Microscopic Dynamics of Plasmas and Chaos

Elskens, Yves; Escande, Dominique

doi:10.1887/0750306122

Cited by 156 publications

(369 citation statements)

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“…The extension to wave-particle interactions [6] should not be too difficult. The mean-field description of the twodimensional point vortices model [51] has been rigorously obtained in a series of papers [13,52,53].…”

Section: Discussionmentioning

confidence: 99%

Large Deviation Techniques Applied to Systems with Long-Range Interactions

et al. 2005

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We discuss a method to solve models with long-range interactions in the microcanonical and canonical ensemble. The method closely follows the one introduced by R.S. Ellis, Physica D 133, 106 (1999), which uses large deviation techniques. We show how it can be adapted to obtain the solution of a large class of simple models, which can show ensemble inequivalence. The model Hamiltonian can have both discrete (Ising, Potts) and continuous (HMF, Free Electron Laser) state variables. This latter extension gives access to the comparison with dynamics and to the study of non-equilibrium effects. We treat both infinite range and slowly decreasing interactions and, in particular, we present the solution of the α-Ising model in one-dimension with 0 ≤ α < 1. Keywords: Long-range interactions, Large deviation techniques, Mean-field limit.PACS numbers: 05.20.-y Classical statistical mechanics A system with long-range interactions is characterized by an interparticle potential V (r) which decreases at large distances r slower than a power r −α with α < d, d being the dimension of the embedding space [1]. Classical examples are self-gravitating [2] and Coulomb [3] systems, vortices in two-dimensional fluid mechanics [4], waveparticles interaction [5,6] and trapped charged particles [7]. The behaviour of such systems is interesting both from the dynamical point of view, because they display peculiar quasi-stationary states that are related to the underlying Vlasov equations [8], and from the static point of view, because equilibrium statistical mechanics shows new types of phase transitions and cases of ensemble inequivalence [9]. In this paper we will restrict ourselves to the second aspect.In long-range interacting systems, essentially all the particles contribute to the local field: the fluctuations around the mean value are small because of the law of large numbers. This explains qualitatively why the mean-field scaling, which amounts to let the number of particles go to infinity at fixed volume [10,11], is usually extremely good. However, let us remind that for long-range interacting systems, microcanonical and canonical ensembles are not necessarily equivalent in the mean-field limit [12,13,14]. Moreover, because of the non additivity of the energy, the usual construction of the canonical ensemble cannot be applied. This is the reason why the microcanonical ensemble is considered by some authors [2,15] as the only "physically motivated" one. Hence, it is extremely important to develop rigorous techniques to solve non trivial physical models in the microcanonical ensemble. One finds in books the solution for the perfect gas, but generalizations to interacting particle systems are difficult.The goal of this paper is to advocate the use of large deviation techniques as a tool to explicitly derive microcanonical and canonical equilibrium solutions for a wide class of models. As a first step in this direction, we will discuss here a general solution method to treat in full detail mean-field models without short distance singu...

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Section: Discussionmentioning

confidence: 99%

Large Deviation Techniques Applied to Systems with Long-Range Interactions

et al. 2005

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“…The same closure for the long time behavior of the spectral cumulants can be derived without a priori assumptions on the statistic of the process, e.g., Newel et al (2001), Elskens & Escande (2003). should be made.…”

Section: Weak Turbulencementioning

confidence: 99%

MHD Dynamos and Turbulence

Tobias

Cattaneo

Boldyrev

2012

Ten Chapters in Turbulence

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“…This model was derived from the N -body description of the beam-plasma system [1]. More recently, this was done again in a heuristic way (see section 2.1 of [48]), and in a rigorous one by a series of controlled approximations (see the remaining of chapter 2 of [48]), which enables replacing the many particles of the bulk by their collective vibrations. So, in figure 3 the blue central part of the distribution is no longer present as particle degrees of freedom; if one is interested in the evolution of the red bump, one may incorporate the left green wing into the bulk too.…”

Section: Describing Plasma Dynamics With Finite Dimensional Hamimentioning

confidence: 99%

“…In reality, a typical initial perturbation excites also a wealth of beam modes. When their contribution is properly taken into account, Yves Elskens found (section 3.8.3 of reference [48]) that an initial perturbation with amplitude 1 evolves in time according to the time-reversible expression e γjLt + e −γjLt − e −γjL|t| : the beam modes act subtractively to compensate the damped eigenmode, and to restitute the Vlasovian solution. This apparent intricacy corresponds to experimental reality.…”

Section: A Recovering Vlasovian Linear Theory With a Mechanical Undementioning

confidence: 99%

How to Face the Complexity of Plasmas?

Escande

2013

Nonlinear Systems and Complexity

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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 sub-parts. 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 sub-part considers possible methodological improvements, some of them specific to plasma physics, 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 data bases, stimulating the scientific discussion of published results, diversifying the way results are made available, assessing more quality than quantity, 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.Non quia difficilia sunt, non audemus, sed quia non audemus, difficilia sunt. It is not because things are difficult that we do not dare, it is because we do not dare that things a...

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Microscopic Dynamics of Plasmas and Chaos

Cited by 156 publications

References 0 publications

Large Deviation Techniques Applied to Systems with Long-Range Interactions

Large Deviation Techniques Applied to Systems with Long-Range Interactions

MHD Dynamos and Turbulence

How to Face the Complexity of Plasmas?

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