{
  "id": "1512.02855",
  "version": "v1",
  "published": "2015-12-09T13:37:55.000Z",
  "updated": "2015-12-09T13:37:55.000Z",
  "title": "Inconsistency of microcanonical entropy: the case of chemical potential",
  "authors": [
    "Arash Tavassoli",
    "Afshin Montakhab"
  ],
  "comment": "5 pages and one figure",
  "categories": [
    "cond-mat.stat-mech",
    "physics.chem-ph"
  ],
  "abstract": "Recently, there has been a considerable amount of debate regarding a correct and \"consistent\" definition of microcanonical entropy. Consistent thermodynamics is based on the assumption of adiabatic invariance of microcanonical entropy. Such an assumption is equivalent to a consistency relation which equates derivatives of the entropy to ensemble average of corresponding quantity in micro-state space (phase space or Hilbert space). It has been argued that the Gibbs (volume) entropy satisfies such a consistency relation under general condition, and is thus preferred over the Boltzmann (surface) entropy, which is only consistent for large macroscopic systems. In this work we propose to re-examine such a consistency relation when the number of particles ($N$) is considered as the independent thermodynamic variable. In other words, we investigate the consistency relation for the chemical potential which is a fundamental thermodynamic quantity. We show both by simple analytical calculations as well as model examples that \\emph{both} definitions of entropy (Gibbs as well as Boltzmann) lead to inconsistent results when one considers such a relation for the chemical potential for finite or infinite system sizes. Our results cast doubt on the validity of the adiabatic invariance as a required property of thermodynamic entropy. We close by providing commentary on the derivation of thermodynamics from mechanics which typically leads to controversial and inconsistent results. Considering entropy as an \\emph{emergent} property of macroscopic systems may provide a way out of such inconsistencies.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2015-12-09T13:37:55.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "microcanonical entropy",
      "chemical potential",
      "consistency relation",
      "inconsistency",
      "adiabatic invariance"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 5,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}