Several aims guided me while I wrote. My first goal was to build from the familiar to the abstract and still get to entropy, conceived microscopically, in the second chapter. I sought to keep the book crisp and lean: derivations were to be succinct and simple; topics were to be those essential for physics and astronomy. From the professor's perspective, a semester is a short time, and few undergraduate curricula can devote more than a semester to thermal physics.

Modularity was another aim. Instructors' tastes vary greatly, and so I sought maximal flexibility in what to teach and when to cover it. The book's logical structure is displayed in figure P1. Chapters 1 to 3 develop topics that appear in the typical fat textbook for introductory physics but are rarely assimilated by students in that course, if the instructor even gets to the topics. Thus the book presumes only an elementary knowledge of classical mechanics and some rudimentary ideas from quantum theory, primarily the de Broglie relationship p = h/λ and the idea of energy eigenstates.

A benefit of modularity is that one can study chapter 13—the classical theory—any time after chapter 5. I placed the classical theory so far back in the book because I think students should get to use the quantum machinery of chapters 4 and 5 on some important physics before they face the development of more formalism.

If we know the temperature of a system and the values of its external parameters, how can we estimate its physical properties, such as energy, pressure, magnetic moment, and distribution of molecular velocities? The question is answered in this chapter: we derive the canonical probability distribution, learn some techniques for applying it efficiently, and work out two major examples.

Probabilities

Probabilities enter into thermal physics because the available data are insufficient to determine the individual properties of 1020 molecules. Moreover, even if such finely detailed data were available, no person or computer could cope with it. Out of necessity, one turns to a statistical analysis. A handful of data and some plausible reasoning lead to predictions whose success is nothing short of astonishing.

Before we look at what a “probability” means, we should note that probabilities always arise in a context. For example, the probability of a 4 appearing, given that I roll a die once, is 1/6. The probability of a four appearing, if I were to count the number of letters that come in my daily mail, would be quite different. Sometimes the context is made explicit; at other times, it is left implicit; but always there is a context.

In a loose sense, this chapter is about the coexistence of solids, liquids, and gases, usually taken two at a time. What physical condition must be met if coexistence is to occur? What other relations follow? The chapter provides some answers. And in its last section, it develops a classic equation—the van der Waals equation of state—that was the first to describe gas and liquid in coexistence.

Phase diagram

Figure 12.1 displays three phases of water as a function of pressure and temperature. By the word phase, one means here a system or portion of a system that is spatially homogenous and has a definite boundary. [In turn, homogeneity means that the chemical composition (including relative amounts), the crystalline structure (if any), and the mass density are uniform in space. A continuous variation produced by gravity, however, is allowed, as in the case of a finite column of air in the Earth's gravitational field.] The liquid phase is typified by a glass of water; the solid, by an ice cube; and the vapor, by the “dry” steam (that is, steam without water droplets) in the turbine of a power plant, as was illustrated in figure 3.1.

Of more interest, however, are the curves in the drawing, locations where two phases coexist. When a pond freezes over and develops a 20 centimeter layer of ice, the lower surface of the ice and the top of the remaining liquid water coexist at a temperature and pressure that lie along the leftward (and upward) tilted curve that emanates from the triple point.

By now, the classical ideal gas should be a familiar system. We can turn to the full quantum theory of an ideal gas. The applications are numerous, and some will be examined in chapter 9. The present chapter develops an effective technique for working with a quantum ideal gas.

Coping with many particles all at once

Every macroscopic system consists of a great many individual particles. Coping with all of them simultaneously, even if only statistically, can be a daunting task. A flow chart will help us to see some of our accomplishments and to see what steps lie ahead. Figure 8.1 displays such a chart on the supposition that the system consists of one species of particle only, for example, N electrons or N helium atoms (of a specific isotope).

