INDEX
PREFACE
This book is based on lectures prepared for a year-long course of the same title, designed as a central part of a new curriculum in applied physics. It was offered for the first time in the academic year 1971–1972 and has been repeated annually since.
There is a good deal of doubt about precisely what applied physics is, but a reasonably clear picture has emerged of how an applied physics student ought to be educated, at least here at Caltech. There should be a rigorous education in basic physics and related sciences, but one centered around the macroscopic world, from the atom up rather than from the nucleus down: that is, physics, with emphasis on those areas where the fruits of research are likely to be applicable elsewhere. The course from which this book arose was designed to be consistent with this concept.
The course level was designed for first-year graduate students in applied physics, but in practice it has turned out to have a much wider appeal. The classroom is shared by undergraduates (seniors and an occasional junior) in physics and applied physics, plus graduate students in applied physics, chemistry, geology, engineering, and applied mathematics. All are assumed to have a reasonable undergraduate background in mathematics, a course including electricity and magnetism, and at least a little quantum mechanics.
The basic outline of the book is simple. After a chapter designed to start everyone off at the same level in thermodynamics and statistical mechanics, we have the basic states—gases, solids, and liquids—a few special cases, and, finally, phase transitions. What we seek in each case is a feeling for the essential nature of the stuff, and how one goes about studying it. In general, the book should help give the student an idea of the language and ideas that are reasonably current in fields other than the one in which he or she will specialize. In short, this is an unusual beast: an advanced survey course.
All the problems that appear at the ends of the chapters were used as either homework or examination problems during the first three years in which the course was taught. Some are exercises in applying the material covered in the text, but many are designed to uncover or illuminate various points that arise, and are actually an integral part of the course. Such exercises are usually referred to at appropriate places in the text.
There is an annotated bibliography at the end of each chapter. The bibliographies are by no means meant to be comprehensive surveys even of the textbooks, much less of the research literature of each field. Instead they are meant to guide the student a bit deeper if he wishes to go on, and they also serve to list all the material consulted in preparing the lectures and this book. There are no footnotes to references in the text.
The history of science is used in a number of places in this book, usually to put certain ideas in perspective in one way or another. However, it serves another purpose, too: the study of physics is essentially a humanistic enterprise. Much of its fascination lies in the fact that these mighty feats of the intellect were performed by human beings, just like you and me. I see no reason why we should ever try to forget that, even in an advanced physics course. Physics, I think, should never be taught from a historical point of view—the result can only be confusion or bad history—but neither should we ignore our history. Let me hasten to acknowledge the source of the history found in these pages: Dr. Judith Goodstein, the fruits of whose doctoral thesis and other research have insinuated themselves into many places in the text.
Parts of the manuscript in various stages of preparation have been read and criticized by some of my colleagues and students, to whom I am deeply grateful. Among these I would like especially to thank Jeffrey Greif, Professors T. C. McGill, C. N. Pings and H. E. Stanley, David Palmer, John Dick, Run-Han Wang, and finally Deepak Dhar, a student who contributed the steps from Eq. (4.5.20) to Eq. (4.5.24) in response to a homework assignment. I am indebted also to Professor Donald Langenberg, who helped to teach the course the first time it was offered. The manuscript was typed with great skill and patience, principally by Mae Ramirez and Ann Freeman. Needless to say, all errors are the responsibility of the author alone.
DAVID L. GOODSTEIN
Pasadena, California
April 5, 1974
ONE
THERMODYNAMICS AND STATISTICAL MECHANICS
1.1INTRODUCTION: THERMODYNAMICS AND STATISTICAL MECHANICS OF THE PERFECT GAS
Ludwig Boltzmann, who spent much of his life studying statistical mechanics, died in 1906, by his own hand. Paul Ehrenfest, carrying on the work, died similarly in 1933. Now it is our turn to study statistical mechanics.
Perhaps it will be wise to approach the subject cautiously. We will begin by considering the simplest meaningful example, the perfect gas, in order to get the central concepts sorted out. In Chap. 2 we will return to complete the solution of that problem, and the results will provide the foundation of much of the rest of the book.
The quantum mechanical solution for the energy levels of a particle in a box (with periodic boundary conditions) is
where m 
is Planck's constant, and q (which we shall call the wave vector) has three components, x, y, and z, given by
where
and
L is the dimension of the box, whose volume is L 
are each required to specify the state, so that there are a number of states for each energy of the single particle.
