Intro to modern statistical mechanics chandler pdf download






















Electrons in Metals 97 Classical Ideal Gases, the Classical Limit A Dilute Gas of Atoms Dilute Gas of Diatomic Molecules 4. Statistical Mechanical Theory of Phase Transitions 5 1.

Ising Model Lattice Gas Broken Symmetry and Range of Correlations Mean Field Theory Variational Treatment of Mean Field Theory Renormalization Group RG Theory Monte Carlo Method in Statistical Mechanics A Monte Carlo Trajectory Classical Fluids 7 1 Averages in Phase Space. Reversible Work Theorem Thermodynamic Properties from g r Measurement of g r by Diffraction Solvation and Chemical Equilibrium in Liquids Molecular Liquids Functions Application: Chemical Kinetics Another Application: Self-Diffusion Fluctuation-Dissipation Theorem Response Functions Absorption For the most part, those placed within the text are meant to be worked out immediately as simple exercises.

Occasionally, however, I have engaged in the pedagogical device of asking questions that require development of a sophisticated idea in order to obtain the answer. These Exercises are marked with an asterisk, and there are three ways of handling them.

First, you might figure out the answer on the spot in which case you " " are more than pretty good! Second, you could cheat and look for the required techniques in other textbooks a kind of "cheating" I hope to inspire. Third, you can keep on thinking about the problem but proceed with studying the text. In the last case, you will often find the techniques for the solution will gradually appear later. Finally, we have produced a diskette with text related computer programs that run on IBM-PC compatibles.

Press, New York, It is the ubiquitous presence of fluctuations that makes observations interesting and worthwhile. Indeed, without such random processes, liquids would not boil, the sky would not scatter light, indeed every dynamic process in life would cease. It is also true that it is the very nature of these fluctuations that continuously drives all things toward ever- increasing chaos and the eventual demise of any structure.

Fortun- ately, the time scales for these eventualities are often very long, and the destruction of the world around us by natural fluctuations is not something worth worrying about.

Statistical mechanics and its macroscopic counterpart, thermodynamics, form the mathematical theory with which we can understand the magnitudes and time scales of these fluctuations, and the concomitant stability or instability of structures that spontaneous fluctuations inevitably destroy.

The presence of fluctuations is a consequence of the complexity of the systems we observe. Macroscopic systems are composed of many particles-so many particles that it is impossible to completely control or specify the system to an extent that would perfectly prescribe the evolution of the system in a deterministic fashion. Ignorance, therefore, is a law of nature for many particle systems, and this ignorance leads us to a statistical description of observations and the admittance of ever-present fluctuations.

Even those observed macroscopic properties we conceive of as being static are irrevocably tied to the statistical laws governing dynamical fluctuations. In Chapter 3, we will show that this equation is equivalent to a formula for the mean square density fluctuations in the gas. The equation can be regarded entirely as a consequence of a particular class of statistics in this case, the absence of correlations between density fluctuations occurring at different points in space , and need not be associated with any details of the molecular species in the system.

Further, if these uncorre- lated density fluctuations ceased to exist, the pressure would also vanish. As we will see later in Chapter 8, we can also consider the correlation or influence of a fluctuation occurring at one instant with those occurring at other points in time, and these considerations will tell us about the process of relaxation or equilibration from nonequi- librium or unstable states of materials. But before we venture deeply into this subject of characterizing fluctuations, it is useful to begin by " " considering what is meant by equilibrium and the energetics associated with removing macroscopic systems from equilibrium.

This is the subject of thermodynamics. While many readers of this book may be somewhat familiar with this subject, we take this point as our beginning because of its central importance to statistical mechanics. As we will discuss in Chapter 3, the reversible work or energetics associated with spontaneous fluctuations determines the likelihood of their occurrence.

In fact, the celebrated second law of thermo- dynamics can be phrased as the statement that at equilibrium, all fluctuations consistent with the same energetics are equally likely. Before discussing the second law, however, we need to review the first law and some definitions too. The quantity, to which we give the symbol E, is defined as the total energy of the system.

We postulate that it obeys two properties. First internal , energy is extensive. That means it is additive. For example consider , the composite system pictured in Fig. Due to this additivity, extensive properties depend linearly on the size of the system.

