Concepts of quantum statistical thermodynamics

In this article we have used some of the concepts of quantum-statistical mechanics. These concepts can, of course, be found in the textbooks (Ter Haar, 1966 Feynman, 1972 McQuarrie, 1976), but the ideas that are most relevant to this paper are summarized in this appendix. In particular, we prove the thermodynamic variation principle, which has been applied several times. 

The dominant view currently held about the physical significance of thermodynamics is based on the interpretation of a "thermodynamic state" as a composite that best describes the knowledge of an observer possessing only partial information about the "actual state" of the system. The "actual state" at any instant of time is defined as a wave function (a pure state or a projection operator) of quantum mechanics. The theories that have recently evolved pursuant to this view have been called informational, though the same concept is the foundation of all statistical thermodynamics. 

We conclude this brief discussion of deviations from the third law by stating that, although the cases of nonconformity are frequent, we can usually understand their origin with the aid of molecular concepts and quantum statistics. The latter discipline permits calculation of thermodynamic quantities, thereby providing a useful check on experimental data indeed, it often supplies answers of greater accuracy. In this way, it is possible to use the third law to build up tables of absolute entropies of chemical substances. 

Being a contemporary of the nineteenth century, Gibbs could obviously not have had any concept of quantum mechanics and its role in laying a sound foundation of modern statistical thermodynamics. In this modern formulation, the classic Gibbsian version of statistical thermodjmamics does, however, emerge as a limiting case as our discussion in Section 2.5 reveals. 

The development of theoretical chemistry ceased at about 1930. The last significant contributions came from the first of the modern theoretical physicists, who have long since lost interest in the subject. It is not uncommon today, to hear prominent chemists explain how chemistry is an experimental science, adequately practiced without any need of quantum mechanics or the theories of relativity. Chemical thermodynamics is routinely rehashed in the terminology and concepts of the late nineteenth century. The formulation of chemical reaction and kinetic theories take scant account of statistical mechanics and non-equilibrium thermodynamics. Theories of molecular structure are entirely classical and molecular cohesion is commonly analyzed in terms of isolated bonds. Holistic effects and emergent properties that could... 

Recently there has emerged the beginning of a direct, operational link between quantum chemistry and statistical thermodynamic. The link is obtained by the ability to write E = V Vij—namely, to write the output of quantum-mechanical computations as the standard input for statistical computations, It seems very important that an operational link be found in order to connect the discrete description of matter (X-ray, nmr, quantum theory) with the continuous description of matter (boundary conditions, diffusion). The link, be it a transformation (probably not unitary) or other technique, should be such that the nonequilibrium concepts, the dissipative structure concepts, can be used not only as a language for everyday biologist, but also as a tool of quantitation value, with a direct, quantitative and operational link to the discrete description of matter. 

The placement of statistical mechanics in the sequence is another issue. I think that careful treatments of thermodynamics and quantum mechanics should precede the presentation of statistical mechanics. This can be accomplished with thermodynamics in the first semester, quantum mechanics in the second semester, followed by statistical mechanics near the end of the course. If statistical mechanics is taught before thermodynamics or quantum mechanics, you must either provide a brief introduction to some of the concepts of these subjects at the beginning of the treatment or integrate it into the treatment. 

Next, we review findings of educational research about the main areas of physical chemistry. Most of the work done was in the areas of basic thermodynamics and electrochemistry, and some work on quantum chemistry. Other areas, such as chemical kinetics, statistical thermodynamics, and spectroscopy, have not so far received attention (although the statistical interpretation of entropy is treated in studies on the concepts of thermodynamics). Because many of the basics of physical chemistry are included in first-year general and inorganic courses (and some even in senior high school), many of the investigations have been carried out at these levels. 

