Abstract
The subject matter of this paper concerns the paraxial approximation for the transport of charged particles. We focus on the magnetic confinement properties of charged particle beams. The collisions between particles are taken into account through the Boltzmann kernel. We derive the magnetic high field limit and we emphasize the main properties of the averaged Boltzmann collision kernel, together with its equilibria.
Highlights
The charged particle beams play a major role in many applications : particle physics experiments, particle therapy, astrophysics, etc
Observe that the Larmor radius scales like ε2 since both the typical perpendicular velocity and the cyclotronic period are of orders ε. This explains our choice for the space unit in the perpendicular directions in (3) : we focus on the finite Larmor radius regime i.e., the space unit in the perpendicular directions and the Larmor radius are of the same order [14], [16]
Up to a factor depending on x3, the equilibrium f writes exp − |v|2 + (v3 − w3(x3))2 exp − |ωcx + ⊥v − w(x3)|2 − |v|2
Summary
The charged particle beams play a major role in many applications : particle physics experiments, particle therapy, astrophysics, etc. Most of them have been obtained by linearization around Maxwellians, expecting that the Maxwellians belong to the equilibria of the averaged collision kernels. It happens that this fails to be true, at least in the finite Larmor radius regime. The equilibria of the averaged kernel follow thanks to a H type theorem, see Section 4. Fluid models around these equilibria are investigated as well.
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