification of Kinks: 400
• 1134 Zigzag Kinks: 405
• 1135 Applications: 413
• 114 SpringandBall Vector 2D Models: 415
• 1141 The Ground State: 417
• 1142 Excitation Spectrum: 420
• 115 Vector 2D FK Model: 422
• 1151 LockedtoSliding Transition: 423
• 1152 FuseSafety Device on an Atomic Scale: 429
• 12 Conclusion: 431
• 13 Historical Remarks: 435
• References: 441
• Index: 465

Bibliographic information

• Title: The Frenkel-Kontorova Model: Concepts, Methods, and Applications; Physics and Astronomy Online Library; Texts and Monographs in Physics, ISSN 0172-5998; Theoretical and Mathematical Physics
• Authors: Oleg M. Braun, Yuri S. Kivshar
• Edition: illustrated
• Publisher: Springer, 2004
• ISBN: 3540407715, 9783540407713
• Length: 472 pages
• Subjects: Science / Physics / Condensed Matter; Science / Physics / General; Science / Physics / Mathematical & Computational; Science / Spectroscopy & Spectrum Analysis; Technology & Engineering / Materials Science