If the forces between the particles must be included, then the analysis is difficult. Moreover, there is no general algorithm for proceeding, not even by computer. To be sure, the canonical probability distribution usually is applicable and provides estimates in principle, but the actual evaluation of the partition function, say, defies a direct approach. We have, however, successfully dealt with one such difficult situation. The Debye model for a solid incorporates the inter-particle forces, for example, those between adjacent copper atoms in a crystal. The forces ensure that the motions of adjacent atoms are correlated, and the correlation leads to sound waves, the theoretical basis of the Debye model.

All too easily, a book on thermal physics leaves the impression of a collection of applications: blackbody radiation, Debye theory, conduction electrons, van der Waals equation, and so on. Those applications are vital; they connect the theoretical and experimental worlds. Yet even more important is to come away from the book with a sense of its underlying theoretical structure. What is its equivalent of the equation F = ma in a mechanics text? Or of Maxwell's equations in a book on electromagnetism?

In this book, the key theoretical ideas are the Second Law of Thermodynamics, the canonical probability distribution, the partition function, and the chemical potential. Of these, the Second Law of Thermodynamics comes first logically and is most nearly the central organizing principle. The majority of the applications, however, found us using the latter three items: P(Ψj) = (1/Z)exp(−Ej/kT), Z, or µ. By comparison with mechanics or electromagnetism, thermal physics suffers in that its most basic principle, the Second Law of Thermodynamics, is separated by layers of secondary theory from the physical applications. Perhaps that reflects the inherent difficulty of coping with 1020 particles all at once, or perhaps the root lies in the diversity of topics that can be addressed.

Here, in brief outline, is the run of the chapter. Section 16.1 presents two experiments to introduce the topic of “critical phenomena,” as that phrase is used in physics. The next section illustrates the mathematical behavior of certain physical quantities near a critical point. Section 16.3 constructs a theoretical model: the Ising model. The following two sections develop methods for extracting predictions from the model. The last three sections draw distinctions, outline some general results, and collect the chapter's essentials.

Achieving a sound theoretical understanding of critical phenomena was the major triumph of thermal physics in the second half of the twentieth century. Thus the topic serves admirably as the culmination of the entire book.

Experiments

Liquid–vapor

We begin with an experiment that you may see as a lecture demonstration, performed either with carbon dioxide or with Freon, a liquid once used in refrigerators. A sealed vertical cylinder contains a carefully measured amount of CO2 (say) under high pressure. To begin with, the system is in thermal equilibrium at room temperature, as sketched in part (a) of figure 16.1. A distinct meniscus separates the clear liquid phase from the vapor phase above it.

Now heat the system with a hair dryer. The meniscus rises (because the liquid expands), and gentle boiling commences. Next the meniscus becomes diffuse, and then it disappears. The CO2 has become spatially homogeneous in density, as illustrated in figure 16.1 (b).

Chapter 2 examines the evolution in time of macroscopic physical systems. This study leads to the Second Law of Thermodynamics, the deepest principle in thermal physics. To describe the evolution quantitatively, the chapter introduces (and defines) the ideas of multiplicity and entropy. Their connection with temperature and energy input by heating provides the chapter's major practical equation.

Multiplicity

Simple things can pose subtle questions. A bouncing ball quickly and surely comes to rest. Why doesn't a ball at rest start to bounce? There is nothing in Newton's laws of motion that could prevent this; yet we have never seen it occur. Why? (If you are skeptical, recall that a person can jump off the floor. Similarly, a ball could—in principle—spontaneously rise from the ground, especially a ball that had just been dropped and had come to rest.)

Or let us look at a simple experiment, something that seems more “scientific.” Figure 2.1 shows the context. When the clamp is opened, the bromine diffuses almost instantly into the evacuated flask. The gas fills the two flasks about equally. The molecules seem never to rush back and all congregate in the first flask. You may say, “That's not surprising.” True, in the sense that our everyday experience tells us that the observed behavior is reasonable. But let's consider this “not surprising” phenomenon more deeply.

