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TL;DR

This paper models the behavior of light rays undergoing Lambertian reflections in a narrow, nearly circular tube, showing that as the tube narrows, the light's angular position converges to a diffusion process with explicitly defined parameters.

Contribution

It provides a rigorous analysis of the limiting diffusion process for light rays in a thin, variable-width tube with Lambertian reflections, including explicit parameter formulas.

Findings

Angular component converges to a diffusion process as tube width approaches zero.

Explicit formulas for diffusion parameters in terms of tube width.

Demonstrates the limiting behavior of light in narrow geometries.

Abstract

We consider random reflections (according to the Lambertian distribution) of a light ray in a thin variable width (but almost circular) tube. As the width of the tube goes to zero, properly rescaled angular component of the light ray position converges in distribution to a diffusion whose parameters (diffusivity and drift) are given explicitly in terms of the tube width.