On the Boltzmann equation for charged particle beams under the effect of strong magnetic fields
type: Item
instance of: scholarly article
title: On the Boltzmann equation for charged particle beams under the effect of strong magnetic fields
zbMATH Open document ID: 1315.35138
author: Mihai Bostan
published in: Discrete and Continuous Dynamical Systems. Series B
publication date: 2015-01-20T00:00:00Z
review text: The next problem is considered \[ \partial_t \mathcal{F} + V\cdot\nabla_X\mathcal{F}+\frac{q}{m}(V\wedge B^{ext})\cdot\nabla_V \mathcal{F}=Q(\mathcal{F},\mathcal{F}),\,\,\, (t,X,V)\in \mathbb{R}_+\times\mathbb{R}^3\times\mathbb{R}^3, \] \[ \mathcal{F}(0,X,V)=\mathcal{F}^{in}(X,V), \,\,\, (X,V)\in \mathbb{R}^3\times\mathbb{R}^3. \] This model was designed for beams which possess an optical axis, assuming that the particles remain close to the optical axis, having about the same kinetic energy. This paper is devoted to the study of the confinement properties of charged particle beams, under the action of strong magnetic fields parallel to the optical axis. The self-consistent electro-magnetic field is neglected but it is taken into account the collisions between particles. The function \(\mathcal{F}= \mathcal{F}(t,X, V)\geqslant 0\) is the presence density of the charged particles in the position-velocity phase space \((X, V).\) The collisions between the particles are taken into account through the Boltzmann kernel: \[ \begin{multlined} Q(\mathcal{F},\mathcal{F})=\int_{S^2}\int_\mathbb{R}^3\sigma(V-V',\omega)\{\mathcal{F}(V-\omega\otimes\omega(V-V'))\mathcal{F}(V'+\omega\otimes\omega(V-V'))\\ -\mathcal{F}(V)\mathcal{F}(V')\}d V'd\omega.\end{multlined} \] The density is searched in the form \[ \mathcal{F}^\varepsilon=\frac{1}{\varepsilon^3}f^\varepsilon(t,\frac{X_1}{\varepsilon^2}, \frac{X_2}{\varepsilon^2},X_3,\frac{V_1}{\varepsilon},\frac{V_2}{\varepsilon},\frac{V_3-u_3}{\varepsilon}). \] by using a formal expansion \(f^\varepsilon=f+\varepsilon f^1+\varepsilon\varepsilon^2 f^2+\dots\) The main goal of this paper is to derive the expression of the averaged version of the Boltzmann collision operator. It is allows to express the equilibria of the averaged Boltzmann collision kernel in terms of six moments.
Mathematics Subject Classification ID: 35Q20
Mathematics Subject Classification ID: 76P05
Mathematics Subject Classification ID: 82D10
Mathematics Subject Classification ID: 78A35
Mathematics Subject Classification ID: 35Q75
Mathematics Subject Classification ID: 82C40
zbMATH DE Number: 6388088
zbMATH Keywords: Boltzmann equation
zbMATH Keywords: averaged operator
zbMATH Keywords: Burgers equation
zbMATH Keywords: Larmor radius
zbMATH Keywords: Boltzmann collision kernel
zbMATH Keywords: H-theorem
describes a project that uses: Vador
MaRDI profile type: Publication
full work available at URL: dcdsb.2015.20.339
OpenAlex ID: W2332781888
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DOI: 10.3934/DCDSB.2015.20.339
description: scientific article; zbMATH DE number 6388088