The perfect gas is a large number of particles in the same box, each of them independently obeying Eqs. (1.1.1) to (1-1-4). The particles occupy no volume, have no internal motions, such as vibration or rotation, and, for the time being, no spin. What makes the gas perfect is that the states and energies of each particle are unaffected by the presence of the other particles, so that there are no potential energies in Eq. (1.1.1). In other words, the particles are noninteracting. However, the perfect gas, as we shall use it, really requires us to make an additional, contradictory assumption: we shall assume that the particles can exchange energy with one another, even though they do not interact. We can, if we wish, imagine that the walls somehow help to mediate this exchange, but the mechanism actually does not matter much as long as the questions we ask concern the possible states of the many-particle system, not how the system contrives to get from one state to another.
From the point of view of quantum mechanics, there are no mysteries left in the system under consideration; the problem of the possible states of the system is completely solved (although some details are left to add on later). Yet we are not prepared to answer the kind of questions that one wishes to ask about a gas, such as: If it is held at a certain temperature, what will its pressure be? The relationship between these quantities is called the equation of state. To answer such a question—in fact, to understand the relation between temperature and pressure on the one hand and our quantum mechanical solution on the other—we must bring to bear the whole apparatus of statistical mechanics and thermodynamics.
This we shall do and, in the course of so doing, try to develop some understanding of entropy, irreversibility, and equilibrium. Let us outline the general ideas briefly in this section, then return for a more detailed treatment.
for each particle such that the total energy comes out right. We have already seen that even a single particle generally has a number of possible states of the same energy; with 10²³ particles, the number of possible quantum states of the set of particles that add up to the same energy can become astronomical. How does the system decide which of these states to choose?
The answer depends, in general, on details that we have not yet specified: What is the past history—that is, how was the energy injected into the box? And how does the system change from one state to another—that is, what is the nature of the interactions? Without knowing these details, there is no way, even in principle, to answer the question.
At this point we make two suppositions that form the basis of statistical mechanics.
1. If we wait long enough, the initial conditions become irrelevant. This means that whatever the mechanism for changing state, however the particles are able to redistribute energy and momentum among themselves, all memory of how the system started out must eventually get washed away by the multiplicity of possible events. When a system reaches this condition, it is said to be in equilibrium.
2. For a system in equilibrium, all possible quantum states are equally likely. This second statement sounds like the absence of an assumption—we do not assume that any particular kind of state is in any way preferred. It means, however, that a state in which all the particles have roughly the same energy has exactly the same probability as one in which most of the particles are nearly dead, and one particle goes buzzing madly about with most of the energy of the whole system. Would we not be better off assuming some more reasonable kind of behavior?
The fact is that our assumptions do lead to sensible behavior. The reason is that although the individual states are equally likely, the number of states with energy more or less fairly shared out among the particles is enormous compared to the number in which a single particle takes nearly all the energy. The probability of finding approximately a given situation in the box is proportional to the number of states that approximate that situation.
The two assumptions we have made should seem sensible; in fact, we have apparently assumed as little as we possibly can. Yet they will allow us to bridge the gap between the quantum mechanical solutions that give the physically possible microscopic states of the system and the thermodynamic questions we wish to ask about it. We shall have to learn some new language, and especially learn how to distinguish and count quantum states of many-particle systems, but no further fundamental assumptions will be necessary.
Let us defer for the moment the difficult problem of how to count possible states and pretend instead that we have already done so. We have N particles in a box of volume V = L³, with total energy E, and find that there are Γ possible states of the system. The entropy of the system, S, is then defined by
where k is Boltzmann's constant
k = 1.38 × 10−16 erg per degree Kelvin
Thus, if we know Γ, we know S, and Γ is known in principle if we know N, V, and E, and know in addition that the system is in equilibrium. It follows that, in equilibrium, S may be thought of as a definite function of E, N, and V,
S = S(E, N, V)
Furthermore, since S is just a way of expressing the number of choices the system has, it should be evident that S will always increase if we increase E, keeping N and V constant; given more energy, the system will always have more ways to divide it. Being thus monotonic, the function can be inverted
E = E(S, N, V)
or if changes occur,
The coefficients of dS, dV, and dN in Eq. (1.1.6) play special roles in thermodynamics. They are, respectively, the temperature
the negative of the pressure
and the chemical potential
These are merely formal definitions. What we have now to argue is that, for example, the quantity T in Eq. (1.1.7) behaves the way a temperature ought to behave.
