{
  "id": "2204.04720",
  "version": "v1",
  "published": "2022-04-10T16:34:54.000Z",
  "updated": "2022-04-10T16:34:54.000Z",
  "title": "The crossover from the Macroscopic Fluctuation Theory to the Kardar-Parisi-Zhang equation controls the large deviations beyond Einstein's diffusion",
  "authors": [
    "Alexandre Krajenbrink",
    "Pierre Le Doussal"
  ],
  "comment": "46 pages",
  "categories": [
    "cond-mat.stat-mech",
    "cond-mat.dis-nn",
    "math-ph",
    "math.MP",
    "math.PR",
    "nlin.SI"
  ],
  "abstract": "We study the crossover from the macroscopic fluctuation theory (MFT) which describes 1D stochastic diffusive systems at late times, to the weak noise theory (WNT) which describes the Kardar-Parisi-Zhang (KPZ) equation at early times. We focus on the example of the diffusion in a time-dependent random field, observed in an atypical direction which induces an asymmetry. The crossover is described by a non-linear system which interpolates between the derivative and the standard non-linear Schrodinger equations in imaginary time. We solve this system using the inverse scattering method for mixed-time boundary conditions introduced by us to solve the WNT. We obtain the rate function which describes the large deviations of the sample-to-sample fluctuations of the cumulative distribution of the tracer position. It exhibits a crossover as the asymmetry is varied, recovering both MFT and KPZ limits. We sketch how it is consistent with extracting the asymptotics of a Fredholm determinant formula, recently derived for sticky Brownian motions. The crossover mechanism studied here should generalize to a larger class of models described by the MFT. Our results apply to study extremal diffusion beyond Einstein's theory.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-04-10T16:34:54.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "macroscopic fluctuation theory",
      "kardar-parisi-zhang equation controls",
      "large deviations",
      "einsteins diffusion",
      "standard non-linear schrodinger equations"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 46,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}