The basic concepts of classical thermodynamics can be summarized by invoking the following four postulates (Callen, 1985):

1. There exist particular states (called equilibrium states) of simple compressible systems that, macroscopically, are characterized completely by the internal energy, U, the volume, V, and the mole or particle numbers, of the chemical components.

2. There exists a function called the entropy, S, of the extensive parameters of any composite system, defined for all equilibrium states and having the following property: The values assumed by the extensive parameters in the absence of an internal constraint are those which maximize the entropy for the composite isolated system.

3. The entropy of a composite system is additive over the constituent subsystems. Moreover, the entropy is a continuous, differentiable, and monotonically increasing function of the internal energy.

4. The entropy of any system vanishes in the state for which (i.e., at the zero of temperature).

Recall that a simple compressible system is defined as one that is macroscopically homogeneous, uncharged, and chemically inert, that is sufficiently large that surface effects can be neglected, and that is not acted on by electric, magnetic, or gravitational fields. Although these four basic postulates are restricted to simple compressible systems, they can readily be extended to more complex systems (Lewis and Randall, 1961).

My intention in this textbook is to provide a self-contained exposition of the fundamentals and applications of statistical thermodynamics for beginning graduate students in the engineering sciences. Especially within engineering, most students enter a course in statistical thermodynamics with limited exposure to statistics, quantum mechanics, and spectroscopy. Hence, I have found it necessary over the years to “start from the beginning,” not leaving out intermediary steps and presuming little knowledge in the discrete, as compared to the continuum, domain of physics. Once these things are done carefully, I find that good graduate students can follow the ideas, and that they leave the course excited and satisfied with their newfound understanding of both statistical and classical thermodynamics.

Nevertheless, a first course in statistical thermodynamics remains challenging and sometimes threatening to many graduate students. Typically, all their previous experience is with the equations of continuum mechanics, whether applied to thermodynamics, fluid mechanics, or heat transfer. For most students, therefore, the mathematics of probability theory, the novelty of quantum mechanics, the confrontation with entropy, and indeed the whole new way of thinking that surrounds statistical thermodynamics are all built-in hills that must be climbed to develop competence and confidence in the subject. For this reason, although I introduce the ensemble method at the beginning of the book, I have found it preferable to build on the related Maxwell–Boltzmann method so that novices are not confronted immediately with the conceptual difficulties of ensemble theory.

To this point, we have dealt exclusively with systems composed of independent particles and thus we have utilized the Maxwell–Boltzmann method of statistical thermodynamics. We know, however, that at a sufficiently high pressure or low temperature any gas will begin demonstrating nonideal behavior. For such real gases, and also for liquids, centralized forces arise among constituent particles owing to shorter intermolecular distances. Consequently, we are eventually confronted with systems composed of dependent rather than independent particles. Such systems mandate that we forsake the Maxwell–Boltzmann method and turn instead to a more robust computational procedure known as the Gibbs or ensemble method of statistical thermodynamics.

The Ensemble Method

We recall from Section 3.2 that an ensemble is a mental collection of a huge number of identical systems, each of which replicates macroscopically the thermodynamic system under investigation. Because such replication occurs at the macroscopic and not at the microscopic level, every member of the ensemble may be associated with a possibly different system quantum state. In essence, the independent particles required for the Maxwell–Boltzmann method are replaced with independent systems for the ensemble method. As a result, when using the latter, we inherently retain independent events proffered for statistical analyses, while accounting for the intermolecular forces needed to model real gases and liquids. In so doing, we shift our focus from a consideration of particle quantum states to a consideration of system quantum states.

As discussed previously, the utility of the Gibbs method rests on two fundamental postulates of statistical thermodynamics.

We found in the previous chapter that the molecular partition function is required to determine the thermodynamic properties of an ideal gas. To evaluate the partition function, specification of pertinent energy levels and degeneracies is necessary. Such knowledge demands that we investigate at least the rudiments of quantum mechanics, and especially those quantum concepts required for subsequent applications to statistical thermodynamics. For this reason, we concentrate in the next few chapters on the Schrödinger wave equation, whose various solutions provide the εj's and gj's needed for the eventual calculation of thermodynamic properties. Depending on your academic background, you might thus consider reviewing classical mechanics (Appendix G) and operator theory (Appendix H) in preparation for your upcoming study of quantum mechanics.

