{
  "id": "2301.06585",
  "version": "v1",
  "published": "2023-01-16T19:57:26.000Z",
  "updated": "2023-01-16T19:57:26.000Z",
  "title": "From Exclusion to Slow and Fast Diffusion",
  "authors": [
    "Patricia Gonçalves",
    "Gabriel Nahum",
    "Marielle Simon"
  ],
  "categories": [
    "math.PR"
  ],
  "abstract": "We construct a nearest-neighbour interacting particle system of exclusion type, which illustrates a transition from slow to fast diffusion. More precisely, the hydrodynamic limit of this microscopic system in the diffusive space-time scaling is the parabolic equation $\\partial_t\\rho=\\nabla (D(\\rho)\\nabla \\rho)$, with diffusion coefficient $ D(\\rho)=m\\rho^{m-1} $ where $ m\\in(0,2] $, including therefore the fast diffusion regime in the range $ m\\in(0,1) $, and the porous {medium} equation for $ m\\in(1,2) $. The construction of the model is based on the generalized binomial theorem, and interpolates continuously in $ m $ the already known microscopic porous medium model with parameter $ m=2 $, the symmetric simple exclusion process with $ m=1 $, going down to a fast diffusion model up to any $ m>0$. The derivation of the hydrodynamic limit for the local density of particles on the one-dimensional torus is achieved via the entropy method -- with additional technical difficulties depending on the regime (slow or fast diffusion) and where new properties of the porous medium model need to be derived.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-01-16T19:57:26.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "symmetric simple exclusion process",
      "hydrodynamic limit",
      "nearest-neighbour interacting particle system",
      "microscopic porous medium model",
      "fast diffusion model"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}