Author:
(1) K. Razi Naqvi, Department of Physics, Norwegian University of Science and Technology (NTNU), 7094 Trondheim, Norway
Table of Links
3 Comments elicited by the solution to Milne’s problem for B-particles
6 Concluding remarks and References
6 Concluding remarks
The claim—motivated by the search for a better boundary condition—that the Lorentz model is a useful tool for investigating bimolecular reactions in solutions became credible only after the publication of Burschka and Titulaer’s numerical solution of the one-dimensional KKE [4]. Immediately prior to that, one critic expressed the “community opinion” by stating in a referee report (on an article co-authored by me) that the LBE “is absolutely useless in dealing with transport in liquids”, and insisted that this task is best handled by solving the KKE. Only then did the need arise for comparing the length scales of inverse and regular Brownian motion. The Trondheim group has shown that inverse Brownian motion, regular Brownian motion and the BGK-model are indistinguishable at the DE-level, provided that one uses the appropriate BC [31], namely that stated in Eq. (20). A Procrustean random walk model that allows no distribution of path lengths seems (to me) unphysical, much like the lattice model that informed Smoluchowksi’s thinking (about the boundary condition at an absorbing surface), and has misinformed generations of students as well as aficionados of chemical kinetics.
Infinitely heavy B-particles, infinitely light L-particles, infinitely inflexible (about the constancy of their pathlenghts) R-particles are all fictions, but some fictions are more fruitful than others, and some are outright useless. Whether the fiction of R-particles will bear fruit (in the setting of diffusion-mediated reactions) or serve as a mere distraction remains to be seen.
A Milne’s problem: calculating the density profiles
The purpose of this appendix is to enable a reader of this article to generate the data used for plotting the density profiles shown in Fig. 1.
The density data for L-particles were generated with the aid of a variational calculation [32], in which ℓ was used a the unit of length. The corresponding data for B-particles were computed by improving the results obtained by the Trondheim group through a half-range treatment [10], in which the 𝑁th order approximation for the particle density 𝑛 was expressed in the form
and values of 𝑥0 (1.459877Λ), 𝑥𝑖 and 𝜆𝑖 (for 𝑖 =1–8) resulting from a ninth-order approximation (𝑁 = 8) were reported, and the values of 𝑛(𝑥) close to the wall were compared with those found by Marshall and Watson (M&W) on the basis of their exact analytical treatment [33]. The improvements consists of three minor changes: the value of 𝑥0 has been replaced by 𝑥0 = 1.460354Λ (the first seven figures of the exact result), one more term has been added, and the values of 𝑥𝑖 and 𝜆𝑖 for the last three terms (𝑖 =7–9) have been optimised in a least-squares fit to the numbers in column (A) of Table 1 of M&W. The complete set of {𝑥𝑖 , 𝜆𝑖} values (of mostly-analytical-partly-empirical origin) is displayed in Table 1, the upper part of which is identical with Table II of ref. [10]. For plotting the density of B-particles in Fig. 1, the length scale was changed from Λ to ℓ.
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