How do we expect a temperature to behave? There are two requirements. One is merely a question of units, and we have already taken care of that by giving the constant k a numerical value; T will come out in degrees Kelvin. The other, more fundamental point is its role in determining whether two systems are in equilibrium with each other. In order to predict whether anything will happen if we put two systems in contact (barring deformation, chemical reactions, etc.), we need only know their temperatures. If their temperatures are equal, contact is superfluous; nothing will happen. If we separate them again, we will find that each has the same energy it started with.
Let us see if T defined in Eq. (1.1.7) performs in this way. We start with two systems of perfect gas, each with some E, N, V, each internally in equilibrium, so that it has an S and a T; use subscripts 1 and 2 for the two boxes. We establish thermal contact between the two in such a way that the N's and V's remain fixed, but energy is free to flow between the boxes. The question we ask is: When contact is broken, will we find that each box has the same energy it started with?
During the time that the two systems are in contact, the combined system fluctuates about among all the states that are allowed by the physical circumstances. We might imagine that at the instant in which contact is broken, the combined system is in some particular quantum state that involves some definite energy in box 1 and the rest in box 2; when we investigate later, these are the energies we will find. The job, then, is to predict the quantum state of the combined system at the instant contact is broken, but that, of course, is impossible. Our fundamental postulate is simply that all states are equally likely at any instant, so that we have no basis at all for predicting the state.
The precise quantum state is obviously more than we need to know in any case—it is the distribution of energy between the two boxes that we are interested in. That factor is also impossible to predict exactly, but we can make progress if we become a bit less particular and ask instead: About how much energy is each box likely to have? Obviously, the larger the number of states of the combined system that leave approximately a certain energy in each box, the more likely it is that we will catch the boxes with those energies.
When the boxes are separate, either before or after contact, the total number of available choices of the combined system is
It follows from Eq. (1.1.5) that the total entropy of the system is
Now suppose that, while contact exists, energy flows from box 1 to box 2. This flow has the effect of decreasing Γ1 and increasing Γ2. By our argument, we are likely to find that it has occurred if the net result is to have increased the total number of available states Γ1Γ2, or, equivalently, the sum S1 + S2. Obviously, the condition that no net energy flow be the most likely circumstance is just that the energy had already been distributed in such a way that Γ1Γ2, or S1 + S2, was a maximum. In this case, we are more likely to find the energy in each box approximately unchanged than to find that energy flowed in either direction.
The differences between conditions in the boxes before contact is established and after it is broken are given by
from Eqs. (1.1.6) to (1-1-9), with the N's and V's fixed, and
since energy is conserved overall. Thus,
If S1 + S2 was already a maximum, then, for whatever small changes do take place, the total will be stationary,
so that Eq. (1.1.15) reduces to
which is the desired result.
It is easy to show by analogous arguments that if the individual volumes are free to change, the pressures must be equal in equilibrium, and that if the boxes can be exchange particles, the chemical potentials must be equal. We shall, however, defer formal proof of these statements to Secs. 1.2f and 1.2g, respectively.
We are now in a position to sketch a possible procedure for answering the prototype question suggested earlier: At a given temperature, what will the pressure be? Given the quantities E, N, V for a box of perfect gas, we count the possible states to compute S. Knowing E(S, V, N), we can then find
and finally arrive at P(T, V, N) by eliminating S between Eqs. (1.1.18) and (1-1-19). That is not the procedure we shall actually follow—there will be more convenient ways of doing the problem—but the very argument that that procedure could, in principle, be followed itself plays an important role. It is really the logical underpinning of everything we shall do in this chapter. For example, Eqs. (1.1.6) to (1-1-9) may be written together:
Our arguments have told us that not only is this equation valid, and the meanings of the quantities in it, but also that it is integrable; that is, there exists a function E(S, V, N) for a system in equilibrium. All of equilibrium thermodynamics is an elaboration of the consequences of those statements.
