{
  "id": "1102.3969",
  "version": "v2",
  "published": "2011-02-19T05:09:34.000Z",
  "updated": "2011-04-25T04:52:45.000Z",
  "title": "Stochastic thermodynamics for delayed Langevin systems",
  "authors": [
    "Huijun Jiang",
    "Tiejun Xiao",
    "Zhonghuai Hou"
  ],
  "comment": "16 pages, 5 figures",
  "doi": "10.1103/PhysRevE.83.061145",
  "categories": [
    "cond-mat.stat-mech"
  ],
  "abstract": "Stochastic thermodynamics (ST) for delayed Langevin systems are discussed. By using the general principles of ST, the first-law-like energy balance and trajectory-dependent entropy s(t) can be well-defined in a similar way as that in a system without delay. Since the presence of time delay brings an additional entropy flux into the system, the conventional second law $<{\\Delta {s_{tot}}}> \\ge 0$ no longer holds true, where $\\Delta {s_{tot}}$ denotes the total entropy change along a stochastic path and $<...>$ stands for average over the path ensemble. With the help of a Fokker-Planck description, we introduce a delay-averaged trajectory-dependent dissipation functional $\\eta [{\\chi(t)}]$ which involves the work done by a delay-averaged force $\\bar F({x,t})$ along the path $\\chi (t)$ and equals to the medium entropy change $\\Delta {s_m}[ {x(t)}]$ in the absence of delay. We show that the total dissipation functional R = \\Delta s + \\eta, where $\\Delta s$ denotes the system entropy change along a path, obeys $< R > \\ge 0$, which could be viewed as the second law in the delayed system. In addition, the integral fluctuation theorem $< <e^(-R)>=1 also holds true. We apply these concepts to a linear Langevin system with time delay and periodic external force. Numerical results demonstrate that the total entropy change $< {\\Delta {s_{tot}}} >$ could indeed be negative when the delay feedback is positive. By using an inversing-mapping approach, we are able to obtain the delay-averaged force $\\bar F({x,t})$ from the stationary distribution and then calculate the functional $R$ as well as its distribution. The second law $< R > \\ge 0$ and the fluctuation theorem are successfully validated.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2011-04-25T04:52:45.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "05.70.-a",
      "02.30.Ks",
      "05.40.-a"
    ],
    "keywords": [
      "delayed langevin systems",
      "stochastic thermodynamics",
      "additional entropy flux",
      "integral fluctuation theorem",
      "conventional second law"
    ],
    "tags": [
      "journal article"
    ],
    "publication": {
      "publisher": "APS",
      "journal": "Physical Review E",
      "year": 2011,
      "month": "Jun",
      "volume": 83,
      "number": 6,
      "pages": "061144"
    },
    "note": {
      "typesetting": "TeX",
      "pages": 16,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2011PhRvE..83f1144J"
    }
  }
}