Fractional Diffusion Limits of Non-Classical Transport Equations

TL;DR

This paper investigates how non-classical transport equations behave under certain scalings, revealing that their diffusion limits can be either regular or fractional, depending on the tail behavior of the path-length distribution, using Fourier and moment methods.

Contribution

It establishes the asymptotic diffusion limits of non-classical transport equations, including fractional diffusion limits, based on the tail behavior of the path-length distribution.

Findings

Diffusion limits depend on tail behavior of path-length distribution.

Both regular and fractional diffusion equations are derived as limits.

Analysis employs Fourier transform and moment methods.

Abstract

We establish asymptotic diffusion limits of the non-classical transport equation derived in [E. W. Larsen, A generalized Boltzmann equation for non-classical particle transport, Joint international topical meeting on mathematics & computation and supercomputing in nuclear applications, 2007]. By introducing appropriate scaling parameters, the limits will be either regular or fractional diffusion equations depending on the tail behaviour of the path-length distribution. Our analysis uses the Fourier transform combined with a moment method. We conclude with remarks on the diffusion limit of the periodic Lorentz gas equation.