On the effects of a totally reflecting barrier on an unbiased 1D random walk
Bibliographic record
Abstract
Abstract This paper is devoted to discussing the behavior of an unbiased 1D random walk in a semi‐infinite lattice confined by a certain type of boundary, termedtotally reflecting, and in particular to proving the validity of the so‐called “reflection principle” for computing the probability mass function in such systems. This is motivated by the publication of an earlier paper by [Orlowski, Phys. Status Solidi B239, 158–161 (2003)], which denied the validity of the reflection principle, and proposed a different, inconsistent method of computing the walker's probability mass, and also by the fact that earlier authors published arguments for the reflection principle that themselves contained errors or were not entirely rigorous. This paper provides a new, rigorous argument for the validity of the reflection principle, and also shows where the error in Orlowski's analysis lies. It further contextualizes these arguments with respect to some existing literature on similar systems where the probability of reflection may be less than unity, and discusses the proper relationship between the discrete random walk, the diffusion equation, and approches which employ a different, master equation‐based discretization technique. The spatial relationship between the discrete and continuous formulations is discussed, and an existing derivation [van Kampen and Oppenheim, J. Math. Phys.13, 842–849 (1972)] is extended. Finally, it is shown that the reflection principle, as outlined in this paper, preserves initially uniform concentrations for all time (in contrast to Orlowski's proposed method).
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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 | 0.001 | 0.002 |
| Scholarly communication | 0.002 | 0.002 |
| Open science | 0.001 | 0.002 |
| Research integrity | 0.002 | 0.001 |
| Insufficient payload (model declined to judge) | 0.005 | 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".