A Generalized Kinetic Approach for the Study of Relativistic Electron Beams
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A Generalized Kinetic Approach for the Study of Relativistic Electron Beams M.C. Carrisi
Received: 7 December 2011 / Accepted: 17 February 2012 / Published online: 23 May 2012 © Springer Science+Business Media B.V. 2012
Abstract Recently a macroscopic extended model with many moments for the description of mono atomic gases has been proposed and an exact closure, up to whatever order with respect to thermodynamical equilibrium has been determined. Afterwards a further condition arising from the imposition of the entropy principle to extended models of fluids under the influence of an external electro-magnetic field has been considered and a closure compatible with this condition has been found, by using a macroscopic approach. Such condition was never imposed before in literature, in the context of relativistic Extended Thermodynamics with many moments. In this paper an alternative closure for this last model, compatible with the further condition, is proposed. It is obtained by using a generalized kinetic approach. If one fixes a certain order with respect to equilibrium the closure is determined in terms of a polynomial and two functions of one scalar. Keywords Extended thermodynamics · Fluid models · Relativistic gases · Entropy principle
1 Introduction Relativistic charged particle beams are of great interest in many areas of physics and astrophysics as in the study of solar flares [1] or pulsar magnetospheres [2], or in technology, as in the study of laser devices (see [3] or [4], for example). Relativistic electron beams have been recently detected also in thunderclouds (see [5] and [6], for example). Particle beams are usually described by the Vlasov or Vlasov-Boltzmann equation. It gives a microscopic description of the beam by using a kinetic approach. If one needs or wants to give a macroscopic description of the beam it is necessary to move to a fluid model. It could be done, for example, by using a moment method as in paper [7], where Amendt and Weitzner proposed a moment model with 14 independent variables based on the Vlasov M.C. Carrisi () Dipartimento di Matematica ed Informatica, Università di Cagliari, Via Ospedale 72, 09124 Cagliari, Italy e-mail: [email protected]
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M.C. Carrisi
equation and they found an approximate closure for such model. Afterwards, the use of the methods of extended thermodynamics (see [8] for the relativistic case) allowed to delve deeper into the closure problem of that model and to find in [9] an exact closure up to the second order with respect to equilibrium, and in [10] an exact closure up to whatever order with respect to equilibrium. In [11] and [12], as suggested by Extended Thermodynamics, the model for relativistic electron beams has been extended by considering an arbitrary but fixed number of moments. The result is a many moments model appropriate for mono atomic gases of which an exact closure, up to whatever order with respect to thermodynamical equilibrium, has been found. Such model is capable to describe fluids that are not under the influ
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