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

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Springer Science & Business Media, Jan 9, 2004 - Science - 472 pages
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Theoretical physics deals with physical models. The main requirements for a good physical model are simplicity and universality. Universal models which can be applied to describe a variety of different phenomena are very rare in physics and, therefore, they are of key importance. Such models attract the special attention of researchers as they can be used to describe underlying physical concepts in a simple way. Such models appear again and again over the years and in various forms, thus extending their applicability and educa tional value. The simplest example of this kind is the model of a pendulum; this universal model serves as a paradigm which encompasses basic features of various physical systems, and appears in many problems of very different physical context. Solids are usually described by complex models with many degrees of freedom and, therefore, the corresponding microscopic equations are rather complicated. However, over the years a relatively simple model, known these days as the Prenkel-K ontorova model, has become one of the fundamental and universal tools of low-dimensional nonlinear physics; this model describes a chain of classical particles coupled to their neighbors and subjected to a pe riodic on-site potential.
  

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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
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

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
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
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