{
  "id": "1802.09440",
  "version": "v1",
  "published": "2018-02-26T16:36:55.000Z",
  "updated": "2018-02-26T16:36:55.000Z",
  "title": "Reaction-diffusion on the fully-connected lattice: probability distributions and extreme value statistics for $A+B\\rightarrow$ Ø",
  "authors": [
    "Loïc Turban"
  ],
  "comment": "28 pages, 16 figures. Continuation of arXiv:1711.01248",
  "categories": [
    "cond-mat.stat-mech"
  ],
  "abstract": "We study the two-species diffusion-annihilation process, $A+B\\rightarrow$ \\O, on the fully-connected lattice. Probability distributions for the number of particles and the reaction time are obtained for a finite-size system using a master equation approach. Mean values and variances are deduced from generating functions. When the reaction is far from complete, i.e., for a large number of particles of each species, mean-field theory is exact and the fluctuations are Gaussian. In the scaling limit the reaction time displays extreme-value statistics in the vicinity of the absorbing states. A generalized Gumbel distribution is obtained for unequal initial densities, $\\rho_A>\\rho_B$. For equal or almost equal initial densities, $\\rho_A\\simeq\\rho_B$, the fluctuations of the reaction time near the absorbing state are governed by a probability density involving derivatives of $\\vartheta_4$, the Jacobi theta function.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2018-02-26T16:36:55.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "extreme value statistics",
      "probability distributions",
      "fully-connected lattice",
      "reaction time displays extreme-value statistics",
      "reaction-diffusion"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 28,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}