Chaos, Complexity and Transport: Theory and Applications

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Review talks

Review talks

      Y. Couder  (ENS Paris, France) - To be announced

      T. Evans  (General Atomics, USA) - Implications of Topological Complexity and Hamiltonian Chaos in the Edge Magnetic Field of Toroidal Plasmas*

    Magnetically confined toroidal plasmas, such as in tokamaks and stellarators, are particularly attractive for fusion research because of their potential for obtaining steady-state plasma pressures and confinement times needed to achieve substantially self-driven burning plasmas. Fusion plasma discharges, in the first generation of experimental tokamak reactors such as ITER, are expected to last up to 500 s with thermal output powers of 400 MW. This implies steady-state thermal loads, on the solid surfaces of power exhaust components, reaching 5 MW/m^2 under ideal conditions. Using the most heat tolerant materials currently available, these components will fail at 10–15 MW/m^2 leaving only a factor of 2-3 for error.
Research in the current generation of tokamak has shown that non-ideal effects such as small resonant and non-resonant magnetic perturbation from external field coils and internal MHD modes significantly alter the edge magnetic topology producing relatively complex 3D structures. These structures create asymmetries in the magnetic footprints that define heat flux distributions on power exhaust components and imply peak power loads that approach steady-state material failure limits (10–15 MW/m^2) in fusion reactors. The topology of these structures plays an important role in the nature of the transport and the stability of the boundary plasmas. For example, resonant magnetic fields applied to high confinement (H-mode) plasmas in poloidally diverted plasmas create magnetic footprints consistent with bifurcations in the topology of the separatrix and stochastic field line excursion. This results in a reduction of the effective particle confinement without affecting the energy confinement and stabilizes large, repetitive, MHD instabilities know as edge localized modes (ELMs) that drive heat and particle impulses capable of quickly damaging power exhaust components in fusion reactors. In addition, observations of edge plasmas in tokamaks, using high speed cameras, reveal an array of complex dynamic processes such as the formation of rapidly growing coherent filaments during ELMs and active bursts of intermittent turbulence associated with the rapid ejection of high density plasma clumps.
The topological properties of the complex structures, observed in the active region of the plasma edge, are qualitatively consistent with that of homoclinic tangles defined by invariant manifolds of the system and Hamiltonian chaos resulting from intersections of stable and unstable manifolds. Nevertheless, a quantitative description of the dynamics involved in these processes requires developing a better understanding of the plasma response to such complex topologies. A review of the recent experimental observations and progress on 3D fluid, kinetic and extended MHD modeling of the edge plasma in tokamaks will be given in this talk. These will be related to the topological structure of the invariant manifolds and the chaotic structure of the field lines based on conservative dynamical system theory.

*Work supported by the U.S. Department of Energy under DE-FC02-04ER54698.

      S. Fauve  (ENS Paris, France) - Chaotic dynamos generated by turbulent flows

    We first report the observation of a magnetic field generated by a turbulent von Karman flow of liquid sodium (VKS dynamo experiment). Small changes of the flow driving parameters generate different dynamical regimes of the magnetic field: oscillations, intermittent bursts and field reversals. In a second part, we discuss the effect of turbulence on the nature of the dynamo bifurcation, on the geometry of the generated magnetic field and on scaling laws for the magnetic energy density. We show that the dynamics of the large scale magnetic field result from a small number of competing modes.

      C. Jaffé  (University of West Virginia, USA) - A Review of Transition State Theory with Applications

    In this talk I will review the the mathematical foundations of transition state theory and will present a variety of applications. These will include mass transport in the solar system, ionization of atomic systems by crossed electric and magnetic fields, and laser driven chemical reactions.

      J. Laskar  (IMCCE, France) - To be announced

      S. Ruffo  (University of Florence, Italie) - Quasi-stationary states in mean-field dynamics

    I will review the topic of quasi-stationary states. These states are often encountered in mean-field Hamiltonian dynamics and are robust to external and stochastic perturbations. Examples of systems where they appear are: the Hamiltonian Mean-Field (HMF) model, the Colson-Bonifacio model of a free-electron laser, Escande's wave-particle Hamiltonian, mean-field self-gravitating systems, 2D hydrodynamics and Onsager's vortices.
A feature of such states, which makes them truly thermodynamic, is that their lifetime typically increases as a power of system size.
Besides that, they are "attractive", since one observes covergence from a different initial state. Recently, the use of Lynden-Bell entropy has been advocated to describe such states. In this talk, I will discuss the merits and drawbacks of this approach.

