Confusion concerning the extrapolated endpoint. When will it ever end?
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arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866913345191280640 |
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| author | Naqvi, K. Razi |
| author_facet | Naqvi, K. Razi |
| contents | 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 $γ$ 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 $γ$ 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2405_02448 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Confusion concerning the extrapolated endpoint. When will it ever end? Naqvi, K. Razi Statistical Mechanics 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 $γ$ 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 $γ$ 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. |
| title | Confusion concerning the extrapolated endpoint. When will it ever end? |
| topic | Statistical Mechanics |
| url | https://arxiv.org/abs/2405.02448 |