{
  "id": "1709.08906",
  "version": "v1",
  "published": "2017-09-26T09:19:22.000Z",
  "updated": "2017-09-26T09:19:22.000Z",
  "title": "Definitions and Evolutions of Statistical Entropy for Hamiltonian Systems",
  "authors": [
    "Xiangjun Xing"
  ],
  "comment": "30 preprint pages, 4 pdf figures",
  "categories": [
    "cond-mat.stat-mech"
  ],
  "abstract": "Regardless of studies and debates over a century, the statistical origin of the second law of thermodynamics still remains illusive. One essential obstacle is the lack of a proper theoretical formalism for non-equilibrium entropy. Here I revisit the seminal ideas about non-equilibrium statistical entropy due to Boltzmann and due to Gibbs, and synthesize them into a coherent and precise framework. Using this framework, I clarify the anthropomorphic principle of entropy, and analyze the evolution of entropy for classical Hamiltonian systems under different experimental setups. I find that evolution of Boltzmann entropy obeys a Stochastic H-Theorem, which relates probability of Boltzmann entropy increasing to that of decreasing. By contrast, the coarse-grained Gibbs entropy is monotonically increasing, if the microscopic dynamics is locally mixing, and the initial state is a Boltzmann state. These results clarify the precise meaning of the second law of thermodynamics for classical systems, and demonstrate that it is the initial condition as a Boltzmann state that is ultimately responsible for the arrow of time.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2017-09-26T09:19:22.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "second law",
      "definitions",
      "boltzmann state",
      "boltzmann entropy obeys",
      "seminal ideas"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}