{
  "id": "2012.15607",
  "version": "v1",
  "published": "2020-12-31T13:54:43.000Z",
  "updated": "2020-12-31T13:54:43.000Z",
  "title": "Zeroth Law in Quantum Thermodynamics at Strong Coupling: `in Equilibrium', not `Equal Temperature'",
  "authors": [
    "Jen-Tsung Hsiang",
    "Bei-Lok Hu"
  ],
  "comment": "22 pages, 2 figures",
  "categories": [
    "cond-mat.stat-mech",
    "hep-th",
    "quant-ph"
  ],
  "abstract": "The zeroth law of thermodynamics involves a transitivity relation (pairwise between three objects) expressed either in terms of `equal temperature' (ET), or `in equilibrium' (EQ) conditions. In conventional thermodynamics conditional on vanishingly weak system-bath coupling these two conditions are commonly regarded as equivalent. In this work we show that for thermodynamics at strong coupling they are inequivalent: namely, two systems can be in equilibrium and yet have different effective temperatures. A recent result \\cite{NEqFE} for Gaussian quantum systems shows that an effective temperature $T^{*}$ can be defined at all times during a system's nonequilibrium evolution, but because of the inclusion of interaction energy, after equilibration the system's $T^*$ is slightly higher than the bath temperature $T_{\\textsc{b}}$, with the deviation depending on the coupling. A second object coupled with a different strength with an identical bath at temperature $T_{\\textsc{b}}$ will not have the same equilibrated temperature as the first object. Thus $ET \\neq EQ $ for strong coupling thermodynamics. We then investigate the conditions for dynamical equilibration for two objects 1 and 2 strongly coupled with a common bath $B$, each with a different equilibrated effective temperature. We show this is possible, and prove the existence of a generalized fluctuation-dissipation relation under this configuration. This affirms that `in equilibrium' is a valid and perhaps more fundamental notion which the zeroth law for quantum thermodynamics at strong coupling should be based on. Only when the system-bath coupling becomes vanishingly weak that `temperature' appearing in thermodynamic relations becomes universally defined and makes better physical sense.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2020-12-31T13:54:43.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "zeroth law",
      "strong coupling",
      "quantum thermodynamics",
      "equal temperature",
      "effective temperature"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 22,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}