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Section: Scientific Foundations

Keywords : plasma physics, beam physics, kinetic models, reduced models, Vlasov equation, modeling, mathematical analysis, asymptotic analysis, existence, uniqueness.

Kinetic models for plasma and beam physics

Plasmas and particle beams can be described by a hierarchy of models including N -body interaction, kinetic models and fluid models. Kinetic models in particular are posed in phase-space and involve specific difficulties. We perform a mathematical analysis of such models and try to find and justify approximate models using asymptotic analysis.

Models for plasma and beam physics

The plasma statecan be considered as the fourth state of matter, obtained for example by bringing a gas to a very high temperature ( 10 4K or more). The thermal energy of the molecules and atoms constituting the gas is then sufficient to start ionization when particles collide. A globally neutral gas of neutral and charged particles, called plasma, is then obtained. Intense charged particle beams, called non neutral plasmas by some authors, obey similar physical laws.

The hierarchy of models describing the evolution of charged particles within a plasma or a particle beam includes N -body models where each particle interacts directly with all the others, kinetic models based on a statistical description of the particles and fluid models valid when the particles are at a thermodynamical equilibrium.

In a so-called kinetic model , each particle species s in a plasma or a particle beam is described by a distribution function Im1 ${f_s{(\#119857 ,\#119855 ,t)}}$ corresponding to the statistical average of the particle distribution in phase-space corresponding to many realizations of the physical system under investigation. The product Im2 ${f_s~d\#119857 ~d\#119855 }$ is the average number of particles of the considered species, the position and velocity of which are located in a bin of volume Im3 ${d\#119857 ~d\#119855 }$ centered around Im4 ${(\#119857 ,\#119855 )}$ . The distribution function contains a lot more information than what can be obtained from a fluid description, as it also includes information about the velocity distribution of the particles.

A kinetic description is necessary in collective plasmas where the distribution function is very different from the Maxwell-Boltzmann (or Maxwellian) distribution which corresponds to the thermodynamical equilibrium, otherwise a fluid description is generally sufficient. In the limit when collective effects are dominant with respect to binary collisions, the corresponding kinetic equation is the Vlasov equation

Im5 ${\mfrac {\#8706 f_s}{\#8706 t}+\#119855 ·\mfrac {\#8706 f_s}{\#8706 \#119857 }+\mfrac qm{(\#119812 +\#119855 ×\#119809 )}·\mfrac {\#8706 f_s}{\#8706 \#119855 }=0,}$

which expresses that the distribution function fs is conserved along the particle trajectories which are determined by their motion in their mean electromagnetic field. The Vlasov equation which involves a self-consistent electromagnetic field needs to be coupled to the Maxwell equations in order to compute this field

Im6 $\mtable{...}$

which describes the evolution of the electromagnetic field generated by the charge density

Im7 ${\#961 {(\#119857 ,t)}=\munder \#8721 sq_s\#8747 f_s{(\#119857 ,\#119855 ,t)}~d\#119855 ,}$

and current density

Im8 ${\#119817 {(\#119857 ,t)}=\munder \#8721 sq_s\#8747 f_s{(\#119857 ,\#119855 ,t)}\#119855 ~d\#119855 ,}$

associated to the charged particles.

When binary particle-particle interactions are dominant with respect to the mean-field effects then the distribution function f obeys the Boltzmann equation

Im9 ${\mfrac {\#8706 f}{\#8706 t}+\#119855 ·\mfrac {\#8706 f}{\#8706 \#119857 }=Q{(f,f)},}$

where Q is the nonlinear Boltzmann collision operator. In some intermediate cases, a collision operator needs to be added to the Vlasov equation.

The numerical solution of the three-dimensional Vlasov-Maxwell system represents a considerable challenge due to the huge size of the problem. Indeed, the Vlasov-Maxwell system is nonlinear and posed in phase space. It thus depends on seven variables: three configuration space variables, three velocity space variables and time, for each species of particles. This feature makes it essential to use every possible option to find a reduced model wherever possible, in particular when there are geometrical symmetries or small terms which can be neglected.

Mathematical and asymptotic analysis of kinetic models

The mathematical analysis of the Vlasov equation is essential for a thorough understanding of the model as well for physical as for numerical purposes. It has attracted many researchers since the end of the 1970s. Among the most important results which have been obtained, we can cite the existence of strong and weak solutions of the Vlasov-Poisson system by Horst and Hunze [74] , see also Bardos and Degond [57] . The existence of a weak solution for the Vlasov-Maxwell system has been proved by Di Perna and Lions [63] . An overview of the theory is presented in a book by Glassey [71] .

Many questions concerning for example uniqueness or existence of strong solutions for the three-dimensional Vlasov-Maxwell system are still open. Moreover, their is a realm of approached models that need to be investigated. In particular, the Vlasov-Darwin model for which we could recently prove the existence of global solutions for small initial data [58] .

On the other hand, the asymptotic study of the Vlasov equation in different physical situations is important in order to find or justify reduced models. One situation of major importance in Tokamaks, used for magnetic fusion as well as in atmospheric plasmas, is the case of a large external magnetic field used for confining the particles. The magnetic field tends to incurve the particle trajectories which eventually, when the magnetic field is large, are confined along the magnetic field lines. Moreover, when an electric field is present, the particles drift in a direction perpendicular to the magnetic and to the electric field. The new time scale linked to the cyclotron frequency, which is the frequency of rotation around the magnetic field lines, comes in addition to the other time scales present in the system like the plasma frequencies of the different particle species. Thus, many different time scales as well as length scales linked in particular to the different Debye length are present in the system. Depending on the effects that need to be studied, asymptotic techniques allow to find reduced models. In this spirit, in the case of large magnetic fields, recent results have been obtained by Golse and Saint-Raymond [72] , [77] as well as by Brenier [61] . Our group has also contributed to this problem using homogenization techniques to justify the guiding center model and the finite Larmor radius model which are used by physicist in this setting [69] , [67] , [68] .

Another important asymptotic problem yielding reduced models for the Vlasov-Maxwell system is the fluid limit of collisionless plasmas. In some specific physical situations, the infinite system of velocity moments of the Vlasov equations can be closed after a few of those, thus yielding fluid models.


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