The model with many moments for relativistic electron beams: A simplified solution

Abstract

In the 1980s, Amendt and Weitzner proposed an interesting model capable to describe relativistic electron beams. It concerned 14 independent variables and the closure was obtained by using the entropy and the Einstein relativity principles. As we know from literature, an extension to many moments allows to achieve an improvement in the results. Three years ago, we exhibited a macroscopic model with an arbitrary but fixed number of moments for relativistic extended thermodynamics. Such model was more general than those previously appeared in literature, so it was applicable even to materials different from an electron beam. Subsequently, we found the closure of such model consistent with the entropy and the Einstein relativity principles, up to whatever order with respect to equilibrium. The solution was determined in terms of a family of arbitrary single variable functions arising from integration. Those results have a very complex shape and are very difficult to handle so a simplification is necessary. In this paper we will reach this goal. Furthermore, we will prove that by fixing a certain order np with respect to equilibrium and a scalar valued single variable function, appearing at that order, then all the terms appearing at orders n ⩽ np are determined without introducing any other function. This result has already been found for the nonrelativistic case but its extension to the relativistic framework is not straightforward and it requires a supplementary mathematical tool: the above mentioned simplification in the shape of the solutions.