In the course of this discussion we have ignored a number of fundamental questions. For example, let us return to the arguments that led to Eq. (1.1.7). As we can see from the argument, even if the temperatures were equal, contact was not at all superfluous. It had an important effect: we lost track of the exact amount of energy in each box. This realization raises two important problems for us. The first is the question: How badly have we lost track of the energy? In other words, how much uncertainty has been introduced? The second is that whatever previous operations put the original amounts of energy into the two boxes, they must have been subject to the same kinds of uncertainties: we never actually knew exactly how much energy was in the boxes to begin with. How does that affect our earlier arguments? Stated differently: Can we reapply our arguments to the box now that we have lost track of its exact energy?
The answer to the first question is basically that the uncertainties introduced into the energies are negligibly, even absurdly, small. This is a quantitative effect, which arises from the large numbers of particles found in macroscopic systems, and is generally true only if the system is macroscopic. We cannot yet prove this fact, since we have not yet learned how to count states, but we shall return to this point later and compute how big the uncertainties (in the energy and other thermodynamic quantities as well) actually are when we discuss thermodynamic fluctuations in Sec. 1.3f. It turns out, however, that in our example, if the temperatures in the two boxes were equal to start with, the number of possible states with energies very close to the original distribution is not only larger than any other possibility, it is also vastly greater than all other possibilities combined. Consequently, the probability of catching the combined system in any other kind of state is very nearly zero. It is due to this remarkable fact that statistical mechanics works.
, where τ is the lifetime. τ may typically be estimated, say, by the time between molecular collisions, or whatever process leads to changes of quantum state. We cannot imagine our box of gas to have had an energy any more definite than δE. δE 
. The fact remains, however, that we must always expect to find a quantum uncertainty in the number of states available to an isolated system.
Having said all this, the point is not as important as it seems; there is actually less than meets the eye. It is true that there are both quantum and thermodynamic uncertainties in the energy of any system, and it is also true that the number of states available to the system is not as exact a concept as it first appeared to be. However, that number of states, for a macroscopic system, turns out to be such a large number that we can make very substantial mistakes in counting without introducing very much error into its logarithm, which is all we are interested in. For example, suppose that we had Γ ~ 10¹⁰⁰. Then even a mistake of a factor of ten in counting Γ introduces an error of only 1% in log Γ, the entropy, which is what we are after, since S = k log (10¹⁰⁰ or 10¹⁰¹) = (100 or 101) log 10. In real systems Γ is more typically of order 10N, where N is the number of particles, so even an error of a factor of N 
and the error in the logarithm is immeasurably small.
The concept that an isolated system has a definite energy and a corresponding definite number of states is thus still a useful and tenable one, but we must understand definite
to mean something a bit less than exact.
Unfortunately, the indeterminacy we are speaking of makes it even more difficult to formulate a way to count the number of states available to an isolated system. However, there is an alternative description that we shall find very useful, and it is closely connected to manipulations of the kind we have been discussing. Instead of imagining a system isolated with fixed energy, we can think of our sample as being held at a constant temperature—for example, by repeatedly connecting it to a second box that is so much larger that its temperature is unaffected by our little sample. In this case, the energy of our sample is not fixed but instead fluctuates about in some narrow range. In order to handle this situation, instead of knowing the number of states at any fixed energy, it will be more convenient to know the number of states per unit range of energies—what we shall call the density of states. Then all we need to know is the density of states as a function of energy, a quantity that is not subject to quantum indeterminacy. This realization suggests the procedure we shall actually adopt. We shall imagine a large, isolated system, of which our sample is a small part (or subsystem). The system will have some energy, and we can make use of the fact that it has, as a result, some number of states, but we will never calculate what that number is. The sample, on the other hand, has a fixed temperature rather than a fixed energy when it is in equilibrium, and we will wind up describing its properties in terms of its density of states. This does not mean that the sample cannot be thought of as having definite energies in definite states—we certainly shall assume that it does—but rather that we shall evade the problem of ever having to count how many states an isolated system can have.
A few more words about formalities may be in order. We have assumed, without stating it, that T and S are nonzero; our arguments would fall through otherwise. Arguments of the type we have used always assume that the system actually have some disposable energy, and hence more than one possible choice of state. This minor but necessary point will later be enshrined within the Third Law of Thermodynamics.
