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E-raamat: From Hamiltonian Chaos to Complex Systems: A Nonlinear Physics Approach

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From Hamiltonian Chaos to Complex Systems: A Nonlinear Physics Approach collects contributions on recent developments in non-linear dynamics and statistical physics with an emphasis on complex systems. This book provides a wide range of state-of-the-art research in these fields. The unifying aspect of this book is demonstration of how similar tools coming from dynamical systems, nonlinear physics, and statistical dynamics can lead to a large panorama of research in various fields of physics and beyond, most notably with the perspective of application in complex systems.



This book reviews progress in non-linear dynamics and statistical physics with an emphasis on complex systems, showing how tools developed for dynamical systems, nonlinear physics and statistical dynamics can open a panorama of research in physics and beyond.
Part I Low Dimensional Chaos
1 Weak Chaos, Infinite Ergodic Theory, and Anomalous Dynamics
3(40)
Rainer Klages
1.1 Introduction
3(3)
1.2 Chaos and Anomalous Dynamics
6(9)
1.2.1 Deterministic Chaos in a Simple Map
6(3)
1.2.2 Weak Chaos and Infinite Ergodic Theory
9(5)
1.2.3 A Generalized Hierarchy of Chaos
14(1)
1.3 Anomalous Diffusion
15(9)
1.3.1 A Simple Model Generating Anomalous Diffusion
16(2)
1.3.2 Continuous Time Random Walk Theory
18(4)
1.3.3 A Fractional Diffusion Equation
22(2)
1.4 Anomalous Fluctuation Relations
24(7)
1.4.1 Fluctuation Relations
24(1)
1.4.2 Fluctuation Relations for Ordinary Langevin Dynamics
25(3)
1.4.3 Fluctuation Relations for Anomalous Langevin Dynamics
28(3)
1.5 Anomalous Dynamics of Biological Cell Migration
31(6)
1.5.1 Cell Migration
31(1)
1.5.2 Experimental Results and Statistical Analysis
32(3)
1.5.3 Stochastic Modeling
35(2)
1.6 Summary
37(6)
References
38(5)
2 Directed Transport in a Stochastic Layer
43(18)
Alexei Vasiliev
2.1 Introduction
43(2)
2.2 External Forcing of Order One
45(4)
2.3 Small External Forcing
49(7)
2.3.1 Main Equations: Diffusion of the Adiabatic Invariant
49(3)
2.3.2 Average Velocity of the Transport
52(4)
2.4 Summary
56(5)
References
57(4)
Part II From Chaos to Kinetics: Application to Hot Plasmas
3 On the Nonlinear Electron Vibrations in a Plasma
61(48)
Didier Benisti
3.1 Introduction
61(2)
3.2 Perturbative Motion of Electrons Acted Upon by an Electrostatic Wave
63(12)
3.2.1 General Formalism
64(4)
3.2.2 Perturbative Analysis
68(7)
3.3 Envelope Equation for a Purely Time-Dependent Wave Amplitude
75(6)
3.3.1 Exponentially Growing Wave
76(1)
3.3.2 Generalized Expression for Χi
77(1)
3.3.3 Symmetric Detrapping
78(1)
3.3.4 Nonlinear Landau Damping Rate
79(2)
3.4 Variational Approach and Generalization to a Space-Dependent Wave Amplitude
81(13)
3.4.1 Physical Discussion of the Previous Results Using a Variational Approach
82(1)
3.4.2 One-Dimensional Variation of the Wave Amplitude
83(8)
3.4.3 Three-Dimensional Space Variation of the Wave Amplitude
91(3)
3.5 Nonlinear Frequency Shift of an SRS-Driven Plasma Wave
94(9)
3.5.1 Derivation of Χr
94(4)
3.5.2 Derivation of αd
98(1)
3.5.3 Comparisons with Results from Vlasov Simulations of Stimulated Raman Scattering and with Previous Theories
99(3)
3.5.4 Discussion of Previously Proposed Nonlinear Dispersion Relations
102(1)
3.6 Conclusion
103(2)
3.7 Appendix: Derivation of ∂ωΧreffr
105(4)
References
106(3)
