Asymptotic analysis of narrow escape problems in nonspherical three-dimensional domains
Bibliographic record
Abstract
Narrow escape problems consider the calculation of the mean first passage time (MFPT) for a particle undergoing Brownian motion in a domain with a boundary that is everywhere reflecting except for at finitely many small holes. Asymptotic methods for solving these problems involve finding approximations for the MFPT and average MFPT that increase in accuracy with decreasing hole sizes. While relatively much is known for the two-dimensional case, the results available for general three-dimensional domains are rather limited. This paper addresses the problem of finding the average MFPT for a class of three-dimensional domains bounded by the level surface of an orthogonal coordinate system. In particular, this class includes spheroids and other solids of revolution. The primary result presented is a two-term asymptotic expansion for the average MFPT of such domains containing an arbitrary number of holes. Steps are taken towards finding higher-order asymptotic expansions for both the average MFPT and the MFPT in these domains. The results for the average MFPT are compared to full numerical calculations performed with the comsol Multiphysics finite element solver for three distinct domains: prolate and oblate spheroids and biconcave disks. This comparison shows good agreement with the proposed two-term expansion of the average MFPT in the three domains.
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How this classification was reachedexpand
Full frame machine prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. The Gemma side is a direct model label for every work in the frame, read from the title-only record. The Codex side is a classifier learned from the 10,348 direct Codex labels and calibrated to design-weighted sample rates; fields without enough sample support carry no Codex call. Candidate is the union of the two sides; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels.
Distilled classifier scores by category (both heads)
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.001 | 0.006 |
| Meta-epidemiology (narrow) | 0.001 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.001 | 0.000 |
| Science and technology studies | 0.000 | 0.002 |
| Scholarly communication | 0.001 | 0.001 |
| Open science | 0.001 | 0.001 |
| Research integrity | 0.001 | 0.001 |
| Insufficient payload (model declined to judge) | 0.001 | 0.000 |
Machine scores (provisional)
The two teacher heads of the student model, read on this work. A score orders the frame for review; it never asserts a category, and the validation status ships verbatim with every row.
Baseline scores from an immature model (maturity gate not passed, 7 training rounds). Scores rank; they never assert a category.
score_only:v0-immature-baseline · verbatim from the scoring run: score_only means the number may rank works, and no category label ships from itClassification
machine, unvalidatedMachine predicted; a candidate call from one source (direct Gemma or distilled Codex), not a consensus.
How this classification was reached, model by model and score by score, is at the end of the page under "How this classification was reached".