Furthermore, we have manipulated the concept of equilibrium in a way that needs to be pointed out and underlined. As we first introduced it, equilibrium was a condition that was necessary before we could even begin to discuss such ideas as entropy and temperature; there was no way, for example, that the temperature could even be defined until the system had been allowed to forget its previous history. Later on, however, we found ourselves asking a different kind of question: What is the requirement on the temperature that a system be in equilibrium? This question necessarily implies that the temperature be meaningful and defined when the system is not in equilibrium. We accomplished this step by considering a restricted kind of disequilibrium. We imagined the system (our combined system) to be composed of subsystems (the individual boxes) that were themselves internally in equilibrium. For each subsystem, then, the temperature, entropy, and so on are well defined, and the specific question we ask is: What are the conditions that the subsystems be in equilibrium with each other? When we speak of a system not in equilibrium, we shall usually mean it in this sense; we think of it as composed of various subsystems, each internally in equilibrium but not necessarily in equilibrium with each other. For systems so defined, it follows by a natural extension of Eqs. (1.1.10) and (1-1-11) that the entropy of the system, whether in equilibrium or not, is the sum of the entropies of its subsystems, and by an extension of the succeeding arguments that a general condition that the system be in equilibrium is that the temperature be uniform everywhere.
We have also seen that a system or subsystem in equilibrium is not in any definite state in the quantum mechanical sense. Instead, it is free to be in any of a very large number of states, and the requirement of equilibrium is really only that it be equally free to be in any of those states. The system thus fluctuates about among its various states. It is important to remember that these fluctuations are not fluctuations out of equilibrium but rather that the equilibrium is the averaged consequence of these fluctuations.
We now wish to carry out our program, which means that we must learn how to count the number of states it is possible for a system to have or, more precisely, how to avoid having to count that number. This is a formidable task, and we shall need some powerful tools. Accordingly, we shall devote the next section to discussing the quantities that appear in thermodynamics, and the methods of manipulating such quantities.
1.2THERMODYNAMICS
a. The Laws of Thermodynamics
Thermodynamics is basically a formal system of logic deriving from a set of four axioms, known as the Laws of Thermodynamics, all four of which we arrived at, or at least flirted with, in our preliminary discussion of the previous section. We shall not be concerned here with formalities, but let us, without rigor and just for the record, indicate the sense of the four laws. Being a logical system, the four laws are called, naturally, the Zeroth, First, Second, and Third. From the point of view of thermodynamics, these laws are not to be arrived at, as we have done, but rather are assumptions to be justified (and are amply justified) by their empirical success.
The Zeroth Law says that the concept of temperature makes sense. A single number, a scalar, assigned to each subsystem, suffices to predict whether the subsystems will be found to be in thermal equilibrium should they be brought into contact. Equivalently, we can say that if bodies A and B are each separately found to be in thermal equilibrium with body C (body C, if it is small, may be called a thermometer), then they will be in equilibrium with each other.
The First Law is the thermodynamic statement of the principle of conservation of energy. It is usually stated in such a way as to distinguish between two kinds of energy—heat and work—the changes of energy in a system being given by
where Q is heat and R is work.
We can easily relate Eq. (1.2.1) to Eq. (1.1.6). Suppose that our box of perfect gas were actually a cylinder with a movable piston of cross-sectional area A 
and displace the piston an amount dx, we are doing an amount of mechanical work on the gas inside given by
, and volume, dV = −A dx (the sign tells us that the volume decreases when we do positive work on the gas),
Fig. 1.2.1
If the process we have described was done in isolation, so that energy was unable to leak in or out in any other form, Eqs. (1.2.1) and (1-2-2) together tell us that
Comparing this result to Eqs. (1.1.6) and (1-1-8) of the previous section, we see that the pressure we are using here, which is just the force per unit area, is the same as the pressure as defined there, provided that, for our fixed number of particles, holding Q constant means that S has been held constant. We can show that such is the case. As we push the piston in, the quantitative values of the energies of the single-particle states, given by some relation like Eqs. (1.1.1) and (1-1-2), will change because the dimensions of the box (the value of L in the x direction) are changing. However, the enumeration of the single-particle states, the number of them and their separate identities, does not change. Consider a particular state of the system—a particular distribution of the particles among their various single-particle states, using up all the available energy—before the displacement. When the displacement occurs, each single-particle state shifts its energy a bit, but we can still identify one single-particle state of the new system with the state it came from in the old. If we induce the displacement slowly, each particle will stay in the state it is in, and so the work done just goes into changing the energies of all the occupied single-particle levels. The same statement is true of each of the possible states of the system, and so although the energy of the system changes in the process, the number of possible states, and hence the entropy, does not.
