Confusion concerning the extrapolated endpoint. When will it ever end?

TL;DR

This paper investigates discrepancies in extrapolation length values across different physical models, highlighting the influence of length scales and jump length variance on these measurements.

Contribution

It clarifies the reasons behind conflicting extrapolation length results in radiative transfer and particle absorption models, emphasizing the roles of length scale disparity and jump length variance.

Findings

Disparity in length scales causes differences in extrapolation length values.

Zero variance in jump lengths affects the extrapolation length.

The independence of b3 from R for certain l values is inconsistent with classical equations.

Abstract

In a paper on ``the Brownian motion analog of the well-known Milne problem in radiative transfer theory'' [\textit{J Stat Phys} 25 (1981) 569--82], Burschka and Titulaer reported: ``The value we find for this `Milne extrapolation length' is, in the appropriate dimensionless units, approximately twice the value found in the radiative transfer problem.'' A study by Ziff [\textit{J Stat Phys} 65 (1991) 1217--33], concerned with the absorption of particles executing a Rayleigh flight (randomly directed displacements of equal length $l$) by a black sphere of radius $R$, led to a value for the extrapolation length $\gamma$ about half as small as the benchmark result (for $R\gg l)$. The first discrepancy is shown to result from the disparity of the two length scales; the second, from the zero variance of the jump lengths. Ziff's finding that $\gamma$ is independent of $R$ for $0<l\leq 2R$, cannot be reconciled with studies based on the Lorentz-Boltzmann equation and the Klein-Kramers equation.