{
  "abstract": "We study the mean first-reaction time (MFRT) of a Brownian particle in a bounded three-dimensional domain $Ω$ with a reflecting boundary $\\partialΩ$, that contains a thin reactive filament. The filament is a tube of cross-sectional diameter $\\varepsilon\\ll 1$ around a smooth closed curve $Γ$, and its surface is partially reactive with reactivity $b$. As $\\varepsilon\\to 0$ the filament reduces to a line source in the outer problem. Its cross-section enters only through a logarithmic reactive capacitance $d_c(b)$, so that the MFRT is $\\mathcal{O}(-\\log[\\varepsilon d_c])$. This contrasts with the $\\mathcal{O}(\\eps^{-1})$ scaling for spatially localized traps of radius $\\eps$. Using matched asymptotic expansions, we derive a Fredholm integral equation on $Γ$ for the line density of the reactive flux. It sums all powers of $ν\\equiv -1/\\log[\\varepsilon d_c]$ and involves two geometric quantities: the regular part $R$ of the Neumann Green's function, which encodes the interaction of $Γ$ with $\\partialΩ$, and an intrinsic self-energy of $Γ$ that depends only on its shape. The framework extends to multiple filaments, splitting probabilities, a spatially heterogeneous diffusivity, and the principal eigenvalue of the Laplacian. For the unit sphere we solve the integral equation with a simple trapezoidal Nyström scheme, and we validate the predictions for rings, linked rings and torus knots against full 3-D finite-element solutions.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.38715",
  "author_total": 4,
  "authors": [
    {
      "name": "Denis S. Grebenkov"
    },
    {
      "name": "Theodore Kolokolnikov"
    },
    {
      "name": "Alan E. Lindsay"
    },
    {
      "name": "Michael J. Ward"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/9z1hbf0a2chfz.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.38715",
  "fulltext_url": null,
  "id": "9z1hbf0a2chfz",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.38715",
  "language": null,
  "licence": "LicenseRef-arXiv-nonexclusive-1.0",
  "licence_label": "arXiv licence",
  "licence_source": "article",
  "licence_url": "http://arxiv.org/licenses/nonexclusive-distrib/1.0/",
  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "math.AP"
    ],
    [
      "arxiv",
      "physics.bio-ph"
    ],
    [
      "arxiv",
      "q-bio.QM"
    ]
  ],
  "openalex": null,
  "pdf_url": "https://arxiv.org/pdf/2610.38715",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-28",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/9z1hbf0a2chfz.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:36:23Z",
      "native_id": "2610.38715",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.38715"
    }
  ],
  "subjects": [
    {
      "field": "biology",
      "origin": "arxiv:q-bio.QM",
      "subfield": "bioinformatics"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.AP",
      "subfield": "analysis-pdes"
    },
    {
      "field": "physics",
      "origin": "arxiv:physics.bio-ph",
      "subfield": "biological-medical"
    }
  ],
  "title": "The Mean First-Reaction Time on a Thin Filament in a 3-D Domain",
  "title_text": "The Mean First-Reaction Time on a Thin Filament in a 3-D Domain",
  "type": "preprint",
  "updated_at": "2026-11-04T15:20:53Z",
  "url": "https://research.draconitic.com/paper/9z1hbf0a2chfz",
  "venue": "arXiv",
  "versions": []
}