4 How to Face the Complexity of Plasmas?
109(50)
Dominique F. Escande
4.1 Introduction
110(4)
4.1.1 What This
Chapter Is About
110(2)
4.1.2 Plasma Physics
112(2)
4.2 Facing Plasma Complexity
114(20)
4.2.1 Present Status of the Description of Plasma Complexity
114(8)
4.2.2 Possible Methodological Improvements
122(12)
4.3 Describing Plasma Dynamics with Finite-Dimensional Hamiltonian Systems
134(15)
4.3.1 Recovering Vlasovian Linear Theory with a Mechanical Understanding
137(3)
4.3.2 Quasilinear Theory
140(1)
4.3.3 Dynamics When the Distribution Is a Plateau
141(1)
4.3.4 Diffusion in a Given Spectrum of Waves
142(4)
4.3.5 A Crucial Numerical Simulation
146(1)
4.3.6 New Analytical Calculations
147(2)
4.4 Conclusion
149(1)
4.5 Appendix 1: Extended Summary
150(3)
4.6 Appendix 2: First Example of a Claim Section
153(1)
4.7 Appendix 3: Second Example of a Claim Section
153(6)
References
155(4)
5 First Principle Transport Modeling in Fusion Plasmas: Critical Issues for ITER
159(32)
Yanick Sarazin
5.1 Transport Issues in Controlled Fusion Devices
159(3)
5.1.1 Magnetic Configuration and Main Plasma Parameters
159(2)
5.1.2 Transport and Fusion Performance
161(1)
5.1.3 Transport and Turbulence
162(1)
5.2 Turbulence Modeling: The Need for a kinetic Description
162(6)
5.2.1 Collisionless Fluid Approaches "a la Hammett-Perkins"
163(3)
5.2.2 Gyrokinetic Description
166(2)
5.3 Main Micro-Instabilities in Fusion Plasmas
168(11)
5.3.1 Physical Understanding of Drift-Wave and Interchange Instabilities
168(4)
5.3.2 Simple Model for Drift-Wave and Interchange Instabilities
172(3)
5.3.3 Bump-on-Tail Instability
175(4)
5.4 Critical Issues in Turbulent Transport Modeling
179(7)
5.4.1 Gradient-Versus Flux-Driven Models
179(1)
5.4.2 Profile Relaxation and Turbulence Trapping
180(3)
5.4.3 Large Scale Flows and Transport Barriers
183(3)
5.5 Conclusion
186(5)
References
187(4)
Part III From Kinetics to Fluids and Solids
6 Turbulent Thermal Convection and Emergence of Isolated Large Single Vortices in Soap Bubbles
191(16)
Hamid Kellay
6.1 Introduction
191(1)
6.2 Isolated Vortices
192(5)
6.3 Statistical Properties of the Temperature and Velocity Fields
197(8)
6.4 Conclusion
205(2)
References
205(2)
7 On the Occurrence of Elastic Singularities in Compressed Thin Sheets: Stress Focusing and Defocusing
207(26)
Alain Pocheau
7.1 Introduction
208(2)
7.2 On Singularity Occurrence in Sheet Elasticity: From Elastica to Crumpled Paper
210(2)
7.3 Basics on Linear Elasticity of Sheets
212(5)
7.3.1 Sheet Elastic Energy
213(1)
7.3.2 Gaussian Curvature and Theorema Egregium
214(2)
7.3.3 Sheet Equilibrium and Foppl-von Karman's Equation
216(1)
7.4 Experiment
217(5)
7.4.1 Setup
217(1)
7.4.2 Compression Route
218(3)
7.4.3 Defocusing Scaling
221(1)
7.5 Energy Criterion for Stress Focusing and Scalings
222(2)
7.5.1 Energy Criterion for Stress Focusing or Defocusing
222(1)
7.5.2 Scalings
223(1)
7.6 Phase Diagram and Nature of Singularities
224(6)
7.6.1 Scale-Invariance and Defocusing
225(1)
7.6.2 Scalings and Phase Diagram for Singularities
226(3)
7.6.3 Plasticity
229(1)
7.7 Conclusion
230(3)
References
231(2)
8 Transport Properties in a Model of Quantum Fluids and Solids
233(36)
Christophe Josserand
8.1 Introduction: One Equation, Many Contexts
233(6)
8.1.1 Bose-Einstein Condensates
234(2)
8.1.2 Superfluid Helium
236(1)