In other words, if we double the size of the system keeping other things fixed, the energy of the system will double. Composite system.

This means that if the energy of a system changes, it must be as a result of doing something to the system-that is, allowing some form of energy to flow into or out of the system. One thing we can do is perform mechanical work. What else is there? Empirically we know that the energy of a system can be changed by doing work on the system, or by allowing heat to flow into the system.

In it, ijlW is the differential work done on the system manipulating mechanical constraints , and 4Q is the differential heat flow into the system. In general, there are many mechanical extensive variables, and their changes involve work. The definition of heat is really not complete, however, unless we postulate a means to control it. Adiabatic walls are the constraints that prohibit the passage of heat into the system. Another important point to keep in mind is that work and heat are forms of energy transfer.

Exercise 1. Experimentally we know that isolated systems tend to evolve spontaneously toward simple terminal states. These states are called " " equilibrium states. By simple we mean that macroscopically they can be characterized by a small number of variables.

In particular, the equilibrium state of a system is completely characterized macro- scopically by specifying E and X. For a system in which the relevant mechanical extensive variables are the volume and the numbers of molecules, the variables that characterize the system are E y V, Hi,.

By the way in the case of , electric and magnetic fields, care is required in the development of an extensive electrical and magnetic energy. Can you think of the source of the difficulty? An illustrative system. This observation can be used to verify that composition variables play a mathematically equivalent role to the standard mechanical extensive variables.

See Exercise 1. The complete list of relevant variables is sometimes a difficult experimental issue. But whatever the list is, the most important feature of the macroscopic equilibrium state is that it is characterized by a very small number of variables, small compared to the overwhelming number of mechanical degrees of freedom that are necessary to describe in general an arbitrary non-equilibrium state of a macroscopic many particle system. Virtually no system of physical interest is rigorously in equi- librium.

However, many are in a metastable equilibrium that usually can be treated with equilibrium thermodynamics. Generally, if in the course of observing the system, it appears that the system is independent of time, independent of history, and there are no flows of energy or matter, then the system can be treated as one which is at equilibrium, and the properties of the system can be characterized by E V, «! Ultimately, however, one is never sure that y the equilibrium characterization is truly correct, and one relies on the internal consistency of equilibrium thermodynamics as a guide to the correctness of this description.

An internal inconsistency is the signature of nan-equilibrium behavior or the need for additional macroscopic variables and not a failure of thermodynamics. What can thermodynamics tell us about these equilibrium states? Consider a system in equilibrium state I formed by placing certain constraints on the system. One or more of these constraints can be removed or changed and the system will evolve to a new terminal state II.

The determination of state II can be viewed as the basic task of thermodynamics. As examples, consider the system pictured in Fig. Let piston move around. Punch holes in piston perhaps permeable only to one species. Remove adiabatic wall and let system exchange heat with surroundings. What terminal states will be produced as a result of these changes?

To answer this question, a principle is needed. This principle is the second law of thermodynamics. While this motivation to consider the second law is entirely macroscopic, the principle has a direct bearing on microscopic issues, or more precisely, the nature of fluctuations. The reasoning is as follows: Imagine that the constraints used to form the initial state I have just been removed and the system has begun its relaxation to state II.

After the removal of the constraints, it becomes impossible to discern with certainty whether the formation of state I was the result of applied constraints now removed or the result of a spontaneous fluctuation. Therefore, the analysis of the basic task described above will tell us about the energetics or thermodynamics of spontaneous fluctuations, and we shall see that this information will tell us about the likelihood of fluctuations and the stability of state II.

With this foreshadowing complete, let us now turn to the principle that provides the algorithm for this analysis. As our remarks have already indicated, the second law is intimately related to and indeed a direct consequence of reasonable and simple statistical assumptions concerning the nature of equilibrium states. We will consider this point of view in Chapter 3. But for now we , present this law as the following postulate: There is an extensive function of state, S E, X , which is a monotonically increasing function of E, and if state B is adiabatically accessible from state A, then SB SA.

Notice that if this state B was reversibly accessible from state A then , the process B- A could be carried out adiabatically too. In that case, the postulate also implies that SA SB. A reversible process is one that can be exactly retraced by infinitesimal changes in control variables.