dx type) has been done, and, furthermore, the single-particle states are not affected. But the amount of energy available to be divided among the particles has increased, the number of ways of dividing it has increased as well, and thus the entropy has increased. From Eqs. (1.2.1) of this section with dR = 0, and (1-1-6) and (1-1-7) of the previous section, we see that
for changes that take place in equilibrium, so that (1.1.6) is applicable. Thus, for a fixed number of particles, we can write the first law for equilibrium changes in the form
Although changes in heat are always equal to T dS, there are kinds of work other than P dV: magnetic work, electric work, and so on. However, it will be convenient for us to develop the consequences of thermodynamics for this kind of work—that is, for mechanical work alone—and return to generalize our results later in this chapter.
According to the celebrated Second Law of thermodynamics, the entropy of a system out of equilibrium will tend to increase. This statement means that when a system is not in equilibrium, its number of choices is restricted—some states not forbidden by the design of the system are nevertheless unavailable due to its past history. As time goes on, more and more of these states gradually become available, and once a state becomes available to the random fluctuations of a system, it never again gets cut off; it always remains a part of the system's repertory. Systems thus tend to evolve in certain directions, never returning to earlier conditions. This tendency of events to progress in an irreversible way was pointed out by an eleventh-century Persian mathematician, Omar Khayyam (translated by Edward Fitzgerald):
The Moving Finger writes; and, having writ,
Moves on: nor all thy Piety nor Wit
Shall lure it back to cancel half a line,
Nor all thy Tears Wash out a Word of it.
There have been many other statements of the Second Law, all of them less elegant.
If disequilibrium means that the entropy tends to increase, equilibrium must correspond to a maximum in the entropy—the point at which it no longer can increase. We saw this in the example we considered in the previous section. If the two boxes, initially separate, had been out of equilibrium, one of them (the hotter one) would have started with more than its fair share of the energy of the combined system. Owing to this accident of history, the number of states available to the combined system would have been smaller than if the cooler box had a fairer portion of the total energy to share out among its particles. The imbalance is redressed irreversibly when contact is made. In principle, a fluctuation could occur in which the initially cooler box had even less energy than it started with, but such a fluctuation is so utterly unlikely that there is no need to incorporate the possibility of it into our description of how things work in the real world. Quite the opposite, in fact: we have dramatic success in describing the real world if we assume that such fluctuations are impossible. That is just what the entropy principle does.
The Third and final Law states that at the absolute zero of temperature, the entropy of any body is zero. In this form it is often called Nernst's theorem. An alternative formulation is that a body cannot be brought to absolute zero temperature by any series of operations. In this form the law basically means that all bodies have the same entropy at zero degrees. According to the earlier statement, if a body had no disposable energy, so that its temperature were zero, it would have only one possible state: Γ = 1 and thus S = k log Γ = 0. This amounts to asserting that the quantum ground state of any system is nondegenerate. Although there are no unambiguous counter-examples to this assertion in quantum mechanics, there is no formal proof either. In any case, there may be philosophical differences, but there are no practical differences between the two ways of stating the law. Among other things, the Third Law informs us that thermodynamic arguments should always be restricted to nonzero temperatures.
b. Thermodynamic Quantities
As we see in Eq. (1.2.5), the energy of a system at equilibrium with a fixed number of particles may be developed in terms of four variables, which come in pairs, T and S, P and V. Pairs that go together to form energy terms are said to be thermodynamically conjugate to each other. Of the four, two, S and V, depend directly on the size of the system and are said to be extensive variables. The others, T and P, are quite independent of the size (if a body of water is said to have a temperature of 300°K, that tells us nothing about whether it is a teaspoonful or an ocean) and these are called intensive variables.
We argued in Sec. 1.1 that if we know the energy as a function of S and V, then we can deduce everything there is to know, thermodynamically speaking, about the body in question: P = −(∂E/∂V)S, T = (∂E/∂S)V, and T(V, S), together with P(V, S), gives us T(P, V); if we want the entropy, T(V, S) can be inverted to give S(T, V) and so on. If, on the other hand, we know the energy as a function, say of T and V, we generally do not have all the necessary information. There may, for example, be no way to find the entropy. For this reason, we shall call S and V the proper independent variables of the energy: given in terms of these variables the energy tells the whole story.
