{
  "id": "1901.09384",
  "version": "v1",
  "published": "2019-01-27T14:34:53.000Z",
  "updated": "2019-01-27T14:34:53.000Z",
  "title": "A giant disparity and a dynamical phase transition in large deviations of the time-averaged size of stochastic populations",
  "authors": [
    "Pini Zilber",
    "Naftali R. Smith",
    "Baruch Meerson"
  ],
  "comment": "10 pages, 7 figures",
  "categories": [
    "cond-mat.stat-mech"
  ],
  "abstract": "We study large deviations of the time-averaged size of stochastic populations described by a continuous-time Markov jump process. When the expected population size $N$ in the steady state is large, the large deviation function (LDF) of the time-averaged population size can be evaluated by using a WKB (after Wentzel, Kramers and Brillouin) method, applied directly to the master equation for the Markov process. For a class of models that we identify, the direct WKB method predicts a giant disparity between the probabilities of observing an unusually small and an unusually large values of the time-averaged population size. The disparity results from a qualitative change in the \"optimal\" trajectory of the underlying classical mechanics problem. The direct WKB method also predicts, in the limit of $N\\to \\infty$, a singularity of the LDF, which can be interpreted as a second-order dynamical phase transition. The transition is smoothed at finite $N$, but the giant disparity remains. The smoothing effect is captured by the van-Kampen system size expansion of the exact master equation near the attracting fixed point of the underlying deterministic model. We describe the giant disparity at finite $N$ by developing a different variant of WKB method, which is applied in conjunction with the Donsker-Varadhan large-deviation formalism and involves subleading-order calculations in $1/N$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-01-27T14:34:53.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "dynamical phase transition",
      "giant disparity",
      "large deviation",
      "stochastic populations",
      "time-averaged size"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 10,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}