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, the NTE has had a somewhat scattered treatment in mathematical literature with a variety of different approaches. Within a probabilistic framework it has somewhat undeservingly received little attention in recent years; nonetheless, a few probabilistic treatments can be found. 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 c-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 (Cox et al. 2019; Harris et al. 2018; Horton et al. 2018). The methodology used in this paper harmonises the culture of expectation semigroup analysis from the theory of stochastic processes against c-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.