The Frenkel-Kontorova Model : Concepts, Methods, and Applications (Texts and Monographs in Physics) (2004. XVIII, 472 p.)

The Frenkel-Kontorova Model : Concepts, Methods, and Applications (Texts and Monographs in Physics) (2004. XVIII, 472 p.)

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  • 製本 Hardcover:ハードカバー版/ページ数 472 p.
  • 商品コード 9783540407713

Full Description


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. The book 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, and nonequilibrium dynamics of interacting many-body systems.

Table of Contents

  1 Introduction                                   1  (8)
1.1 The Frenkel-Kontorova Model 1 (4)
1.2 The Sine-Gordon Equation 5 (4)
2 Physical Models 9 (22)
2.1 General Approach 9 (1)
2.2 A Mechanical Model 10 (2)
2.3 Dislocation Dynamics 12 (2)
2.4 Surfaces and Adsorbed Atomic Layers 14 (4)
2.5 Incommensurate Phases in Dielectrics 18 (2)
2.6 Crowdions and Lattice Defects 20 (1)
2.7 Magnetic Chains 21 (2)
2.8 Josephson Junctions 23 (2)
2.9 Nonlinear Models of the DNA Dynamics 25 (2)
2.10 Hydrogen-Bonded Chains 27 (2)
2.11 Models of Interfacial Slip 29 (2)
3 Kinks 31 (68)
3.1 The Peierls-Nabarro Potential 31 (7)
3.2 Dynamics of Kinks 38 (9)
3.2.1 Effective Equation of Motion 38 (2)
3.2.2 Moving Kinks 40 (2)
3.2.3 Trapped Kinks 42 (2)
3.2.4 Multiple Kinks 44 (3)
3.3 Generalized On-Site Potential 47 (19)
3.3.1 Basic Properties 48 (2)
3.3.2 Kink Internal Modes 50 (4)
3.3.3 Nonsirmsoidal On-Site Potential 54 (4)
3.3.4 Multiple-Well Potential 58 (5)
3.3.5 Multi-Barrier Potential 63 (3)
3.4 Disordered Substrates 66 (9)
3.4.1 Effective Equation of Motion 68 (4)
3.4.2 Point Defects 72 (1)
3.4.3 External Inhomogeneous Force 73 (2)
3.5 Anharmonic Interatomic Interaction 75 (24)
3.5.1 Short-Range Interaction 77 (5)
3.5.2 Nonconvex Interatomic Potentials 82 (7)
3.5.3 Kac-Baker Interaction 89 (3)
3.5.4 Long-Range Interaction 92 (4)
3.5.5 Compactors Kinks 96 (3)
4 Breathers 99 (42)
4.1 Perturbed Sine-Gordon Breathers 99 (4)
4.1.1 Large-Amplitude Breathers 99 (3)
4.1.2 Small-Amplitude Breathers 102(1)
4.2 Breather Collisions 103(8)
4.2.1 Many-Soliton Effects 105(2)
4.2.2 Fractal Scattering 107(2)
4.2.3 Soliton Cold Gas 109(2)
4.3 Impurity Modes 111(10)
4.3.1 Structure and Stability 111(5)
4.3.2 Soliton Interactions with Impurities 116(5)
4.4 Discrete Breathers 121(15)
4.4.1 General Remarks 121(1)
4.4.2 Existence and Stability 122(3)
4.4.3 The Discrete NLS Equation 125(6)
4.4.4 Dark Breathers 131(3)
4.4.5 Rotobreathers 134(2)
4.5 Two-Dimensional Breathers 136(2)
4.6 Physical Systems and Applications 138(3)
5 Ground State 141(54)
5.1 Basic Properties 141(8)
5.2 Fixed-Density Chain 149(16)
5.2.1 Commensurate Configurations 149(10)
5.2.2 Incommensurate Configurations 159(6)
5.3 Free-End Chain 165(9)
5.3.1 Frank-van-der-Merwe Transition 167(4)
5.3.2 Devil's Staircase and Phase Diagram 171(3)
5.4 Generalizations of the FK Model 174(21)
5.4.1 On-Site Potential of a General Form 174(3)
5.4.2 Anharmonic Interatomic Potential 177(7)
5.4.3 Nonconvex Interaction 184(11)
6 Statistical Mechanics 195(48)
6.1 Introductory Remarks 195(2)
6.2 General Formalism 197(5)
6.3 Weak-Bond Limit: Glass-Like Properties 202(9)
6.3.1 Ising-Like Model 202(3)
6.3.2 Configurational Excitations 205(3)
