Fermi–Dirac Statistics

The main application of Fermi–Dirac Statistics is to calculate the properties of electrons. This chapter explains how the properties of fermions account for the behavior of metals. The Fermi energy is introduced and shown to correspond to a very high temperature, so that most properties can be obtained from low-temperature expansions. Both discrete and continuous densities of states are discussed. The Sommerfeld expansion is derived explicitly. The low-temperature specific heat and compressibility are derived. The most important fermions are electrons, and understanding the properties of electrons is central to understanding the properties of all materials. In this chapter we will study the ideal Fermi gas, which turns out to explain many of the properties of electrons in metals.

Phase Transitions

Phase transitions are introduced using the van der Waals gas as an example. The equations of state are derived from the Helmholtz free energy of the ideal gas. The behavior of this model is analyzed, and an instability leads to a liquid-gas phase transition. The Maxwell construction for the pressure at which a phase transition occurs is derived. The effect of phase transition on the Gibbs free energy and Helmholtz free energy is shown. Latent heat is defined, and the Clausius–Clapeyron equation is derived. Gibbs' phase rule is derived and illustrated.

Thermodynamic Processes

This chapter begins by defining terms critical to understanding thermodynamics: reversible, irreversible, and quasi-static. Because heat engines are central to thermodynamic principles, they are described in detail, along with their operation as refrigerators and heat pumps. Various expressions of efficiency for such engines lead to alternative expressions of the second law of thermodynamics. A Carnot cycle is discussed in detail as an example of an idealized heat engine with optimum efficiency. A special case, called negative temperatures, where temperatures actually exceed infinity, provides further insights. In this chapter we will discuss thermodynamic processes, which concern the consequences of thermodynamics for things that happen in the real world.

Perturbations of Thermodynamic State Functions

Because small changes in thermodynamic quantities will play a central role in much of the development of thermodynamics, the key concepts are introduced in this short chapter. The First Law (conservation of energy) can be expressed simply in terms of infinitesimal quantities: a small change in the energy of a system is equal to the heat added plus the work done on the system. The theories of statistical mechanics and thermodynamics deal with the same physical phenomena. Exact and inexact differentials are defined, along with the important concept of an integrating factor that relates them. The useful equation relating small changes in heat to corresponding changes in entropy is derived.

Continuous Random Numbers

The theory of probability developed in Chapter 3 for discrete random variables is extended to probability distributions, in order to treat the continuous momentum variables. The Dirac delta function is introduced as a convenient tool to transform continuous random variables, in analogy with the use of the Kronecker delta for discrete random variables. The properties of the Dirac delta function that are needed in statistical mechanics are presented and explained. The addition of two continuous random numbers is given as a simple example. An application of Bayesian probability is given to illustrate its significance. However, the components of the momenta of the particles in an ideal gas are continuous variables.

Phase Transitions and the Ising Model

Chapter 17 presented one example of a phase transition, the van der Waals gas. This chapter provides another, the Ising model, a widely studied model of phase transitions. We first give the solution for the Ising chain (one-dimensional model), including the introduction of the transfer matrix method. Higher dimensions are treated in the Mean Field Approximation (MFA), which is also extended to Landau theory. The Ising model is deceptively simple. It can be defined in a few words, but it displays astonishingly rich behavior. It originated as a model of ferromagnetism in which the magnetic moments were localized on lattice sites and had only two allowed values.

Irreversibility

The phenomenon of irreversibility is explained on the basis of an analysis by H. L. Frisch. The history of the debate over irreversibility is briefly discussed, including Boltzmann’s H-theorem, Zermelo's Wiederkehreinwand, Poincaré recurrences, Loschmidt's Umkehreinwand and Liouville’s theorem. The derivation of irreversible behavior for the ideal gas position distribution is carried out explicitly. Using this derivation, the Wiederkehreinwand and the Umkehreinwand are revisited and explained. The first thing we must establish is the meaning of the term ‘irreversibility’. This is not quite as trivial as it might seem. The irreversible behavior I will try to explain is that which is observed. Every day we see that time runs in only one direction in the real world,.

Refining the Definition of Entropy

If a macroscopic system as ever been in thermal contact with another macroscopic system, the width of the energy distribution is not zero. This is in contrast to the approximation made in Chapter 7 that the energy dependence of the entropy is given by a delta function. The width is very narrow (proportional to the inverse square root of the number of particles), but this leads to small errors in the predictions of the entropy. Massieu functions are used to derive the canonical entropy because they allow the extension to non-monotonic densities of states, which will be needed in later chapters. The grand canonical entropy is defined similarly. The canonical entropy and the grand canonical entropy of the classical ideal gas are calculated as examples.

Thermodynamic Identities

Many of the calculations in thermodynamics concern the effects of small changes. To carry out such calculations, we often need to evaluate first and second partial derivatives of some thermodynamic quantities with respect to other thermodynamic quantities. Although there are many such partial second derivatives, they are related by thermodynamic identities. This chapter explains the most straightforward way of deriving the needed thermodynamic identities. After explaining the derivation of Maxwell relations and how to find the right one for any given problem, Jacobian methods are introduced, with an accolade to their simplicity and utility. Several examples of the derivation of thermodynamic identities are given, along with a systematic guide for solving general problems.

The Postulates and Laws of Thermodynamics

The points of view about the behavior of macroscopic systems arising from thermodynamics and statistical mechanics are compared and contrasted. The concept of a state function is explained, and the postulates of thermodynamics, which describe fundamental properties of the entropy, are presented. Some are essential postulates, which are always valid, while others, called optional postulates, are very useful when they do apply. The Laws of Thermodynamics are derived from these postulates. With this chapter we begin the formal study of thermodynamics, which represents a detour from the development of the foundations of classical statistical mechanics that we began in Part I.