Concentration phenomena for neutronic multigroup diffusion in random environments

TL;DR

This paper investigates the asymptotic behavior of the principal eigenvalue in a multigroup neutron diffusion model within random media, revealing concentration phenomena and connecting eigenvalues to homogenized Hamilton-Jacobi equations.

Contribution

It extends the analysis of multigroup diffusion systems from periodic to random environments, providing new homogenization results and insights into eigenfunction concentration.

Findings

Asymptotic characterization of the principal eigenvalue in random media

Connection between eigenvalue problems and viscous Hamilton-Jacobi equations

Identification of concentration phenomena of eigenfunctions

Abstract

We study the asymptotic behavior of the principal eigenvalue of a weakly coupled, cooperative linear elliptic system in a stationary ergodic heterogeneous medium. The system arises as the so-called multigroup diffusion model for neutron flux in nuclear reactor cores, the principal eigenvalue determining the criticality of the reactor in a stationary state. Such systems have been well-studied in recent years in the periodic setting, and the purpose of this work is to obtain results in random media. Our approach connects the linear eigenvalue problem to a system of quasilinear viscous Hamilton-Jacobi equations. By homogenizing the latter, we characterize the asymptotic behavior of the eigenvalue of the linear problem and exhibit some concentration behavior of the eigenfunctions.