6.3.3 Two-Level Systems and Specific Heat 208(3)
6.4 Strong-Bond Limit: Gas of 211(9)
Quasiparticles
6.4.1 Sharing of the Phase Space and 214(1)
Breathers
6.4.2 Kink-Phonon Interaction 215(3)
6.4.3 Kink-Kink Interaction 218(1)
6.4.4 Discreteness Effects 218(2)
6.5 Statistical Mechanics of the FK Chain 220(23)
6.5.1 Transfer-Integral Method 220(5)
6.5.2 The Pseudo-Schrodinger Equation 225(2)
6.5.3 Susceptibility 227(6)
6.5.4 Hierarchy of Superkink Lattices 233(1)
6.5.5 Equal-Time Correlation Functions 234(5)
6.5.6 Generalized FK Models 239(4)
7 Thermalized Dynamics 243(48)
7.1 Basic Concepts and Formalism 243(14)
7.1.1 Basic Formulas 245(2)
7.1.2 Mori Technique 247(2)
7.1.3 Diffusion Coefficients 249(2)
7.1.4 Noninteracting Atoms 251(2)
7.1.5 Interacting Atoms 253(4)
7.2 Diffusion of a Single Kink 257(11)
7.2.1 Langevin Equation 258(3)
7.2.2 Intrinsic Viscosity 261(2)
7.2.3 Anomalous Diffusion 263(2)
7.2.4 Kink Diffusion Coefficient 265(3)
7.3 Dynamic Correlation Functions 268(4)
7.4 Mass Transport Problem 272(19)
7.4.1 Diffusion in a Homogeneous Gas 273(3)
7.4.2 Approximate Methods 276(5)
7.4.3 Phenomenological Approach 281(3)
7.4.4 Self-Diffusion Coefficient 284(2)
7.4.5 Properties of the Diffusion 286(5)
Coefficients
8 Driven Dynamics 291(52)
8.1 Introductory Remarks 291(1)
8.2 Nonlinear Response of Noninteracting 292(8)
Atoms
8.2.1 Overdamped Case 293(1)
8.2.2 Underdamped Case 294(6)
8.3 Overdamped FK Model 300(6)
8.4 Driven Kink 306(2)
8.5 Instability of Fast Kinks 308(8)
8.6 Supersonic and Multiple Kinks 316(7)
8.7 Locked-to-Sliding Transition 323(5)
8.7.1 Commensurate Ground States 323(1)
8.7.2 Complex Ground States and Multistep 323(5)
Transition
8.8 Hysteresis 328(2)
8.9 Traffic Jams 330(4)
8.10 Periodic Forces: Dissipative Dynamics 334(5)
8.11 Periodic Driving of Underdamped Systems 339(4)
9 Ratchets 343(22)
9.1 Preliminary Remarks 343(2)
9.2 Different Types of Ratchets 345(11)
9.2.1 Supersymmetry 345(1)
9.2.2 Diffusional Ratchets 346(7)
9.2.3 Inertial Ratchets 353(3)
9.3 Solitonic Ratchets 356(7)
9.3.1 Symmetry Conditions 357(1)
9.3.2 Rocked Ratchets 357(4)
9.3.3 Pulsating Ratchets 361(2)
9.4 Experimental Realizations 363(2)
10 Finite-Length Chain 365(18)
10.1 General Remarks 365(1)
10.2 Ground State and Excitation Spectrum 366(8)
10.2.1 Stationary States 366(3)
10.2.2 Continuum Approximation 369(1)
10.2.3 Discrete Chains 370(2)
10.2.4 Vibrational Spectrum 372(2)
10.3 Dynamics of a Finite Chain 374(7)
10.3.1 Caterpillar-Like Motion 374(1)
10.3.2 Adiabatic Trajectories 375(4)
10.3.3 Diffusion of Short Chains 379(2)
10.3.4 Stimulated Diffusion 381(1)
10.4 Nonconvex Potential 381(2)
11 Two-Dimensional Models 383(48)
11.1 Preliminary Remarks 383(2)
11.2 Scalar Models 385(7)
11.2.1 Statistical Mechanics 389(2)
11.2.2 Dynamic Properties 391(1)
11.3 Zigzag Model 392(23)
11.3.1 Ground State 394(3)
11.3.2 Aubry Transitions 397(3)
11.3.3 Classification of Kinks 400(5)
11.3.4 Zigzag Kinks 405(8)
11.3.5 Applications 413(2)
11.4 Spring-and-Ball Vector 2D Models 415(7)
11.4.1 The Ground State 417(3)
11.4.2 Excitation Spectrum 420(1)
11.4.3 Dynamics 420(2)
11.5 Vector 2D FK Model 422(9)
11.5.1 Locked-to-Sliding Transition 423(6)
11.5.2 "Fuse-Safety Device" on an Atomic 429(2)
Scale
12 Conclusion 431(4)
13 Historical Remarks 435(6)
References 441(24)
Index 465