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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.



Low Dimensional Chaos


Chapter 1. Weak Chaos, Infinite Ergodic Theory, and Anomalous Dynamics

This book chapter introduces to the concept of weak chaos, aspects of its ergodic theory description, and properties of the anomalous dynamics associated with it. In the first half of the chapter we study simple one-dimensional deterministic maps, in the second half basic stochastic models, and eventually an experiment. We start by reminding the reader of fundamental chaos quantities and their relation to each other, exemplified by the paradigmatic Bernoulli shift. Using the intermittent Pomeau–Manneville map the problem of weak chaos and infinite ergodic theory is outlined, defining a very recent mathematical field of research. Considering a spatially extended version of the Pomeau–Manneville map leads us to the phenomenon of anomalous diffusion. This problem will be discussed by applying stochastic continuous time random walk theory and by deriving a fractional diffusion equation. Another important topic within modern nonequilibrium statistical physics are fluctuation relations, which we investigate for anomalous dynamics. The chapter concludes by showing the importance of anomalous dynamics for understanding experimental results on biological cell migration.

Rainer Klages

Chapter 2. Directed Transport in a Stochastic Layer

We consider a problem of transport in a spatially periodic potential under the influence of a slowly time-dependent unbiased periodic external force. Using methods of the adiabatic perturbation theory we show that for a periodic external force of general kind the system demonstrates directed (ratchet) transport in the chaotic domain on very long time intervals and obtain a formula for the average velocity of this transport. Two cases are studied: the case of the external force of small amplitude and the case of the external force with amplitude of order one.

Alexei Vasiliev

From Chaos to Kinetics: Application to Hot Plasmas


Chapter 3. On the Nonlinear Electron Vibrations in a Plasma

Many applications, including the control of parametric instabilities detrimental for inertial confinement fusion, which motivates the present work, require an accurate kinetic description of the electron vibrations in a plasma, henceforth called

electron plasma waves

. This issue actually gave rise to a countless number of papers, even beyond the plasma physics community, due to some fascinating effects like Landau damping, which is the most famous example of collisionless dissipation. However, very few theoretical results are available when the wave is so intense that it deeply traps a significant fraction of the electrons in its potential, and these results are mostly restricted to academic situations. By contrast, in this chapter we provide a description of nearly monochromatic electron plasma waves valid from the linear to the strongly nonlinear regime, using hypotheses general enough to address a real physics situation like stimulated Raman scattering in a fusion plasma. Completely new theoretical results are obtained regarding the collisionless dissipation and the dispersion relation of an electron plasma wave, whose accuracy was tested against very careful kinetic simulations of stimulated Raman scattering.

Didier Bénisti

Chapter 4. How to Face the Complexity of Plasmas?

This paper has two main parts. The

first part

is subjective and aims at favoring a brainstorming in the plasma community. It discusses the present theoretical description of plasmas, with a focus on hot weakly collisional plasmas. It comprises two subparts. The first one deals with the present status of this description. In particular, most models used in plasma physics are shown to have feet of clay, there is no strict hierarchy between them, and a principle of simplicity dominates the modeling activity. At any moment the description of plasma complexity is provisional and results from a collective and somewhat unconscious process. The second subpart considers possible methodological improvements, some of them specific to plasma physics and some others of possible interest for other fields of science. The proposals for improving the present situation go along the following lines: improving the way papers are structured and the way scientific quality is assessed in the referral process, developing new databases, stimulating the scientific discussion of published results, diversifying the way results are made available, assessing more quality than quantity, and making available an incompressible time for creative thinking and non-purpose-oriented research. Some possible improvements for teaching are also indicated. The suggested improvement of the structure of papers would be for each paper to have a “claim section” summarizing the main results and their most relevant connection to previous literature. One of the ideas put forward is that modern nonlinear dynamics and chaos might help revisiting and unifying the overall presentation of plasma physics. The

second part

of this chapter is devoted to one instance where this idea has been developed for three decades: the description of Langmuir wave–electron interaction in one-dimensional plasmas by a finite-dimensional Hamiltonian. This part is more specialized and is written like a classical scientific paper. This Hamiltonian approach enables recovering Vlasovian linear theory with a mechanical understanding. The quasilinear description of the weak warm beam is discussed, and it is shown that self-consistency vanishes when the plateau forms in the tail distribution function. This leads to consider the various diffusive regimes of the dynamics of particles in a frozen spectrum of waves with random phases. A recent numerical simulation showed that diffusion is quasilinear when the plateau sets in and that the variation of the phase of a given wave with time is almost non-fluctuating for random realizations of the initial wave phases. This led to new analytical calculations of the average behavior of the self-consistent dynamics when the initial wave phases are random. Using Picard iteration technique, they confirm numerical results and exhibit a spontaneous emission of spatial inhomogeneities.

