Revisiting Discrepancies in Diffusion Theory: Preliminary Material

Table of Links

Abstract and 1 Introduction

2 Preliminary material

3 Comments elicited by the solution to Milne’s problem for B-particles

4 Flux to a spherical trap

5 Molecular motion in liquids

6 Concluding remarks and References

2 Preliminary material

2.1 Statement of the Milne problem and terminology

A homogeneous, semi-infinite, non-absorbing medium occupies the half-space 𝑥 > 0, and sustains a constant current of test particles in the negative 𝑥 direction. The medium (or the host) itself contains no sources or sinks, and the plane boundary at 𝑥 = 0 acts as a black wall that absorbs all particles incident on it. The test particles obey either the one-speed LorentzBoltzmann equation (LBE) of neutron transport or radiative transfer, in which case one is faced with the Milne problem, or the Klein-Kramers equation (KKE), which leads one to the Brownian analog of the Milne problem. The problem is to determine 𝑛(𝑥), the density of the particles inside the medium (𝑥 > 0).

As two versions of the KKE have been brought to bear on the Milne problem, the label Klein-Kramers equation needs some elaboration. When Fermi analyzed Milne’s problem, in the context of neutron diffusion [9, pp. 980–1016], he decided to simplify the analysis “by considering the fictitious case in which the neutrons move in one dimension along a line, instead of being free in space.” The choice, he explained, “has the the advantage that we can easily obtain approximate expressions valid for the actual three-dimensional case.” The equation used by B&T applies to true one-dimensional Brownian motion. The Trondheim group (myself and two colleagues) have investigated this version [10] as well as the complete analog of the Milne problem, in which particles move in three-dimensional space, but plane symmetry prevails and the cocentration depends on only one cartesian coordinate [11]. Menon, Kumar and Sahni [12] pointed out that, as regards particle density, the two versions stand on an equal footing, but their difference does manifest itself when other physical moments are computed.

The models described by the KKE and the LBE lie at opposite poles. The latter has been called “inverse Brownian motion” [14, 16], and Hoare [17] refined the terminology by referrring to the former regular Brownian motion. A diffusing particle will be called a “B-particle” or an “L-particle” according as it obeys the KKE or LBE. Neither of these models is regarded as too remote to resemble a real physical system; the KKE is widely believed to provide a serviceable description of the thermal wanderings of a large particle (such as a colloid suspended in a liquid), and the LBE has found numerous applications in the transport of photons through a turbid medium or of a neutron through a moderator.

2.2 Results for the particle density