{
  "abstract": "We extend Mark Kac's kinetic framework from single-species Euclidean spaces to a two-species, non-linear interaction space on the open cone $(0, \\infty)^2$ governed by the Matsumoto--Yor rational collision involution. We identify a non-linear conserved energy $E(x,y) = x + y + 1/x$ and an invariant conic metric measure $dμ= \\frac{dx \\, dy}{x(x+y)}$. By establishing a two-species $H$-theorem, we prove that entropy production vanishes if and only if post-collision velocities are statistically independent ($I(U;V)=0$). In contrast to the transform-domain proof of Letac and Wesołowski~\\cite{letac2000}, which relies on joint Laplace transforms of general measures, our approach yields a direct physical derivation in state space via the transformation law for logarithmic gradients and separation of variables. This uncouples the functional-differential system to uniquely characterize the Generalized Inverse Gaussian (GIG) and Gamma distributions as the non-equilibrium thermodynamic equilibria.",
  "abstract_shown": true,
  "affiliations": [],
  "article_type": null,
  "arxiv": "2610.18892",
  "author_total": 1,
  "authors": [
    {
      "name": "Yoshiaki Itoh"
    }
  ],
  "bibtex": "https://research.draconitic.com/paper/5784znv1db9fh.bib",
  "cited_by": 0,
  "doi": "10.48550/arxiv.2610.18892",
  "fulltext_url": null,
  "id": "5784znv1db9fh",
  "issns": [],
  "landing_url": "https://arxiv.org/abs/2610.18892",
  "language": null,
  "licence": "LicenseRef-arXiv-nonexclusive-1.0",
  "licence_label": "arXiv licence",
  "licence_source": "article",
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  "metadata_licence": "CC0-1.0",
  "native_subjects": [
    [
      "arxiv",
      "cond-mat.stat-mech"
    ],
    [
      "arxiv",
      "math-ph"
    ],
    [
      "arxiv",
      "math.MP"
    ],
    [
      "arxiv",
      "math.PR"
    ],
    [
      "openalex",
      "3109|Statistical Mechanics and Entropy"
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  ],
  "openalex": "W7221923418",
  "pdf_url": "https://arxiv.org/pdf/2610.18892",
  "pmcid": null,
  "pmid": null,
  "published": "2026-10-22",
  "publisher": null,
  "ris": "https://research.draconitic.com/paper/5784znv1db9fh.ris",
  "sources": [
    {
      "first_seen": "2026-10-30T12:34:34Z",
      "native_id": "2610.18892",
      "source": "arxiv",
      "url": "https://arxiv.org/abs/2610.18892"
    }
  ],
  "subjects": [
    {
      "field": "mathematics",
      "origin": "arxiv:math-ph",
      "subfield": "mathematical-physics"
    },
    {
      "field": "mathematics",
      "origin": "arxiv:math.PR",
      "subfield": "probability"
    },
    {
      "field": "physics",
      "origin": "arxiv:cond-mat.stat-mech",
      "subfield": "statistical-nonlinear"
    }
  ],
  "title": "Two-Species Boltzmann--Kac Kinetics with Matsumoto--Yor Nonlinear Collisions: Conserved Energy and Conic Equilibria",
  "title_text": "Two-Species Boltzmann--Kac Kinetics with Matsumoto--Yor Nonlinear Collisions: Conserved Energy and Conic Equilibria",
  "type": "preprint",
  "updated_at": "2026-10-30T12:34:38Z",
  "url": "https://research.draconitic.com/paper/5784znv1db9fh",
  "venue": "arXiv",
  "versions": []
}