Dominique F. Escande

Chapter 5. First Principle Transport Modeling in Fusion Plasmas: Critical Issues for ITER

Tokamaks aim at confining hot plasmas by means of strong magnetic fields in view of reaching a net energy gain through fusion reactions. Plasma confinement turns out to be governed by small-scale instabilities which saturate nonlinearly and lead to turbulent fluctuations of a few percent. This paper recalls the basic equations for modeling such weakly collisional plasmas. It essentially relies on the kinetic, or more precisely the gyrokinetic, description, although some attempts are made to incorporate some of the kinetic properties, namely, wave-particle resonances, in fluid models by means of collisionless closures. Three main types of micro-instabilities are detailed and studied linearly, namely, drift waves, interchange, and bump-on-tail. Finally, some of the main critical issues in turbulence modeling are addressed: flux-driven versus gradient-driven models, the subsequent impact of mean profile relaxation on turbulent transport dynamics, and the role of large-scale flows, either at equilibrium or turbulence driven, on turbulence saturation and on the possible triggering of transport barriers. The significant progress in understanding and prediction of turbulent transport in tokamak plasmas thanks to first-principle simulations is highlighted.

Yanick Sarazin

From Kinetics to Fluids and Solids


Chapter 6. Turbulent Thermal Convection and Emergence of Isolated Large Single Vortices in Soap Bubbles

Experiments using a novel thermal convection cell consisting of half a soap bubble heated at the equator to study turbulent thermal convection and the movement of isolated vortices are reviewed. The soap bubble, subject to stratification, develops thermal convection at its equator. A particular feature of this cell is the emergence of isolated vortices. These vortices resemble hurricanes or cyclones and similarities between these structures and their natural counterparts are found. This is brought forth through a study of the mean square displacement of these objects showing signs of superdiffusion. In addition to these features, the study of the statistical properties of the turbulence engendered in these soap bubbles shows a clear indication for the existence of the so-called Bolgiano–Obukhov scaling both for the temperature and the velocity fluctuations. A remarkable transition is uncovered: the temperature and the velocity structure functions show intermittency for small temperature gradients; this intermittency then disappears for large gradients.

Hamid Kellay

Chapter 7. On the Occurrence of Elastic Singularities in Compressed Thin Sheets: Stress Focusing and Defocusing

Compressing thin sheets usually yields the formation of singularities which focus curvature and stretch on points or lines. In particular, following the common experience of crumpled paper where a paper sheet is crushed in a paper ball, one might guess that elastic singularities should be the rule beyond some compression level. In contrast, we show here that, somewhat surprisingly, compressing a sheet between cylinders makes singularities spontaneously disappear at large compression. This “stress-defocusing” phenomenon is qualitatively explained from scale invariance and further linked to a criterion based on a balance between stretch and curvature energies on defocused states. This criterion is made quantitative using the scalings relevant to sheet elasticity and compared to experiment. These results are synthesized in a phase diagram completed with plastic transitions. They end up with a renewed vision of elastic singularities as a thermodynamic condensed phase where stress is focused, in competition with a regular diluted phase where stress is defocused. Different compression routes may be followed in this diagram by managing differently the two principal curvatures of a sheet, as experimentally achieved here. In practice, besides the famous Elastica and crumpled paper routes, this offers interesting alternatives for compressing a sheet with an amazing spontaneous regularization of geometry and stress that repels the occurrence of plastic damages.

Alain Pocheau

Chapter 8. Transport Properties in a Model of Quantum Fluids and Solids

We discuss the general transport properties of the non-linear Schrödinger equation in the context of quantum fluid models. In particular, we will discuss two striking behaviors described within this model: the nucleation of quantized vortices and the non-classical rotational inertia.

Christophe Josserand

Beyond Physics: Examples of Complex Systems


Chapter 9. Spatial and Temporal Order Beyond the Deterministic Limit: The Role of Stochastic Fluctuations in Population Dynamics

Modeling the self-consistent dynamics of an ensemble made of microscopic constituents can be tackled via deterministic or alternatively stochastic viewpoints. The latter enables one to respect the discrete nature of the scrutinized medium, a possibility which is conversely prevented when dealing with the former idealized approximation. As we shall here discuss, stochastic finite-size fluctuations can drive the emergence of regular spatiotemporal cycles that persist for moderate and even large sizes of the population and which are not captured within the mean-field descriptive scenario. The van Kampen system-size expansion is an elegant mathematical approach that allows one to investigate the key role played by the inherent stochasticity. We here provide a pedagogical introduction to such a method and discuss its application to a model of autocatalytic reactions.

Duccio Fanelli

Chapter 10. An Ising Model for Road Traffic Inference

We review some properties of the “belief propagation” algorithm, a distributed iterative map used to perform Bayesian inference and present some recent work where this algorithm serves as a starting point to encode observation data into a probabilistic model and to process large-scale information in real time. A natural approach is based on the linear response theory and various recent instantiations are presented. We will focus on the particular situation where the data have many different statistical components, representing a variety of independent patterns. As an application, the problem of reconstructing and predicting traffic states based on floating car data is then discussed.

Cyril Furtlehner


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