From thermonuclear fusion to Hamiltonian chaos

Escande, Dominique

doi:10.1140/epjh/e2016-70063-5

Cited by 10 publications

(14 citation statements)

References 73 publications

(96 reference statements)

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“…With this change of variables, the new Hamiltonian H(P, ν) does not depend on Q. If the value of ν is fixed, the action variable P (p, q, ν) is a constant of motion under the dynamics of the Hamiltonian H. In the model defined by equations (5)(6), the parameter ν follows a stochastic differential equation. This implies that the action variable P also follows a stochastic differential equation, and evolves on the same timescale as ν(t).…”

Section: Formulation Of the Model And Theoretical Resultsmentioning

confidence: 99%

“…On the other hand, we assume that chaos is weak enough in the regular regions 4 and 5 of figure (4) to keep the system to an average value very close to p within the time τ av . The local diffusion model of second order is represented in figure (6). It consists of three patches of infinite diffusion coefficient D in p-space.…”

Section: Averaging Of the Dynamics: The Local Diffusion Modelmentioning

confidence: 99%

“…The local diffusion model of second order is represented in figure (6). It consists of three patches of infinite diffusion coefficient D in p-space.…”

Section: Averaging Of the Dynamics: The Local Diffusion Modelmentioning

confidence: 99%

“…Transport in Hamiltonian systems and is a classical field of dynamical system theory [1,2,3,4], with a huge number of applications [5,6]. Beyond deterministic dynamical systems, a lot of work has been devoted in the past to study the effect of random perturbations [7], more specifically on Hamiltonian systems and on area preserving maps, for instance in the context of plasma physics [8,9].…”

Section: Introductionmentioning

confidence: 99%

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Transport in Hamiltonian systems with slowly changing phase space structure

Bouchet

Woillez

2020

Communications in Nonlinear Science and Numerical Simulation

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Transport in Hamiltonian systems with weak chaotic perturbations has been much studied in the past. In this paper, we introduce a new class of problems: transport in Hamiltonian systems with slowly changing phase space structure that are not order one perturbations of a given Hamiltonian. This class of problems is very important for many applications, for instance in celestial mechanics. As an example, we study a class of onedimensional Hamiltonians that depend explicitly on time and on stochastic external parameters. The variations of the external parameters are responsible for a distortion of the phase space structures: chaotic, weakly chaotic and regular sets change with time. We show that theoretical predictions of transport rates can be made in the limit where the variations of the stochastic parameters are very slow compared to the Hamiltonian dynamics. Exact asymptotic results can be obtained in the classical case where the Hamiltonian dynamics is integrable for fixed values of the parameters. For the more interesting chaotic Hamiltonian dynamics case, we show that two mechanisms contribute to the transport. For some range of the parameter variations, one mechanism -called "transport by migration with the mixing regions" -is dominant. We are then able to model transport in phase space by a Markov model, the local diffusion model, and to give reasonably good transport estimates.

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Section: Formulation Of the Model And Theoretical Resultsmentioning

confidence: 99%

Section: Averaging Of the Dynamics: The Local Diffusion Modelmentioning

confidence: 99%

“…The local diffusion model of second order is represented in figure (6). It consists of three patches of infinite diffusion coefficient D in p-space.…”

Section: Averaging Of the Dynamics: The Local Diffusion Modelmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Transport in Hamiltonian systems with slowly changing phase space structure

Bouchet

Woillez

2020

Communications in Nonlinear Science and Numerical Simulation

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“…This task requires a new approach, which will be the matter of a future work. It is worthwhile to mention that in the nonlinear regime, the saturation of the weak warm beam-plasma instability generates in phase space a type of turbulence, which has a very close connection to Hamiltonian chaos theory (see, e.g., [8,35] and references therein). A complete treatment of the self-consistent deterministic case remains an open issue, the proof of which must be based at least (but not only) on the same ingredients than those used for proving the Landau damping [48], and more particularly on the control of nonlinear wave-wave interactions (e.g.…”

