Multi-species neutron transport equation

Abstract

The Neutron Transport Equation (NTE) describes the flux of neutrons through inhomogeneous fissile medium. Whilst well treated in the nuclear physics literature (cf. [9, 27]), the NTE has had a somewhat scattered treatment in mathematical literature with a variety of different approaches (cf. [8, 25]). Within a probabilistic framework it has somewhat undeservingly received little attention in recent years; nonetheless, probabilistic treatments can be found see for example [19, 26, 24, 29, 4, 3]. In this article our aim is threefold. First we want to introduce a slightly more general setting for the NTE, which gives a more complete picture of the different species of particle and radioactive fluxes that are involved in fission. Second we consolidate the classical c0-semigroup approach to solving the NTE with the method of stochastic representation which involves expectation semigroups. Third we provide the leading asymptotic of our multi-species NTE, which will turn out to be crucial for further stochastic analysis of the NTE in forthcoming work [6, 5]. The methodology used in this paper harmonises the culture of expectation semigroup analysis from the theory of stochastic processes against c0-semigroup theory from functional analysis. In this respect, our presentation is thus part review of existing theory and part presentation of new research results based on generalisation of existing results.