{
  "id": "2304.13719",
  "version": "v1",
  "published": "2023-04-26T17:55:57.000Z",
  "updated": "2023-04-26T17:55:57.000Z",
  "title": "Towards Microscopic Theory of Decaying Turbulence",
  "authors": [
    "Alexander Migdal"
  ],
  "comment": "8 pages, 4 figures",
  "categories": [
    "physics.flu-dyn",
    "math-ph",
    "math.MP",
    "nlin.SI"
  ],
  "abstract": "We develop a quantitative microscopic theory of decaying Turbulence by studying the dimensional reduction of the Navier-Stokes loop equation for the velocity circulation. We have found a degenerate family of solutions of the \\NS{} loop equation\\cite{M93, M23PR} for the Wilson loop in decaying Turbulence in arbitrary dimension $d >2$. This family of solutions corresponds to a nonlinear random walk in complex space $\\bC^d$, described by an algebraic equation between consecutive positions. The probability measure is explicitly constructed in terms of products of conventional measures for orthogonal group $SO(d)$ and a sphere $\\bS^{d-3}$. We compute a prediction for the three-dimensional Wilson loop for a circle of radius $R$ as a universal function of $\\frac{R}{\\sqrt{2 \\nu t}}$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-04-26T17:55:57.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "decaying turbulence",
      "navier-stokes loop equation",
      "nonlinear random walk",
      "three-dimensional wilson loop",
      "orthogonal group"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 8,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}