Entropy: Thermodynamic definition and quantum expression

Abstract

Numerous expressions exist in the scientific literature purporting to represent entropy. Are they all acceptable? To answer this question, we review the thermodynamic definition of entropy, and establish eight criteria that must be satisfied by it. The definition and criteria are obtained by using solely the general, nonstatistical statements of the first and second laws presented in Thermodynamics: Foundations and Applications [Elias P. Gyftopoulos and Gian Paolo Beretta (Macmillan, New York, 1991)]. We apply the eight criteria to each of the entropy expressions proposed in the literature and find that only the relation S=-kTr\ensuremath{\rho}ln \ensuremath{\rho} satisfies all the criteria, provided that the density operator \ensuremath{\rho} corresponds to a homogeneous ensemble of identical systems, identically prepared. Homogeneous ensemble means that every member of the ensemble is described by the same density operator \ensuremath{\rho} as any other member, that is, the ensemble is not a statistical mixture of projectors (wave functions).