{
  "abstract": "Metadynamics is a class of enhanced-sampling methods that is widely used in molecular modeling. Here, we consider metadynamics with a one-dimensional collective variable and explore the limitations of using Gaussian hills in light of existing convergence results. We reduce the corresponding evolution problem to a replicator-type differential equation and analyze its long-time behavior. For a periodic collective variable, we prove the convergence of metadynamics. However, the situation is fundamentally different when the collective variable is defined on a finite interval. In this case, the stationary solution only exists in the weak sense as a finite atomic measure. The solution to the differential equation weakly converges to it. Consequently, metadynamics is ineffective due to the absence of a clear quasi-stationary state. A quasi-stationary, transient solution can only be captured in the \"INTERVAL\" framework if the free energy outside the finite interval remains nearly constant across sufficiently large regions compared to $σ$.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2609.01934",
  "author_total": 2,
  "authors": [
    {
      "name": "Andrey Badanin"
    },
    {
      "name": "Olga Rogacheva"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/4gr79kxzqv981.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2609.01934",
  "fulltext_url": null,
  "id": "4gr79kxzqv981",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2609.01934",
  "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-ph"
    ],
    [
      "arxiv",
      "math.MP"
    ],
    [
      "openalex",
      "1606|Advanced Physical and Chemical Molecular Interactions"
    ]
  ],
  "openalex": "W7207530769",
  "pdf_url": "https://arxiv.org/pdf/2609.01934",
  "pmcid": null,
  "pmid": null,
  "published": "2026-09-01",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/4gr79kxzqv981.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:18:16Z",
      "native_id": "2609.01934",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2609.01934"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math-ph",
      "subfield": "mathematical-physics"
    }
  ],
  "title": "On the Convergence of Metadynamics with Gaussian Hills",
  "title_text": "On the Convergence of Metadynamics with Gaussian Hills",
  "type": "preprint",
  "updated_at": "2026-10-30T12:18:20Z",
  "url": "https://research.draconitic.com/paper/4gr79kxzqv981",
  "venue": "arXiv",
  "versions": []
}