{
  "id": "1110.6303",
  "version": "v1",
  "published": "2011-10-28T11:48:18.000Z",
  "updated": "2011-10-28T11:48:18.000Z",
  "title": "Towards a large deviation theory for statistical-mechanical complex systems",
  "authors": [
    "Guiomar Ruiz",
    "Constantino Tsallis"
  ],
  "comment": "6 pages, 4 figures",
  "categories": [
    "cond-mat.stat-mech",
    "math-ph",
    "math.MP",
    "math.PR"
  ],
  "abstract": "The theory of large deviations constitutes a mathematical cornerstone in the foundations of Boltzmann-Gibbs statistical mechanics, based on the additive entropy $S_{BG}=- k_B\\sum_{i=1}^W p_i \\ln p_i$. Its optimization under appropriate constraints yields the celebrated BG weight $e^{-\\beta E_i}$. An elementary large-deviation connection is provided by $N$ independent binary variables, which, in the $N\\to\\infty$ limit yields a Gaussian distribution. The probability of having $n \\ne N/2$ out of $N$ throws is governed by the exponential decay $e^{-N r}$, where the rate function $r$ is directly related to the relative BG entropy. To deal with a wide class of complex systems, nonextensive statistical mechanics has been proposed, based on the nonadditive entropy $S_q=k_B\\frac{1- \\sum_{i=1}^W p_i^q}{q-1}$ ($q \\in {\\cal R}; \\,S_1=S_{BG}$). Its optimization yields the generalized weight $e_q^{-\\beta_q E_i}$ ($e_q^z \\equiv [1+(1-q)z]^{1/(1-q)};\\,e_1^z=e^z)$. We numerically study large deviations for a strongly correlated model which depends on the indices $Q \\in [1,2)$ and $\\gamma \\in (0,1)$. This model provides, in the $N\\to\\infty$ limit ($\\forall \\gamma$), $Q$-Gaussian distributions, ubiquitously observed in nature ($Q\\to 1$ recovers the independent binary model). We show that its corresponding large deviations are governed by $e_q^{-N r_q}$ ($\\propto 1/N^{1/(q-1)}$ if $q>1$) where $q= \\frac{Q-1}{\\gamma (3-Q)}+1 \\ge 1$. This $q$-generalized illustration opens wide the door towards a desirable large-deviation foundation of nonextensive statistical mechanics.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2011-10-28T11:48:18.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "37A60",
      "37A35",
      "37H05"
    ],
    "keywords": [
      "large deviation theory",
      "statistical-mechanical complex systems",
      "nonextensive statistical mechanics",
      "gaussian distribution",
      "independent binary variables"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 6,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2011arXiv1110.6303R"
    }
  }
}