This chapter has several goals. One is to increase your understanding of the chemical potential. Another is to describe the changes that arise in basic thermodynamic laws when particles may enter or leave “the system.” A third goal is to study additional properties of the Helmholtz free energy, which first appeared in chapter 7. And a fourth goal is to introduce another free energy, the Gibbs free energy, and to explore its properties. Clearly, a lot is going on, but the section structure should enable you to maintain your bearings.

Generalities about an open system

In this section, we examine changes that arise in basic thermodynamic laws when particles may enter (or leave) what one calls “the system.” What might be examples of such entering or leaving? The migration of ions in an aqueous solution provides an example. So does the motion of electrons from one metal to another at a thermocouple junction. When one studies the equilibrium between argon in the gaseous phase and argon adsorbed on glass (as we did in section 7.4), one may take the adsorbed atoms to constitute “the system.” Then atoms may enter the system from the vapor phase and also leave it. The coexistence of two phases–solid, liquid, or vapor–provides a final example in which one may take a specific phase as “the system.” Atoms or molecules may enter that phase or leave it.

Chapter 1 is meant as a review, for the most part. Indeed, if you have taken a good general physics course, then much of chapters 2 and 3 will be review also. Thermal physics has some subtle aspects, however, so it is best that we recapitulate basic ideas, definitions, and relationships. We begin in section 1.1 with the ideas of heating something and of temperature.

Heating and temperature

Suppose you want to fry two eggs, sunny-side up. You turn on the electric range and put the copper-bottomed frying pan on the metal coil, which soon glows an orangish red. The eggs begin to sizzle. From a physicist's point of view, energy is being transferred by conduction from the red-hot coil through the copper-bottomed pan and into the eggs. In a microscopic description of the process, one would say that, at the surface of contact between iron coil and copper pan, the intense jiggling of the iron atoms causes the adjacent copper atoms to vibrate more rapidly about their equilibrium sites and to pass such an increase in microscopic kinetic energy along through the thickness of the pan and finally into the eggs.

Meanwhile, your English muffin is in the toaster oven. Near the oven's roof, two metal rods glow red-hot, but there is no direct contact between them and the muffin. Rather, the hot metal radiates electromagnetic waves (of a wide spectrum of frequencies but primarily in the infrared region); those waves travel 10 centimeters through air to the muffin; and the muffin absorbs the electromagnetic waves and acquires their energy. The muffin is being heated by radiation.

Two paragraphs set the scene for the entire chapter. Recall that the molecules of a gas are in continual, irregular motion. Individual molecules possess both energy and momentum, and they transport those quantities with them. What is the net transport of such quantities? In thermal equilibrium, it is zero. If, however, the system has macroscopic spatial variations in temperature, being hotter in some places than in others, then net transport of energy may arise (even from irregular molecular motion). Or if the locally averaged velocity is nonzero and varies spatially (as it does in fluid flow through a pipe), then net transport of momentum may arise.

Our strategy is first to examine irregular molecular motion in its own right, then to study transport of momentum, and finally to investigate the transport of energy.

Mean free path

Our context is a classical gas. The conditions of temperature and number density are similar to those of air under room conditions. We acknowledge forces between molecules, but we simplify to thinking of the molecules as tiny hard spheres, so that they exert mutually repulsive forces during collisions. Generalization will come later.

When any given molecule wanders among its fellow molecules, its path is a random sequence of long and short “free” paths between collisions. Figure 15.1 illustrates the irregular, broken path. There is a meaningful average distance that the molecule travels between collisions.

Introduction

Numerical experiments are making important contributions to the study of turbulence in the interstellar medium (ISM). Since any numerical simulation is restricted in the range of spatial and temporal scales which it can describe, it is important to develop a prescription for treating the effects of turbulence at the smallest scales, which are generally omitted from this range. Although very little energy resides at the smallest scales, the small scale motions dramatically increase momentum and magnetic flux transport in the ISM, and can also produce rapid thermal and chemical mixing. The most common way to account for these subgridscale effects is to simply assume that the viscosity, electrical resistivity, and other transport coefficients are much larger than their molecular values. The difficult problem of justifying this approach and calculating the so-called eddy diffusivities has received more attention in the atmospheric and stellar turbulence communities than it has, so far, among interstellar turbulence theorists.