      E. Villermaux  (University of Provence, France) - Mixing by random stirring

    The principles of scalar mixing in randomly stirred media are discussed, aiming at describing the overall concentration distribution of the mixture, its shape, and rate of deformation as the mixture evolves towards uniformity. Two distinct experiments, one involving an ever dispersing mixture, the other a mixture confined in a channel, both in high Reynolds, three dimensional flows, behave quite differently. We show how these differences single two fundamental, and concomitant aspects of the process of mixing, namely the distribution of individual histories on one hand, and the interaction of the fluid particles in the medium on the other. The particles, elementary bricks of a mixture, are stretched sheets whose rates of diffusive smoothing and coalescence build up the overall mixture concentration distribution. Consequences of these processes on the spectral, and some geometrical facets of random mixtures are also examined.

      E. Wesfreid  (ESPCI, France) - To be announced

      G. Zaslavsky  (New York University, USA) - Three-dimensional stochastic web with quasicrystal symmetry

    We consider 4-dimensional map provided the weak chaos conditions, when the stochastic web emerges, and discuss the symmetry properties of the web. A brief review will be given for the problem of dynamical generation of different symmetric tiling of plane using the two-dimensional web map. This problem will be generalized to the four-dimensional web map that is different from the Arnold web. An averaging procedure permits to derive a generator of quasicrystal symmetry as a potential in three-dimensional space. In two-dimensional case such potentials are applied to optic lattices. We demonstrate the dynamics along the three-dimensional stochastic webs.

 

Focus talks

      R. Artuso  (University of Cômo, Italy) - Deterministic ratchets and beyond: zeta functions techniques.

    We illustrate the use of non perturbative techniques in the study of deterministic ratchets by working out an exactly solvable model, inspired by the so-called Parrondo games.

      P. H. Chavanis  (University Paul Sabatier, France) - Self-gravitating Brownian particles

    We discuss the dynamics and thermodynamics of a gas of self-gravitating Brownian particles. We consider a strong friction limit and a mean field approximation where the problem reduces to the study of the Smoluchowski-Poisson (SP) system. We show the existence of a canonical phase transition between a gaseous state and a condensed state below a critical temperature T_c. For T<T_c the system undergoes an isothermal collapse. The evolution is self-similar and leads to a finite time singularity. However, ``the singularity contains no mass''. In fact, the collapse continues after the singularity has arisen and a Dirac peak is formed in the post-collapse regime by accreting the mass of the halo. We mention different extensions of this problem: self-gravitating fermions, multi-species systems, generalized thermodynamics, influence of the dimension of space, etc. We also discuss the analogy between the collapse of self-gravitating Brownian particles and the chemotactic aggregation of bacterial populations in biology.

      M. Courbage  (University Paris VII, France) - Complexity and Entropy in Colliding Particle Systems

    We develop quantitative measures of entropy evolution for particle systems undergoing collision process in relation with various unstability properties.

      A. Luo  (Southern Illimois University, USA) - On global transversality and chaos in two-dimensional nonlinear dynamical systems

    In this paper, the global transversality and tangency in two-dimensional nonlinear dynamical systems are discussed, and the exact energy increment function (L-function) for such nonlinear dynamical systems is presented. The Melnikov function is an approximate expression of the exact energy increment. A periodically forced, damped Duffing oscillator with a separatrix is investigated as a sampled problem. The corresponding analytical conditions for the global transversality and tangency to the separatrix are derived. Numerical simulations are carried out for illustrations of the analytical conditions. From analytical and numerical results, the simple zero of the energy increment (or the Melnikov function) may not imply chaos exists. The conditions for the global transversality and tangency to the separatrix may be independent of the Melnikov function. Therefore, the analytical criteria for chaotic motions in nonlinear dynamical systems need to be further developed. The methodology presented in this paper is applicable to nonlinear dynamical systems without any separatrix.

      R. S. MacKay  (University of Warwick, UK) - Manifestations of chaos

    I will describe various systems for which good forms of chaos have been proved relatively recently and discuss their robustness:
1. Two charges of opposite sign in a uniform magnetic field and unequal gyrofrequencies exhibit "second species" chaos;
2. An exact area-preserving tilt map of the cylinder which is mixing and has diffusion constant a certain expression involving 2+sqrt{3}, approximately 0.14434;
3. Three mixing volume-preserving vector fields in 3D containers with no-slip boundaries.

      G. Morfill  (Max-Planck Institute, Germany) - Nonlinear Processes in Complex Plasmas

    "Complex Plasmas" are strongly coupled systems, consisting of ions, electrons, charged microparticles and some neutral gas. The dynamically dominant microparticles are individually visualised and due to their comparatively large mass (many billions atomic masses) characteristic dynamical time scales are in the 10’s of msec range, easily resolved with CCD cameras. Thus complex plasmas are ideal systems for studying some fundamental physics questions, such as the onset of cooperative phenomena, the kinetics of phase transitions and other critical phenomena, nanofluidic properties as well as many other hydrodynamic processes – all at the most elementary kinetic level. After a brief introduction into the properties of this interesting state of matter, some examples of recent research results in the above-mentioned fields will be presented.

