Stochastic Methods for Neutron Transport Equation III: Generational many-to-one and k_eff

The neutron transport equation (NTE) describes the flux of neutrons over time through an inhomogeneous fissile medium. In the recent articles, [A. M. G. Cox et al., J. Stat. Phys., 176 (2019), pp. 425-455; E. Horton, A. E. Kyprianou, and D. Villemonais, Ann. Appl. Probab., 30 (2020), pp. 2573-2612] a probabilistic solution of the NTE is considered in order to demonstrate a Perron-Frobenius type growth of the solution via its projection onto an associated leading eigenfunction. In [S. C. Harris, E. Horton, and A. E. Kyprianou, Ann. Appl. Probab., 30 (2020), pp. 2815-2845; A. M. G. Cox et al., Monte Carlo Methods for the Neutron Transport Equation, https://arxiv.org/abs/2012.02864 (2020)], further analysis is performed to understand the implications of this growth both in the stochastic sense as well as from the perspective of Monte Carlo simulation. Such Monte Carlo simulations are prevalent in industrial applications, in particular where regulatory checks are needed in the process of reactor core design. In that setting, however, it turns out that a different notion of growth takes center stage, which is otherwise characterized by another eigenvalue problem. In that setting, the eigenvalue, sometimes called k-effective (written keff), has the physical interpretation as being the ratio of neutrons produced (during fission events) to the number lost (due to absorption in the reactor or leakage at the boundary) per typical fission event. In this article, we aim to supplement [J. Stat. Phys., 176 (2019), pp. 425-455; Ann. Appl. Probab., 30 (2020), pp. 2573-2612; Ann. Appl. Probab., 30 (2020), pp. 2815-2845; Monte Carlo Methods for the Neutron Transport Equation, https://arxiv.org/abs/2012.02864 (2020)] by developing the stochastic analysis of the NTE further to the setting where a rigorous probabilistic interpretation of keff is given, both in terms of a Perron-Frobenius type analysis as well as via classical operator analysis. To our knowledge, despite the fact that an extensive engineering literature and industrial Monte Carlo software are concentrated around the estimation of keff and its associated eigenfunction, we believe that our work is the first rigorous treatment in the probabilistic sense (which underpins some of the aforesaid Monte Carlo simulations).