{
  "id": "1501.04238",
  "version": "v1",
  "published": "2015-01-17T22:34:33.000Z",
  "updated": "2015-01-17T22:34:33.000Z",
  "title": "Nonequilibrium statistical mechanics of weakly stochastically perturbed system of oscillators",
  "authors": [
    "A. Dymov"
  ],
  "comment": "51 pages",
  "categories": [
    "math-ph",
    "math.MP"
  ],
  "abstract": "We consider a finite region of a $d$-dimensional lattice, $d\\in\\mathbb{N}$, of weakly coupled harmonic oscillators. The coupling is provided by a nearest-neighbour potential (harmonic or not) of size $\\varepsilon$. Each rotator weakly interacts by force of order $\\varepsilon$ with its own stochastic Langevin thermostat of arbitrary positive temperature. We investigate limiting as $\\varepsilon\\rightarrow 0$ behaviour of solutions of the system and of the local energy of oscillators on long-time intervals of order $\\varepsilon^{-1}$ and in a stationary regime. We show that it is governed by an effective equation which is a dissipative SDE with nondegenerate diffusion. Next we assume that the interaction potential is of size $\\varepsilon\\lambda$, where $\\lambda$ is another small parameter, independent from $\\varepsilon$. Solutions corresponding to this scaling describe small law temperature oscillations. We prove that in a stationary regime, under the limit $\\varepsilon\\rightarrow 0$, the main order in $\\lambda$ of the averaged Hamiltonian energy flow is proportional to the gradient of temperature. We show that the coefficient of proportionality, which we call the conductivity, admits a representation through stationary space-time correlations of the energy flow. Most of the results and convergences we obtain are uniform with respect to the number of oscillators in the system.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2015-01-17T22:34:33.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "82C05",
      "82C70"
    ],
    "keywords": [
      "weakly stochastically perturbed system",
      "nonequilibrium statistical mechanics",
      "oscillators",
      "stationary regime",
      "small law temperature oscillations"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 51,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2015arXiv150104238D"
    }
  }
}