      O. Piro  (IMEDEA, Spain) - Dissipative embedding of conservative dynamics

    In a number of apparently unrelated problems, Hamiltonian or volume preserving dynamics takes place on an invariant, eventually attracting, sub-manifold of a dimensionally larger dissipative system. Examples of such situations are a) the motion of neutrally bouyant finite size particles, b) controlled Hamiltonian systems, c) tidally synchronized celestial bodies, etc. I will try to present an integrated view of this class of problems and show a number of phenomena that are likely to appear in all its instances.

      S. Rica  (ENS Paris, France) - Condensation of classical nonlinear waves

    We study the formation of a large-scale coherent structure (a condensate) in classical wave equations by considering the defocusing nonlinear Schr"odinger equation as a representative model. We formulate a thermodynamic description of the condensation process by using a wave turbulence theory with ultraviolet cut-off. In 3 dimensions the equilibrium state undergoes a phase transition for sufficiently low energy density, while no transition occurs in 2 dimensions, in analogy with standard Bose-Einstein condensation in quantum systems.
Numerical simulations show that the thermodynamic limit is reached for systems with ^3$ computational modes and greater.
On the basis of a modified wave turbulence theory, we show that the nonlinear interaction makes the transition to condensation subcritical. The theory is in quantitative agreement with the simulations.

      V. Rom-Kedar  (Institut Weizmann, Israël) - Stable motion in high-dimensional steep repelling potentials

    The appearance of elliptic periodic orbits in families of n-dimensional smooth repelling billiard-like potentials that are arbitrarily steep and close to dispersing billiards is established for any finite n. The width of the stability regions in the parameter space scales as a power-law in 1/n and in the steepness parameter. Thus, it is shown that even though these systems have a uniformly hyperbolic (albeit singular) limit, the ergodicity properties of this limit system are destroyed in the more realistic smooth setting. The considered example is highly symmetric and is not directly linked to the smooth many particle problem. Nonetheless, the possibility of explicitly constructing stable motion in smooth n-degrees of freedom systems limiting to strictly dispersing billiards is now established. Thus, it is shown that the billiards instability mechanism cannot justify by itself the observed ergodicity of smooth many particle systems.

      R. Sanchez  (Oak Ridge National Laboratory, USA) - Non-diffusive modeling of strange transport in magnetically-confined, turbulent plasmas

    The most promising route for the successful production of large amounts of energy from fusion requires the containment of a hot plasma by magnetic fields with the topology of nested, toroidal magnetic surfaces. Understanding the transport properties of these configurations is essential to the success of the fusion energy program, since they can help us to control the leak of particles and energy out of the system. However, multiple experiments carried out during the last two decades in this type of magnetic trap have revealed that the confined plasma is in a strong turbulent state in which transport phenomena exhibit a strange, non-diffusive nature. The observation of a degradation of confinement with increasing external power suggests that the plasma profiles tend to stay close to some critical threshold for the onset of instabilities. In addition, the observed scaling of the energy/particle confinement times and the detection of "avalanche-like" transport events suggest that the dominant transport mechanism may lack a characteristic length scale. Similar evidence also suggests that a characteristic temporal scale may also be missing.
In this focus talk, I will discuss some recent work that aims at improving our understanding of the dynamics of transport in these plasmas as much as at constructing mathematical models better suited to capturing them then the ones currently available. The basic tools used are continuous-time-random walks (CTRWs) and/or fractional differential equations (FDEs), both of which can accommodate this lack of characteristic scales. Numerous examples obtained from numerical simulations and experiments will be used to illustrate the main ideas.

      D. L. Shepelyansky  (Univ. Paul Sabatier, France) - Directed transport and chaos in asymmetric nanostructures

    A board of rigid disks on a triangular lattice has been invented by Galton in 1889 to demonstrate the appearance of statistical laws from dynamical motion. This system, also called periodic Lorentz gas, has been proved to be completely chaotic by Sinai in 60th. The dynamics remains chaotic also for semi-disks scatterers oriented in one direction. In this case the inversion symmetry is broken but the directed transport remains forbidden by the detailed balance principle. This remains true also in presence of polarized monochromatic force produced by microwave radiation. However, when dissipation is present, a new stationary state is born from chaos as shown in [1]. It is characterized by a directed transport which can be efficiently controlled by the microwave polarization even if mean force is zero. Being universal this effect exists for Maxwell or Fermi-Dirac thermostatted gas moving between semi-disks in presence of a microwave field [2-5]. Nowadays technology allows to realize the semi-disk Galton board with a two-dimensional electron gas in a superlattice of micron size antidots. In this case the theory predicts appearance of strong currents induced by microwave fields. This opens new possibilities for creation of room temperature detectors of terahertz radiation.