I describe the multivariate technique of Principal Component Analysis and its application to spectroscopic imaging data of the molecular interstellar medium. The technique identifies differences in line profiles with respect to the noise level at various scales. It is assumed that such differences arise from fluctuations within turbulent flows. From the resultant eigenvectors and eigenimages, a size line width relationship, (δv ∼ τα), can be constructed which describes the relationship between the magnitude of velocity fluctuations and the angular scale over which these occur for a given region. From a sample of selected molecular regions in the outer Galaxy, I find the power law exponent varies from 0.4 to 0.7. Thus, the turbulent flows within molecular regions of the Galaxy do not follow the Kolmogorov-Obukhov relation for incompressible turbulence. Implications of these results are discussed with respect to the injection and dissipation of kinetic energy in molecular regions.

Introduction

In the early, pioneering days of millimeter wave astronomy, the presence of turbulent flows within molecular regions of the Galaxy was inferred from the supersonic line widths of CO spectra. Since that time, telescope and detector technology has advanced such that one can now routinely construct detailed images of molecular emission from which the properties of interstellar turbulence can, in principle, be derived. In practice, statistical descriptions of the observations are required to fully exploit the available information.

Introduction

The density and velocity structure within a molecular cloud is a remnant of its formation environment and the starting point for the creation of stars. It determines how deeply radiation can propagate through the cloud, and is a critical parameter for understanding the evolution of the ISM. How is it best described?

Beginning with Larson (1981), correlations between cloud properties such as linewidth and size have been fit by power laws. Since a power law does not have a characteristic scale, the implication is that clouds are scale-free and self-similar. This has led to statements in the literature that clouds are best described as fractals (e.g. Falgarone, Phillips, & Walker 1991; Elmegreen 1997). On the other hand, other recent studies (Larson 1995; Simon 1997; Goodman et al. 1998; Blitz & Williams 1997) suggest that there are characteristic size and velocity scales in star-forming regions.

Interstellar Turbulence, the second conference organized by the Guillermo Haro International Program on Advanced A strophysical Research, was an excellent forum to review and discuss one of the most intriguing features of cosmic and terrestrial fluids. Turbulence is universal and mysterious, and remains one of the major unsolved problems in physics and astrophysics. It is present in all terrestrial and astrophysical environments: close to our telescopes, it blurs and distorts our view of the skies, and in the interstellar and intergalactic media, somehow, it creates fluctuations and redistributes angular momentum, leading to star formation and large scale structure.

The Guillermo Haro Program was created in 1995 at the Instituto Nacional de Astrofísica, Optica y Electrónica (INAOE), and is named in honor of its founder, the remarkable astronomer-lawyer Guillermo Haro. This second conference was aimed at revising our conceptions on the properties of turbulence, and at summarizing the present status in observational, theoretical, and computational research in interstellar turbulence. It was held in Puebla, México, at the Benemérita Universidad Autónoma de Puebla, during the week of January 12th to 16th, 1998. There were 130 participants, from four continents, and a large fraction of them were very young scientists. The program covered a wide variety of topics, ranging from atmospheric and interstellar turbulent flows, to magnetic fields and cosmic ray transportation, and energy dissipation, fragmentation and star formation.

Introduction

Over the last several decades it has become clear that the dynamics and evolution of star-forming interstellar clouds are difficult to explain without magnetic effects. A principal problem involves support of dense clouds against their own gravity. In general, such clouds are observed to be in approximate virial equilibrium between gravity and internal motions. Seemingly, therefore, they should be stable against collapse. However, observed line widths are almost invariably much greater than the sound speed. Therefore the internal motions that support the clouds are highly supersonic, and simple estimates indicate that shock-induced dissipation of mechanical energy should occur on about the free-fall time. In such a case, non-magnetic turbulence offers no effective support for the clouds (unless, of course, it can somehow be continuously regenerated).