Stochastic Methods for Neutron Transport Equation III: Generational many-to-one and $k_\texttt{eff}$

TL;DR

This paper advances the probabilistic analysis of the neutron transport equation by rigorously interpreting the key eigenvalue keff, crucial for reactor physics, through Perron-Frobenius and operator methods, enhancing understanding of neutron flux growth and Monte Carlo simulations.

Contribution

It provides the first rigorous probabilistic interpretation of keff in the context of neutron transport, complementing existing engineering and simulation approaches.

Findings

Develops stochastic analysis linking keff to Perron-Frobenius theory.

Provides a rigorous probabilistic framework for neutron flux growth.

Bridges operator analysis with Monte Carlo simulation interpretations.

Abstract

The Neutron Transport Equation (NTE) describes the flux of neutrons over time through an inhomogeneous fissile medium. In the recent articles [5, 10], 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 [9, 4], 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 centre stage, which is otherwise characterised by another eigenvalue problem. In that setting, the eigenvalue, sometimes called k-effective (written $k_\texttt{eff}$), 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 [5, 10, 9, 4], 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 is 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).