Mathematical modeling and numerical computation of narrow escape problems
Bibliographic record
Abstract
The narrow escape problem refers to the problem of calculating the mean first passage time (MFPT) needed for an average Brownian particle to leave a domain with an insulating boundary containing N small well-separated absorbing windows, or traps. This mean first passage time satisfies the Poisson partial differential equation subject to a mixed Dirichlet-Neumann boundary condition on the domain boundary, with the Dirichlet condition corresponding to absorbing traps. In the limit of small total trap size, a common asymptotic theory is presented to calculate the MFPT in two-dimensional domains and in the unit sphere. The asymptotic MFPT formulas depend on mutual trap locations, allowing for global optimization of trap locations. Although the asymptotic theory for the MFPT was developed in the limit of asymptotically small trap radii, and under the assumption that the traps are well-separated, a comprehensive study involving comparison with full numerical simulations shows that the full numerical and asymptotic results for the MFPT are within 1% accuracy even when total trap size is only moderately small, and for traps that may be rather close together. This close agreement between asymptotic and numerical results at finite, and not necessarily asymptotically small, values of the trap size clearly illustrates one of the key side benefits of a theory based on a systematic asymptotic analysis. In addition, for the unit sphere, numerical results are given for the optimal configuration of a collection of traps on the surface of a sphere that minimizes the average MFPT. The case of N identical traps and a pattern of traps with two different sizes are considered. The effect of trap fragmentation on the average MFPT is also discussed.
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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 | 0.004 |
| Meta-epidemiology (narrow) | 0.001 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.001 | 0.001 |
| Science and technology studies | 0.001 | 0.001 |
| Scholarly communication | 0.002 | 0.001 |
| Open science | 0.002 | 0.002 |
| Research integrity | 0.002 | 0.001 |
| Insufficient payload (model declined to judge) | 0.003 | 0.001 |
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".