{
  "id": "1410.3923",
  "version": "v1",
  "published": "2014-10-15T04:19:38.000Z",
  "updated": "2014-10-15T04:19:38.000Z",
  "title": "Optimal Transport for Non-Conservative Systems",
  "authors": [
    "L. Chayes",
    "H. K. Lei"
  ],
  "comment": "42 pages. Preliminary posting",
  "categories": [
    "math.AP",
    "math-ph",
    "math.MP"
  ],
  "abstract": "We study, in finite volume, a grand canonical version of the McKean-Vlasov equation where the total particle content is allowed to vary. The dynamics is anticipated to minimize an appropriate grand canonical free energy; we make this notion precise by introducing a metric on a set of positive Borel measures without pre-prescribed mass and demonstrating that the dynamics is a gradient flow with respect to this metric. Moreover, we develop a JKO-scheme suitable for these problems. The latter ideas have general applicability to a class of second order non-conservative problems. For this particular system we prove, using the JKO-scheme, that (under certain assumptions) convergence to the uniform stationary state is exponential with a rate which is independent of the volume. By contrast, in related conservative systems, decay rates scale - at best - with the square of the characteristic length of the system. This suggests that a grand canonical approach may be useful for both theoretical and computational study of large scale systems.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2014-10-15T04:19:38.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "optimal transport",
      "non-conservative systems",
      "appropriate grand canonical free energy",
      "decay rates scale",
      "total particle content"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 42,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2014arXiv1410.3923C"
    }
  }
}