Unfortunately, problems seldom arise in such a way as to make it easy for us to find the functional form, E(S, V). We usually have at hand those quantities that are easy to measure, T, P, V, rather than S and E. It will, therefore, turn out to be convenient to have at our disposal energylike functions whose proper independent variables are more convenient. In fact, there are four possible sets of two variables each, one being either S or T and the other either P or V, and we shall define energy functions for each possible set. The special advantages of each one in particular kinds of problems will show up as we go along.
We define F, the free energy (or Helmholz free energy), by
Together with (1.2.5), this gives
so that
and
In other words, F = F(T, V) in terms of its proper variables, and T and V are obviously a convenient set to use. F also has the nice property that, for any changes that take place at constant temperature,
so that changes in the free energy are just equal to the work done on the system. Recall for comparison that
the work done is equal to the change in energy only if the entropy is held constant or, as we argued in Sec. 1.2a, if the work is done slowly with the system isolated. It is usually easier to do work on a system well connected to a large temperature bath, so that Eq. (1.2.10) applies, than to do work on a perfectly isolated system, as required by Eq. (1.2.11).
The function F(T, V) is so useful, in fact, that we seem to have gotten something for almost nothing by means of the simple transformation, Eq. (1.2.6). However, we have a fairly clear conceptual picture of what is meant by E(S, V) in terms of the counting of states, whereas the connection between the enumeration of the states of a system and F(T, V) is much less obvious. Our job in Sec. 1.3 will be to develop ways of computing F(T, V) from the states of a system, so that we can take advantage of this very convenient function.
The Gibbs potential (or Gibbs free energy), Φ, is defined by
which, combined with Eq. (1.2.7) or (1.2.5), gives
It follows that
and, consequently, Φ = Φ(T, P) in terms of its proper variables. Φ will be the most convenient function for problems in which the size of the system is of no importance. For example, the conditions under which two different phases of the same material, say liquid and gas, can coexist in equilibrium will depend not on how much of the material is present but rather on the intensive variables P and T.
A function of the last remaining pair of variables, S and P, may be constructed by
so that
W is called the enthalpy or heat function. Its principal utility arises from the fact that, for a process that takes place at constant pressure,
which is why it is called the heat function.
A fundamental physical fact underlies all these formal manipulations. The fact is that for any system of a fixed number of particles, everything is determined in principle if only we know any two variables that are not conjugate to each other and also know that the system is in equilibrium. In other words, such a system in equilibrium really has only two independent variables—at a given entropy and pressure, for example, it has no choice at all about what volume and temperature to have. So far we have expressed this fundamental point by defining a series of new energy functions, but the point can be made without any reference to the energy functions themselves by considering their cross-derivatives. For example,
Suppose that we have our perfect gas in a cylinder and piston, as in Fig. 1.2.1, but we immerse our system in a temperature bath, so that changes take place at constant T. If we now change the volume by means of the piston, the entropy will change owing to the changes in the energies of all the single-particle states. Since energy is allowed to leak into or out of the temperature bath at the same time, it would seem difficult to figure out just how much the entropy changes. Equation (1.2.21) tells us how to find out: we need only make the relatively simple measurement of the change in pressure when the temperature is changed at constant volume. It is important to realize that this result is true not only for the perfect gas but also for any system whatsoever; it is perfectly general. Conversely, the fact that this relation holds for some given system tells us nothing at all about the inner workings of that system.
Three more relations analogous to (1.2.21) are easily generated by taking cross-derivatives of the other energy functions. They are
Together these four equations are called the Maxwell relations.
A mnemonic device for the definitions and equations we have treated thus far is given in Appendix A of this chapter.
The machinery developed on these last few pages prepares us to deal with systems of fixed numbers of particles, upon which work may be done only by changing the volume—more precisely, for systems whose energy is a function of entropy and volume only. Let us now see how to generalize this picture.
If the number of particles of the system is free to vary, then, in addition to work and heat, the energy will depend on that number as well. We have already seen, in Eqs. (1.1.6) and (1-1-9), that in this case,
μ is the chemical potential, which we may take to be defined by Eq. (1.2.25), so that
If we now retain all the transformations among the energy functions as we have defined them, then
Similarly,
Thus,
In other words, the effect of adding particles is to change any one of the energy functions by μ dN if the particles were added holding its own proper independent variables fixed:
Notice that the first member of Eq. (1.2.31) is not necessarily relevant. If we manage to change the energy at constant S and V (so that E depends on something besides S and V) and we retain the transformations, Eqs. (1.2.6), (1-2-12), and (1-2-16), then the other functions will change according to
We shall make use of this result later on (to have an application to think about now, you might consider changing the masses of the particles of a perfect gas; see Prob. 1.2).
