Derivation of a Stochastic Neutron Transport Equation

TL;DR

This paper derives stochastic differential equations for neutron transport in three-dimensional media, capturing inherent randomness in scattering, absorption, and sources, and validates the approach with numerical comparisons.

Contribution

It introduces a novel derivation of stochastic neutron transport equations from first principles, including stochastic difference equations and SPDEs, for complex media.

Findings

Stochastic difference equations accurately model neutron density fluctuations.

Derived SPDEs generalize classical neutron transport equations with stochastic terms.

Numerical solutions align well with Monte Carlo simulations.

Abstract

Stochastic difference equations and a stochastic partial differential equation (SPDE) are simultaneously derived for the time-dependent neutron angular density in a general three-dimensional medium where the neutron angular density is a function of position, direction, energy, and time. Special cases of the equations are given such as transport in one-dimensional plane geometry with isotropic scattering and transport in a homogeneous medium. The stochastic equations are derived from basic principles, i.e., from the changes that occur in a small time interval. Stochastic difference equations of the neutron angular density are constructed, taking into account the inherent randomness in scatters, absorptions, and source neutrons. As the time interval decreases, the stochastic difference equations lead to a system of Ito stochastic differential equations (SDEs). As the energy, direction, and position intervals decrease, an SPDE is derived for the neutron angular density. Comparisons between numerical solutions of the stochastic difference equations and independently formulated Monte Carlo calculations support the accuracy of the derivations.