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 by Gyftopoulos and Beretta (1991). We apply the eight criteria to each of the expressions proposed in the literature and, for the first time, find that only the relation S = −kTrρlnρ satisfies all the criteria, provided that the density operator ρ 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 ρ as any other member, that is, the ensemble is not a statistical mixture of projectors (wave functions).
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 by Gyftopoulos and Beretta (1991). We apply the eight criteria to each of the expressions proposed in the literature and, for the first time, find that only the relation S = −kTrρlnρ satisfies all the criteria, provided that the density operator ρ 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 ρ as any other member, that is, the ensemble is not a statistical mixture of projectors (wave functions).
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).
Entropy is the distinguishing and most important concept of our efforts to understand and regularize our observations of a very large class of natural phenomena, and yet, it is one of the most contentious concepts of physics. In this article, we review two expositions of thermodynamics, one without reference to quantum theory, and the other quantum mechanical without probabilities of statistical mechanics. In the first, we show that entropy is an inherent property of any system in any state, and that its analytical expression must conform to eight criteria. In the second, we recognize that quantum thermodynamics: (i) admits quantum probabilities described either by wave functions or by nonstatistical density operators; and (ii) requires a nonlinear equation of motion that is delimited by but more general than the Schroedinger equation, and that accounts for both reversible and irreversible evolutions of the state of the system in time. Both the more general quantum probabilities, and the equation of motion have been defined, and the three laws of thermodynamics are shown to be theorems of this equation.
The prevailing beliefs in the scientific and engineering literature are that: (i) thermodynamics is explained and justified by statistical mechanics; (ii) entropy is a statistical measure of disorder; and (iii) for given values of energy, volume, and amounts of constituents, the largest value of entropy corresponds to both a thermodynamic equilibrium state and the ultimate disorder. In this paper, we provide: (i) a summary of the beliefs as stated by some eminent scientists; (ii) experimental evidence that casts serious doubt about the validity of the beliefs; (iii) an outline of a nonstatistical unified quantum theory of mechanics and thermodynamics; (iv) an outline of a nonquantal, nonstatistical exposition of thermodynamics, valid for all systems (both macroscopic and microscopic), and for all states (both thermodynamic equilibrium and not thermodynamic equilibrium); (v) the definition and analytical expression of the entropy of thermodynamics; (vi) the interpretation of entropy as both a measure of the quantum-theoretic spatial shape of a molecule, and an indicator of order; and (vii) nonstatistical answers to the questions that motivated the introduction of statistical mechanics.
In quantum mechanics, the expression for entropy is usually taken to be -kTr(ρlnρ), where ρ is the density matrix. The convention first appears in Von Neumann's Mathematical Foundations of Quantum Mechanics. The argument given there to justify this convention is the only one hitherto offered. All the arguments in the field refer to it at one point or another. Here this argument is shown to be invalid. Moreover, it is shown that, if entropy is -kTr(ρlnρ), then perpetual motion machines are possible. This and other considerations support the conclusion that this expression is not the quantum-mechanical correlate of thermodynamic entropy. Its usefulness in quantum-statistical mechanics can be explained by its being a convenient quantification of information, but information and entropy are not synonymous. As the present paper shows, one can change while the other is conserved.
In Chapter 3 the introduction of entropy as a state function was facilitated by the realization that there exist possible and impossible processes, and by an examination of the heat and work effects occurring during these processes. From the formal statement of the Second Law of Thermodynamics, as developed from Classical Thermodynamics arguments, it is difficult to assign a physical significance or a physical quality to entropy. In this respect entropy differs from internal energy in spite of the fact that, within the scope of Classical Thermodynamics, both properties are simply mathematical functions of the state of a system. The ready acceptance of the First Law of Thermodynamics, after its enunciation, was due to the easily understood physical significance of internal energy, whereas the lack of corresponding understanding of entropy caused the acceptance of the Second Law of Thermodynamics to be slow. From the classical viewpoint the Second Law is only valid as a direct result of the fact that, to date, human ingenuity has failed to invent a perpetual motion machine. Thus, within the scope of Classical Thermodynamics, the Second Law is a “law” only because it has not yet been disproved. The physical interpretation of entropy had to await the development of quantum theory and statistical mechanics.
Deriving the laws of thermodynamics from a microscopic picture is a central quest of statistical mechanics.This tutorial focuses on the derivation of the first and second law for isolated and open quantum systems far from equilibrium, where such foundational questions also become practically relevant for emergent nanotechnologies.The derivation is based on a microscopic definition of five essential quantities: internal energy, thermodynamic entropy, work, heat, and temperature.These definitions are shown to satisfy the phenomenological laws of nonequilibrium thermodynamics for a large class of states and processes.The consistency with previous results is demonstrated.The framework applies to multiple baths including particle transport and accounts for processes with, e.g., a changing temperature of the bath, which is determined microscopically.An integral fluctuation theorem for entropy production is satisfied.In summary, this tutorial introduces a consistent and versatile framework to understand and apply the laws of thermodynamics in the quantum regime and beyond.