8.1.3 A Model for Supersolidity?
237(1)
8.1.4 Nonlinear Optics
238(1)
8.1.5 Fluid Mechanics
238(1)
8.2 General Properties of the NLS Equation
239(6)
8.2.1 Conserved Quantities and Hamiltonian Structures
240(1)
8.2.2 Invariances of the Equation
241(1)
8.2.3 Integrability, Solitons and Solitary Waves
241(1)
8.2.4 Hydrodynamical Equations
242(2)
8.2.5 Quantized Vortices
244(1)
8.2.6 Dispersion Relation, Spectrum of Excitation and Superfluidity
245(1)
8.3 Vortex Nucleation
245(6)
8.3.1 Around the Transonic Regime
246(2)
8.3.2 The Euler-Tricomi Equation in the Transonic Region
248(2)
8.3.3 From the Euler-Tricomi Equation to Vortex Nucleation?
250(1)
8.4 Nonclassical Rotational Inertia in a Supersolid Model
251(13)
8.4.1 Properties of the Model
252(4)
8.4.2 Ground State of the Gross-Pitaevskii Model
256(5)
8.4.3 A Model Combining Elastic and Superfluid Properties
261(3)
8.5 Conclusion
264(5)
References
264(5)
Part IV Beyond Physics: Examples of Complex Systems
9 Spatial and Temporal Order Beyond the Deterministic Limit: The Role of Stochastic Fluctuations in Population Dynamics
269(24)
Duccio Fanelli
9.1 Introduction
269(1)
9.2 On the Deterministic and Stochastic Viewpoints
270(2)
9.3 The Van Kampen Expansion Applied to a Simple Birth/Death Stochastic Model
272(6)
9.4 A Model of Autocatalytic Reactions
278(1)
9.5 The Aspatial Model: Deterministic and Stochastic Dynamics
279(5)
9.6 Spatial Model: Ordered Patterns Revealed by the van Kampen System Size Expansion
284(5)
9.7 Conclusion
289(4)
References
292(1)
10 An Ising Model for Road Traffic Inference
293(30)
Cyril Furtlehner
10.1 Introduction
293(1)
10.2 The Belief Propagation Algorithm
294(4)
10.3 The Inverse Ising Problem
298(7)
10.3.1 Gibbs Free Energy
300(1)
10.3.2 Plefka's Expansion
300(2)
10.3.3 Linear Response Approximate Solution
302(1)
10.3.4 Bethe Approximation
303(2)
10.4 Application Context
305(6)
10.4.1 Road Traffic Inference
305(1)
10.4.2 An Ising Model for Traffic
305(5)
10.4.3 MRF Model and Pseudo Moment Matching Calibration
310(1)
10.5 Multiple BP Fixed Points for Multiple Traffic Patterns
311(4)
10.6 Experiments with Synthetic and Real Data
315(4)
10.7 Conclusion
319(4)
References
320(3)
Index 323
Xavier Leoncini, Centre de Physique Théorique, xavier.leoncini@cpt.univ-mrs.fr

Marc Leonetti, Aix-Marseille Université, leonetti@irphe.univ-mrs.fr
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