Section: Introductionmentioning

confidence: 99%

Diffusion limit of the Vlasov equation in the weak turbulent regime

Bardos,

Besse

2021

Preprint

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In this paper we study the Hamiltonian dynamics of charged particles subject to a non-selfconsistent stochastic electric field, when the plasma is in the so-called weak turbulent regime. We show that the asymptotic limit of the Vlasov equation is a diffusion equation in the velocity space, but homogeneous in the physical space. We obtain a diffusion matrix, quadratic with respect to the electric field, which can be related to the diffusion matrix of the resonance broadening theory and of the quasilinear theory, depending on whether the typical autocorrelation time of particles is finite or not. In the self-consistent deterministic case, we show that the asymptotic distribution function is homogenized in the space variables, while the electric field converges weakly to zero. We also show that the lack of compactness in time for the electric field is necessary to obtain a genuine diffusion limit. By contrast, the time compactness property leads to a "cheap" version of the Landau damping: the electric field converges strongly to zero, implying the vanishing of the diffusion matrix, while the distribution function relaxes, in a weak topology, towards a spatially homogeneous stationary solution of the Vlasov-Poisson system.

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Basic microscopic plasma physics from N-body mechanics

Escande

Bénisti

Elskens

et al. 2018

Rev. Mod. Plasma Phys.

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Computing is not understanding. This is exemplified by the multiple and discordant interpretations of Landau damping still present after seventy years. For long deemed impossible, the mechanical N -body description of this damping, not only enables its rigorous and simple calculation, but makes unequivocal and intuitive its interpretation as the synchronization of almost resonant passing particles. This synchronization justifies mechanically why a single formula applies to both Landau growth and damping. As to the electrostatic potential, the phase mixing of many beam modes produces Landau damping, but it is unexpectedly essential for Landau growth too. Moreover, collisions play an essential role in collisionless plasmas. In particular, Debye shielding results from a cooperative dynamical self-organization process, where "collisional" deflections due to a given electron diminish the apparent number of charges about it. The finite value of exponentiation rates due to collisions is crucial for the equivalent of the van Kampen phase mixing to occur in the N -body system. The N -body approach incorporates spontaneous emission naturally, whose compound effect with Landau damping drives a thermalization of Langmuir waves. O'Neil's damping with trapping typical of initially large enough Langmuir waves results from a phase transition. As to Coulomb scattering, there is a smooth connection between impact parameters where the two-body Rutherford picture is correct, and those where a collective description is mandatory. The N -body approach reveals two important features of the Vlasovian limit: it is singular and it corresponds to a renormalized description of the actual N -body dynamics.This review deals with the microscopic physics of plasmas, mainly collisionless ones. Its main purpose is to improve the foundations of this physics by laying out, in a pedagogic and elementary manner, a systematic exposition of its approaches using N -body classical mechanics. These approaches enable in particular: (i) a short, yet explicit and rigorous, derivation of Landau damping and growth, (ii) an associated intuitive, yet rigorous, interpretation of these phenomena, (iii) another derivation showing Landau damping to result from phase mixing, as originally proved by van Kampen in a Vlasovian setting, (iv) a third derivation recovering the usual Vlasovian dielectric function and unifying Landau damping and Debye shielding (or screening), (v) unveiling how the microscopic mechanism of Debye shielding is intimately connected to collisions, (vi) a unification of the derivations of spontaneous emission and of Landau damping, (vii) proving the depletion of nonlinearity when there is a plateau in the tail distribution function, (viii) proving that damping with trapping results from a phase transition, (ix) a calculation of Coulomb scattering describing for the first time correctly all impact parameters with no ad hoc cut-off. Item (ii) is important, since the lack of mechanical interpretation prevented the plasma community from accepting the...

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From thermonuclear fusion to Hamiltonian chaos

Cited by 10 publications

References 73 publications

Transport in Hamiltonian systems with slowly changing phase space structure

Transport in Hamiltonian systems with slowly changing phase space structure

Diffusion limit of the Vlasov equation in the weak turbulent regime

Basic microscopic plasma physics from N-body mechanics

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