{
  "annotation_id": "3322d80e-7b63-4a0d-8eeb-8ad2f7bf2700",
  "date_created": "2026-02-17T05:53:08.703000Z",
  "date_modified": "2026-02-17T05:53:08.703000Z",
  "file_hash": "0a06f861928af05bc9981ac8a0d72d6dd1a60225103cbe06a3f4b670e28e6dee",
  "private": false,
  "record": {
    "abstract": "We introduce a {\\it non-linear} generalization of the classical Dobrushin-Lanford-Ruelle (DLR) framework by developing the concept of a $q$-specification and the associated $q$-equilibrium measures. These objects arise naturally from a family of non-linear $q$-stochastic operators acting on the space of probability measures. A $q$-equilibrium measure is characterized as a fixed point of such operators, providing a non-linear analogue of the Gibbs equilibrium in the sense of DLR. We establish general conditions ensuring the existence and uniqueness of $q$-equilibrium measures and demonstrate how quasilocality plays a decisive role in their construction. Moreover, we exhibit examples of $q$-specifications with an empty set of $q$-equilibrium measures. We characterize the set of $q$-equilibrium measures by studying the dynamical systems generated by a class of $q$-stochastic operators. As a concrete application, we show that for the one-dimensional Ising model at sufficiently low temperatures, multiple $q$-equilibrium measures may exist, even though the classical Gibbs measure remains unique. Our results reveal that the $q$-specification formalism extends the DLR theory from linear to non-linear settings and opens a new direction in the study of Gibbs measures and equilibrium states of physical systems.",
    "arxiv_id": "2601.06470",
    "authors": [
      "F. H. Haydarov",
      "B. A. Omirov",
      "U. A. Rozikov"
    ],
    "categories": [
      "math-ph",
      "math.MP"
    ],
    "license": "http://creativecommons.org/licenses/by/4.0/",
    "title": "Non-Linear Generalization of the DLR Equations: $q$-Specifications and $q$-Equilibrium Measures",
    "url": "https://arxiv.org/abs/2601.06470",
    "version": "v1"
  },
  "schema_id": "dorsal/arxiv",
  "source": {
    "execution_id": "04fefe6d-1429-4f1b-bb84-bc2fe91e3eee",
    "id": "arXiv Dataset",
    "type": "Model",
    "variant": "snapshot-2026-01-17",
    "version": "0.1.0"
  },
  "user_id": 1000002
}