The introduction of μ and changes in N thus serve as an example of how to generalize to cases where the energy can depend on something besides S and V, but we wish to retain the transformations between E, F, Φ, and W. However, the chemical potential is a particularly important function in thermodynamics, and its properties deserve further elaboration.
Like P and T, μ is an intensive variable, independent of the size of the system (this point can be seen from the result of Prob. 1.1a, where we see that two subsystems in equilibrium must have the same μ regardless of their sizes), and it is conjugate to an extensive variable, N. Suppose that we think of Eq. (1.2.25) as applying to a piece of matter, and we rewrite it to apply to another piece that differs from the first only in that it is λ times bigger. It could have been made by putting together λ of the small pieces. Then all the extensive quantities, E, S, V, and N, will simply be λ times bigger:
d(λE) = T d(λS) − P d(λV) + μ d(λ N)
or
Since λ is arbitrary, it follows that
If we differentiate this result and subtract Eq. (1.2.25), we get
so that μ, like Φ, is a proper function of T and P. In fact, comparing Eq. (1.2.34) to (1.2.12) (which, remember, is still valid), we have
As we have seen, the general conditions for a system to be in equilibrium will be that T, P, and μ be uniform everywhere.
Equation (1.2.31) tells us that μ is the change in each of the energy functions when one particle is added in equilibrium. That observation might make it seem plausible that μ would be a positive quantity, but it turns out instead that μ is often negative. Why this is so can be seen by studying Eq. (1.2.26). To find μ, we ask: How much energy must we add to a system if we are to add one particle while keeping the entropy and volume constant? Suppose that we add a particle with no energy to the system, holding the volume fixed, and wait for it to come to equilibrium. The system now has the same energy it had before, but one extra particle, giving it more ways in which to divide that energy. The entropy has thus increased. In order to bring the entropy back to its previous value, we must extract energy. The chemical potential—that is, the change in energy at constant S and V—is therefore negative. This argument breaks down if it is impossible to add a particle at zero energy, owing to interactions between the particles; the chemical potential will be positive when the average interaction is sufficiently repulsive, and energy is required to add a particle. It will be negative, for example, for low-density gases and for any solid or liquid in equilibrium with (at the same chemical potential as) its own low density vapor.
Now that we have a new set of conjugate variables, it is possible to define new transformations to new energy functions. For example, the quantity E − μN would be a proper function of the variables S, V, and μ, and W − μN of S, P, and μ. We do not get a new function from Φ − μN, which is just equal to zero: P, T, and μ are not independent variables. Of the possible definitions, one will turn out to be most useful. We define the Landau potential, Ω,
so that
that is, the proper variables of Ω are T, V, and μ. Notice that it follows from Eqs. (1.2.37) and (1-2-34) that
Furthermore, since we have not altered the relations between the other energy functions in defining Ω, arguments just like those leading to Eq. (1.2.32) will yield
c. Magnetic Variables in Thermodynamics
The total energy content of a magnetic field is
where B is the fundamental magnetic field, produced by all currents, and the integral extends over all space. In order to introduce magnetic work into our equilibrium thermodynamic functions, we need to know how much work is done on a sample when magnetic fields change—that is, the magnetic analog of (−P dV). Equation (1.2.41) fails to tell us this for two reasons. First, it does not sort out contributions from the sample and from external sources; B is the total field. Second, there is no way to tell from Eq. (1.2.41) how much work was done in producing B; the work may have been much more than Em with the excess dissipated in heat. Only if all changes took place in equilibrium will Em be equal to the work done in creating the field. In other words, Em is the minimum work (by sample and external sources together) needed to produce the distribution of fields in space.
Even assuming that all changes take place in equilibrium, we must have a way of distinguishing work done on the sample from other contributions—that is, work done on an external source during the same changes. To do so, we decompose B into two contributions
where H is the field due to currents running in external sources and M is the response of the sample, called the magnetization.
To keep things simple, let us set up a definite geometry, which we shall always try to return to when analyzing magnetic problems. We shall imagine H to be a uniform field arising from current
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