{
  "id": "2001.03007",
  "version": "v1",
  "published": "2020-01-09T14:20:29.000Z",
  "updated": "2020-01-09T14:20:29.000Z",
  "title": "Fastest Frozen Temperature for a Degree of Freedom",
  "authors": [
    "X. Y. Zhou",
    "Z. Q. Yang",
    "X. R. Tang",
    "X. Wang",
    "Q. H. Liu"
  ],
  "comment": "6 pages, 4 figures",
  "categories": [
    "cond-mat.stat-mech",
    "quant-ph"
  ],
  "abstract": "The equipartition theorem states that the mean thermal energy for each degree of freedom at a temperature $T$ is given by $k_{B}T/2$ where $k_{B}$ is Boltzmann's constant. However, this theorem for the degree of freedom breaks down at lower enough temperature when the relevant degree of freedom is not thermally excited. The frozen condition can be \\textit{qualitatively} expressed as $T\\ll T_{C}$ which is the characteristic temperature defined via $k_{B}T_{C}\\equiv\\varepsilon_{1}-\\varepsilon_{0}$ in which $\\varepsilon_{0}$ and $\\varepsilon_{1}$ are, respectively, the ground state and first excited energy of the degree of freedom. This characteristic temperature $T_{C}$ so defined is the temperature at which the degree of freedom is almost activated. We report that there is a well-defined temperature at which the specific heat coming from the given degree of freedom \\textit{changes most dramatically} with temperature, i.e., the derivative of the specific heat with respect to temperature reaches its maximum. For some system such as a mixture of ortho- and para- hydrogen molecules, the characteristic temperature is hard to define, the fastest frozen temperature is also straightforward and well-defined. In general, the fastest frozen temperature is a \\emph{quantitative}\\textit{\\ }criterion of the frozen temperature.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-01-09T14:20:29.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "fastest frozen temperature",
      "characteristic temperature",
      "specific heat",
      "mean thermal energy",
      "equipartition theorem states"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 6,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}