In this chapter we will describe how statistical mechanics can be applied to obtain some important results in astrophysics. As an application of classical statistical mechanics we will discuss the Saha ionization formula which plays a role in determining the surface temperature of a star and which will be shown to follow from an analysis of chemical reactions involving ionized particles using statistical mechanics.

We have already emphasized in the last chapter that quantum mechanics has profound implications for the equations of state and, in particular, the stability of matter. In this chapter we will illustrate this effect by considering the collapse of stellar objects. A prominent example is that of white dwarf stars which are stabilized by the Pauli exclusion principle. Understanding white dwarf stars will involve Fermi-Dirac statistics. We will also briefly discuss the fact that neutron stars contain more neutrons than protons and will show that this follows from the analysis of a particular nuclear reaction process treated as a chemical reaction.

In order to present these examples in a suitable setting we start by reviewing a few basic facts about the physics of stellar evolution and we outline the principles that are used to model these objects. This is followed by a qualitative account of stellar evolution. With this background in place the specific examples are considered. We then close this chapter with a qualitative discussion of the cosmic background radiation.

Statistical mechanics is a fundamental part of theoretical physics. Not only does it provide the basic tools for analyzing the behavior of complex systems in thermal equilibrium, but also hints at, and is fully compatible with, quantum mechanics as the theory underlying the laws of nature. In the process one encounters such complex emergent phenomena as phase transitions, superfluidity, and superconductivity which are highly non-trivial consequences of the microscopic dynamics. At the same time statistical mechanics poses conceptual problems such as how irreversibilty can appear from an underlying microscopic system governed by reversible laws.

Historically, statistical mechanics grew out of classical thermodynamics with the aim of providing a dynamical foundation for this phenomenological theory. It thus deals with many-body problems starting from a microscopic model which is typically described by a simple Hamiltonian. The power of statistical mechanics lies in both its simplicity and universality. Indeed the same concept can be applied to a wide variety of systems both classical and quantum mechanical. These include non-interacting and interacting gases, chemical interactions, paramagnetic and spin systems, astrophysics, and solids. On the other hand statistical mechanics brings together a variety of different tools and methods used in theoretical physics, chemistry, and mathematics. Indeed while the basic concepts are easily explained in simple terms a quantitative analysis will quickly involve sophisticated methods.

The purpose of this book is twofold: to provide a concise and self-contained introduction to the key concepts of statistical mechanics and to present the important results from a modern perspective. The book is introductory in character, and should be accessible to advanced undergraduate and graduate students in physics, chemistry, and mathematics.

It is time now to review the progress we have made so far. Our starting point was the fundamental atomic nature of matter. We also assumed that interactions between individual atoms and molecules are governed by the laws of mechanics, either classical or quantum depending on the particular circumstances. In the first chapter we developed a simple qualitative picture of the way molecules interact in a complex system. This qualitative picture allowed us to describe classical thermodynamics. In particular we were able to introduce the key concepts of equilibrium, temperature, entropy, and we were able to point out that complex systems in equilibrium can be well described with only a very small number of state variables. Given that matter is made of very large numbers of independent atoms or molecules this is an extraordinary result.

In the second chapter we began the process of formalizing the qualitative link from mechanics to thermodynamics. The formal development starts of course with mechanics. Mechanics on its own, however, is not enough, as it does not contain the concept of thermal equilibrium. The solution we presented was to define thermal equilibrium probabilistically, and the theory which results is statistical mechanics. This solution requires a fundamentally new idea which is not present in mechanics. Once this idea is accepted, the further development of the subject is straightforward if perhaps technically challenging.

Statistical mechanics is a very successful physical theory. In this book, we have applied it to a variety of systems including non-interacting and interacting gases, paramagnetic and spin systems, quantum systems with both Bose and Fermi statistics, astrophysics, helium superfluids and solids.

The goal of this chapter will be to briefly describe the remarkable properties of helium at low temperatures. After stating some of these properties we will see how they can be understood in terms of the phenomenon of Bose–Einstein condensation described in Chapter 7. We will give the main argument in two different formulations, once using the quasi-particle method of Bogoliubov, and then using a Green function approach.

We start with some experimental facts. Helium is a remarkable element. It was predicted to exist from observations of the Sun before it was found on Earth. It is the only element which remains a liquid at zero temperature and atmospheric pressure. Experimentally the phase diagram of 4He is shown in Figure 10.1. Helium I is a normal fluid and has a normal gas-liquid critical point. Helium II is a mixture of a normal fluid and a superfluid. The superfluid is characterized by the vanishing of its viscosity. Helium I and helium II are separated by a line known as the λ-transition line. At Tλ= 2.18 K, Pλ= 2.29 Pa, helium I, helium II, and helium gas coexist. The specific heat of liquid helium along the vapor transition line forms a logarithmic discontinuity shown in Figure 10.2. The form of this diagram resembles the Greek letter λ and is the reason for calling the transition a λ-transition.

The lack of viscosity of helium II leads to some remarkable experimental consequences, one of which we briefly describe. Let two containers A and B be linked by a thin capillary through which only a fluid with zero (or very low) viscosity can pass freely.

In our discussion so far we described the canonical ensemble of N identical particles or molecules. We found that from the canonical partition sum we can recover the free energy which is one of the thermodynamic potentials introduced in the first chapter. A natural question is whether there are other approaches to statistical mechanics which are in turn related to other state functions such as the entropy. In this chapter we will see that this is indeed the case. We will end up with the complete picture of how different probability measures in statistical mechanics are related to the various potentials in thermodynamics. In the process we will also uncover a simple statistical interpretation of the entropy function in thermodynamics.

The grand canonical ensemble

In the previous chapter we considered a statistical system with a fixed number N of identical molecules. We have argued that although the energy E of the system is a constant its precise value is not known. Hence we considered the probability P(E) that the system had energy E and used it to relate the average value of the energy of the system (involving the microscopic properties of the system) to the macroscopic thermodynamic variable U, the internal energy. In this section we will generalize this approach to include a variable number of molecules, Figure 3.1. We note that the number of particles N in a volume, although a constant, is similarly not precisely known.