We begin this chapter with a historical review of the developments leading to the formulation of quantum mechanics, subsequently focusing on the Bohr model for atomic hydrogen and the de Broglie hypothesis for matter waves. We then introduce the Schrödinger wave equation, the basic postulates of quantum mechanics, and salient insights from these postulates germane to the development of statistical thermodynamics. We next apply the Schrödinger wave equation to the translation energy mode of an atom or molecule. This application conveniently explains both quantum states and quantum numbers, including their relation to our previous notions of microstate and macrostate. We end this chapter by discussing the Heisenberg uncertainty principle, including its utility in defining indistinguishability and symmetry conditions for multiparticle systems.

As indicated in Chapter 18, ensemble theory is especially germane when calculating thermodynamic properties for systems composed of dependent rather than independent particles. Potential applications include real gases, liquids, and polymers. In this chapter, we focus on the thermodynamic properties of nonideal gases. Our overall approach is to develop an equation of state using the grand canonical ensemble. From classical thermodynamics, equilibrium properties can always be determined by suitably operating on such equations of state. As shown later in this chapter, typical evaluations require an accurate model for the intermolecular forces underlying any macroscopic assembly. This requirement is endemic to all applications of ensemble theory, including those for liquids and polymers. As a matter of fact, by mastering the upcoming procedures necessary for the statistics of real gases, you should be prepared for many pertinent applications to other tightly-coupled thermodynamic systems.

The Behavior of Real Gases

As the density of a gas rises, its constituent particles interact more vigorously so that their characteristic intermolecular potential exercises a greater influence on macroscopic behavior. Accordingly, the gas becomes less ideal and more real; i.e., its particles eventually display greater contingency owing to enhanced intermolecular forces. This deviation from ideal behavior is reflected through a more complicated equation of state for real gases.

An equation of state, you recall, describes a functional relation among the pressure, specific volume, and temperature of a given substance.

Book's scope

Large-eddy simulations (LESs) of turbulent flows are extremely powerful techniques consisting in the elimination of scales smaller than some scale Δx by a proper low-pass filtering to enable suitable evolution equations for the large scales to be written. The latter maintain an intense spatio-temporal variability. Large-eddy simulation (LES) poses a very difficult theoretical problem of subgrid-scale modeling, that is, how to account for small-scale dynamics in the large-scale motion equations. LES is an invaluable tool for deciphering the vortical structure of turbulence, since it allows us to capture deterministically the formation and ulterior evolution of coherent vortices and structures. It also permits the prediction of numerous statistics associated with turbulence and induced mixing. LES applies to extremely general turbulent flows (isotropic, free-shear, wall-bounded, separated, rotating, stratified, compressible, chemically reacting, multiphase, magnetohydrodynamic, etc.). LES has contributed to a blooming industrial development in the aerodynamics of cars, trains, and planes; propulsion, turbo-machinery; thermal hydraulics; acoustics; and combustion. An important application lies in the possibility of simulating systems that allow turbulence control, which will be a major source of energy savings in the future. LES also has many applications in meteorology at various scales (small scales in the turbulent boundary layer, mesoscales, and synoptic planetary scales). Use of LES will soon enable us to predict the transport and mixing of pollution. LES is used in the ocean for understanding mixing due to vertical convection and stratification and also for understanding horizontal mesoscale eddies. LES should be very useful for understanding the generation of Earth's magnetic field in the turbulent outer mantle and as a tool for studying planetary and stellar dynamics.

In 1949, in an unpublished report to the U.S. Office of Naval Research, John von Neumann remarked of turbulence that

Few people today would disagree with these comments on the importance of understanding turbulence and, as implied, of its prediction. And, although the turbulence problem has still yet to be “solved,” our understanding of turbulence has significantly advanced since that time; this progress has come through a combination of theoretical studies, often ingenious experiments, and judicious numerical simulations. In addition, from this understanding, our ability to predict, or at least to model, turbulence has greatly improved; methods to predict turbulent flows using large-eddy simulation (LES) are the main focus of the present book.

The impact of von Neumann is still felt today in the prediction of turbulent flows, both in his work on numerical methods and in the people and the research he has influenced. The genesis of the method of large-eddy simulation (or possibly more appropriately, “simulation des grandes échelles”) was in the early 1960s with the research of Joe Smagorinsky. At the time, Smagorinsky was working in von Neumann's group at Princeton, developing modeling for dissipation and diffusion in numerical weather prediction.