      T. H. Solomon  (Bucknell College, USA) - Experimental studies of advection-reaction-diffusion systems

    We present the results of experiments on the effects of chaotic fluid mixing on the dynamics or reacting systems. The flows studied include a blinking vortex flow and a chain of alternating vortices. The reaction is the oscillatory or excitable state of the well-known Belousov-Zhabotinsky chemical reaction. Three sets of experiments are described. (1) Experiments in a blinking vortex flow demonstrate that chemical patterns for the oscillatory reaction reflect the structures predicted by numerical models of the mixing fields for the flow. (2) Synchronization of oscillating reactions in an extended flow (a chain of vortices) is found to be enhanced significantly by the presence of superdiffusive transport characterized by Levy flights that connect different parts of the flow. (3) Front propagation is investigated in a chain of alternating and oscillating vortices that are coupled by chaotic mixing. It is found that the fronts mode-lock onto the frequency of the oscillation of the vortices. The effects of superdiffusion on the mode-locking is also investigated.

      V. Tarasov  (State University of Moscow, Russia) - Fractional Dynamics of Systems with Long-Range Space Interaction and Temporal Memory

    Field equations with time and coordinates derivatives of noninteger order are derived from stationary action principle for the cases of power-law memory function and long-range interaction in systems. The method is applied to obtain a fractional generalization of the Ginzburg-Landau equation and nonlinear Schrodinger equation, and dynamical equations for particles chain with power-law interaction and memory. In more details, we consider two different applications of the action principle: generalized Noether's theorem and Hamiltonian type equations. In the first case, we derive conservation laws in the form of continuity equations that consist of fractional time-space derivatives. Among applications of these results, we consider a chain of coupled oscillators with a power-wise memory function and power-wise interaction between oscillators. In the second case, we consider an example of fractional action and find the corresponding Hamiltonian type equations. The obtained fractional equations and conservation laws can be applied to complex media with/without random parameters or processes.

      J. L. Thiffeault  (Imperial College, UK) - A Topological Theory of Rod Stirring

    In a fluid, stirring is usually necessary to overcome the slow diffusion of most substances. This is important in a wide range of applications, from industry to geophysics. Here I focus on a prototypical application, the stirring of a two-dimensional viscous fluid with rods. Mathematically, stirring rods can be viewed as
'punctures' in a two-dimensional surface: they present topological obstructions to material lines in the fluid. The theory developed by Thurston and Nielsen to classify the possible periodic rod motions in such a system can be used to decide which stirring methods are best, in the sense of generating chaotic trajectoris that lead to chaotic advection. Global aspects of the concentration field of a substance in the stirring device can then be determined via 'train tracks', which are skeletons of an important underlying structure called the unstable foliation. Since there are only a limited number of possible train tracks, all stirring protocols can be classified, as well as their properties. This can be extended to completely general situations where rods are replaced by periodic orbits. I will present experimental and numerical examples of all these concepts. Finally, I will introduce a rod stirring device designed and optimized using these topological principles.

      A. Torcini  (CNR Florence, Italy) - Synchronization in spatially extended systems

    The mechanisms responsible for the synchronization of chaotic extended systems with short-range interactions are revised. Moreover recent results concerning power-law coupled systems are presented. In particular, chaotic synchronization of replica of coupled map lattices is considered. The spatial extension allows to interpret the synchronization transitions (STs) as nonequilibrium critical phenomena: namely, as absorbing phase transitions.
Within this framework two different kind of continuous transitions have been identified for nearest-neighbour coupling : one ruled by linear mechanisms and one by nonlinear effects. In the first case the ST belongs to the multiplicative noise universality class, while if nonlinear effects prevail the critical exponents coincide with those measured for directed percolation (DP). More recently spatially extended chaotic systems with algebraically decaying interactions have been considered.
Also in this situation the STs appear to be continuous, while the critical indexes vary with continuity with the power law exponent characterizing the interaction. Moreover, strong numerical evidences indicate that the transition belongs to the "anomalous directed percolation" family of universality classes previously found for Levy-flight spreading of epidemic processes.

      D. Treschev  (Steklov Mathematical Inst., Russie) Travelling waves in one-dimensional lattices

    We construct a travelling wave in a one-dimensional lattice of identical classical particles, with potential interaction. Some possibilities concerning lattices with several different kinds of particles are discussed.