Today, the situation is just the opposite and it seems at first strange to try to improve our understanding of quantum theory by using methods and techniques developed in statistical mechanics and in thermodynamics. That is, however, what I shall try to do. I shall not go into any technical details which may be found elsewhere.19,21 But I would like to emphasize here the physical ideas behind the formalism. It seems to me that this new development may lead to a clarification of concepts used in widely different fields such as thermodynamics and statistical mechanics of irreversible... 

The functional relation for EhA was derived by Planck by introducing the quantum concept for electromagnetic energy. The derivation is now usually performed by methods of statistical thermodynamics, and Ebx is shown to be related to the energy density of Eq. (8-2) by... 

In Chapter 2, we developed statistical thermodynamics as the central theory that enables ns in principle to calculate thermophysical properties of macroscopic confined flriids. A key feature of statistical thermodynamics is an enormous reduction of information that takes place as one goes from the microscopic world of electrons, photons, atoms, or molecules to the macroscopic world at which one performs measurements of thermophysical properties of interest. This information reduction is effected by statistical concepts such as the most probable distribution of quantum states (see Section 2.2.1). 

Many ideas on chemical bond are rather firmly established now, after more than a century of investigations following the discovery of electron they are accepted by scientists in widely different areas of specialization. We can now confidently distinguish between transient hypotheses and those with a level of stability. When this is done, we will not fail to notice the pattern of interaction between chemistry and physics that I have detailed earlier. Among the chemical bond concepts that scientists (not just chemists) hold indispensable both for explaining what has been experimentally established and for guiding what is yet to be discovered, some can be derived (with appropriate assumptions) from quantum theory, some are conceptually incompatible with quantum theory, and the rest independent of quantum theory. One finds similar relationship in the overlapping between chemistry and physics in thermodynamics and statistical thermodynamics (Vemulapalli and Byerly 1999). 

In this chapter, the foundations of equilibrium statistical mechanics are introduced and applied to ideal and weakly interacting systems. The connection between statistical mechanics and thermodynamics is made by introducing ensemble methods. The role of mechanics, both quantum and classical, is described. In particular, the concept and use of the density of states is utilized. Applications are made to ideal quantum and classical gases, ideal gas of diatomic molecules, photons and the black body radiation, phonons in a harmonic solid, conduction electrons in metals and the Bose—Einstein condensation. Introductory aspects of the density... 

To follow the scale of complexity, the review is divided into three parts. The first two parts deal with the key concept of effective Hamiltonians which describe the dynamical and spectroscopic properties of interfering resonances (Section 2) and resonant scattering (Section 3). The third part. Section 4, is devoted to the resolution of the Liouville equation and to the introduction of the concept of effective Liouvillian which generalizes the concept of effective Hamiltonian. The link between the theory of quantum resonances and statistical physics and thermodynamics is thus established. Throughout this work we have tried to keep a balance between the theory and the examples based on simple solvable models. 

We outline briefly in this section how to link the theory of quantum resonances to statistical physics and thermodynamics by extending the concept of effective Hamiltonian as recently discussed in Ref. [60]. The quantum Liouville-von Neumann equation is written in the form... 

The proposed approach leads directly to practical results such as the prediction—based upon the chemical potential—of whether or not a reaction runs spontaneously. Moreover, the chemical potential is key in dealing with physicochemical problems. Based upon this central concept, it is possible to explore many other fields. The dependence of the chemical potential upon temperature, pressure, and concentration is the gateway to the deduction of the mass action law, the calculation of equilibrium constants, solubilities, and many other data, the construction of phase diagrams, and so on. It is simple to expand the concept to colligative phenomena, diffusion processes, surface effects, electrochemical processes, etc. Furthermore, the same tools allow us to solve problems even at the atomic and molecular level, which are usually treated by quantum statistical methods. This approach allows us to eliminate many thermodynamic quantities that are traditionally used such as enthalpy H, Gibbs energy G, activity a, etc. The usage of these quantities is not excluded but superfluous in most cases. An optimized calculus results in short calculations, which are intuitively predictable and can be easily verified. 

---