This book is based on a course which I have taught over many years to graduate students in several physics departments. Students have been mainly candidates for physics degrees but have included a scattering of people from other departments including chemical engineering, materials science and chemistry. I take a “reductionist” view, that implicitly assumes that the basic program of physics of complex systems is to connect observed phenomena to fundamental physical laws as represented at the molecular level by Newtonian mechanics or quantum mechanics. While this program has historically motivated workers in statistical physics for more than a century, it is no longer universally regarded as central by all distinguished users of statistical mechanics some of whom emphasize the phenomenological role of statistical methods in organizing data at macroscopic length and time scales with only qualitative, and often only passing, reference to the underlying microscopic physics. While some very useful methods and insights have resulted from such approaches, they generally tend to have little quantitative predictive power. Further, the recent advances in first principles quantum mechanical methods have put the program of predictive quantitative methods based on first principles within reach for a broader range of systems. Thus a text which emphasizes connections to these first principles can be useful.

The level here is similar to that of popular books such as those by Landau and Lifshitz, Huang and Reichl. The aim is to provide a basic understanding of the fundamentals and some pivotal applications in the brief space of a year.

General discussion

In general, by a hydrodynamic description of a many body fluid we mean a description valid at long wavelengths and low frequencies and which is based on closure of the local conservation laws of the fluid by use of a linear relation between fluxes and the gradients of densities. The coefficients of the linear relation are transport coefficients and they are phenomenological parameters of the hydrodynamic theory, calculable in principle from a theory describing the system at shorter length and time scales. The resulting hydrodynamic theory is generally a set of nonlinear partial differential equations of which the Navier–Stokes equations for the hydrodynamics of a simple fluid are a familiar example.

The reason that hydrodynamic theories accurately describe slow motions on large length scales is that global conservation laws link long distances to long times. Physically, for example, conservation of mass results in a diffusion equation in which the distance which particles diffuse increases with the square root of the time (see Problem 10.1). Although this link guarantees that some of the slow variables of the system are described by the hydrodynamic equations, it does not ensure that all of the slow variables can be so described. Near critical points associated with second order phase transitions, there are very slow changes in the fluid which are not described by the conservation laws of hydrodynamics, but which arise because of the very slow development and decay of large, almost stable regions looking like one of the (two or more) phases between which the system is slowly fluctuating.

As discussed in Chapter 6, when the temperature is lowered in a classical liquid until the thermal wavelength becomes comparable to the interparticle spacing, then the semiclassical approximation is no longer adequate and quantum effects must be considered. In practice, most classical liquids freeze at all positive pressures before this temperature is reached. The exceptions are the helium liquids (3He and 4He) for which the quantum effects are large enough to prevent freezing as the temperature is lowered while the liquid is kept in equilibrium with its vapor. Conveniently, 3He is a Fermi system and 4He is a Bose system. In these systems as well, phase transitions occur at low enough temperatures. But quantum effects are significant even before these transitions occur. Another system which may for some purposes be regarded as an isotropic liquid with large quantum effects is the collection of electrons in (at least some) metals. Here too, a phase transition to the superconducting state intervenes in many cases at low enough temperatures. Finally neutron stars may contain regions in which neutrons are in a liquid state with large quantum effects and white dwarf stars contain a degenerate electron gas which can be regarded as a quantum liquid. In general, the reason that quantum liquids are so hard to observe is that interactions tend to result in symmetry breaking phase transitions in high density systems at temperatures low enough to permit quantum effects to be observed.

For systems which obey quantum mechanics, the formulation of the problem of treating large numbers of particles is, of course, somewhat different than it is for classical systems. The microscopic description of the system is provided by a wave function which (in the absence of spin) is a function of the classical coordinates {qi}. The mathematical model is provided by a Hamiltonian operator H which is often obtained from the corresponding classical Hamiltonian by the replacement pi → (ħ/i)(∂/∂qi). In other cases the form of the Hamiltonian operator is simply postulated. The microscopic dynamics are provided by the Schrödinger equation i ħ(∂Ψ/∂t) = HΨ which requires as initial condition the knowledge of the wave function Ψ({qi}, t) at some initial time t0. (Boundary conditions on Ψ({qi}, t) must be specified as part of the description of the model as well.) The results of experiments in quantum mechanics are characterized by operators, usually obtained, like the Hamiltonian, from their classical forms and termed observables. Operators associated with observables must be Hermitian. In general, the various operators corresponding to observables do not commute with one another. It is possible to find sets of commuting operators whose mutual eigenstates span the Hilbert space in which the wave function is confined by the Schrödinger equation and the boundary conditions. A set of such (time independent) eigenstates, termed ψν(q), is a basis for the Hilbert space.

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.