As such, it is a quasi-static thermodynamic process carried out arbitrarily slowly enough so that at each stage the system is in equilibrium. In other words, a reversible process progresses within the manifold of equilibrium states. Since these states are so simply characterized by a few variables, any such process can be reversed by controlling those " " variables; hence the name reversible. Natural processes, on the other hand, move out of that manifold through vastly more complex non-equilibrium states requiring, in general, the listing of an enor- mous number of variables perhaps the positions of all particles to characterize these states.

Without controlling all these variables which would be impossible in a general case , it is highly unlikely that in an attempt to reverse such a process, one would observe the system passing through the same points in state space. Hence, the " process is irreversible. As we have already demonstrated, the entropy change for a reversible adiabatic process is zero.

Note also that entropy is a function of state. That means it is defined for those states characterized by E and X. Such states are the thermodynamic equilibrium states. Entropy obeys several other important properties as well. Here, due to reversibility, the "force," f, is a property of the system.

For instance, at equilibrium, the externally applied pressures, pext, are the same as the pressure of the system, p. Since the last equation must hold for all reversible processes i.

To ensure this behavior, the term in square brackets in the last equation must be identically zero. Therefore, the equality holds for nonadiabatic as well as adiabatic processes. This last derivative is defined as the temperature, T. We will see later that this definition is consistent with our physical notion of temperature.

Note that since both E and S are extensive, the temperature is intensive. That is, T is independent of the size of the system. The boxed equations of this section are the fundamental relation- ships that constitute the mathematical statement of the second law of thermodynamics. Which of the two possibilities is acceptable? Composite system illustrating the meaning of internal constraints.

A useful form of the second law, and one that is most closely tied to the energetics of fluctuations, is derived by considering a process in which an internal constraint is applied quasi-statically at constant E and X. Internal constraints are constraints that couple to extensive variables but do not alter the total value of those extensive variables.

For example, consider the system pictured in Fig. But that process would not correspond to the application of an internal constraint. Rather, imagine moving the interior piston. It would require work to do this and the energy of the system would therefore change. But the total volume would not change. This second process does correspond to an application of an internal constraint. With this definition in mind, consider the class of processes depicted in Fig. The application of the internal Internal constraint Natural process occurring Quasi-static after internal constraint process to is turned off constrained equilibrium state X 7 S Equilibrium state with E entropy X Fig.

Entropy changes for a process involving the manipulation of an internal constraint. Illustrative composite system. It will require work to do this, and the requirement that there is no change in energy for the process means that there must also have been a flow of heat.

After attaining and while maintaining this constrained state, we will adiabatically insulate the system. Then, let us imagine what will happen when we suddenly shut off the internal constraint. The system will relax naturally at constant E and X back to the initial state with entropy 5 as depicted in Fig.

This variational principle provides a powerful method for deducing many of the consequences of the second law of thermodynamics. It clearly provides the algorithm by which one may solve the problem we posed as the principal task of thermodynamics. To see why, consider the example in Fig.

We can ask: Given that the system was initially prepared with E partitioned with E in subsystem 1 and isfniiai in subsystem 2, what is the final partitioning of energy? The entropy maximum principle has a corollary that is an energy minimum principle. Such repartitioning lowers the entropy as indicated in the inequality. Note that in computing the entropy of the repartitioned system, we have used the fact that entropy is extensive so that we simply add the entropies of the two separate subsystems.

Now recall that S is a monotonically increasing function of E i. In other words, we can imagine applying internal constraints at constant total S and X, and such processes will necessarily raise the total energy of the system.

This statement is the energy minimum principle to which we have referred. Often, the extremum principles are stated in terms of mathemati- cal variations away from the equilibrium state. Application: Thermal Equilibrium and Temperature An instructive use of the variational form of the second law establishes the criterion for thermal equilibrium and identifies the integrating factor, T, as the property we really conceive of as the temperature.

Heat conducting system. Consider the system in Fig. Let us ask the question: At equilibrium, how are T 1 and r 2 related? To obtain the answer , imagine a small displacement about equilibrium due to an internal constraint.

The variational theorem in the entropy representation is 6S EtX 0. Notice the perspective we have adopted to derive the equilibrium condition of equal temperatures: We assume the system is at an equilibrium state and we learn about that state by perturbing the , system with the application of an internal constraint.

This type of procedure is very powerful and we will use it often. In a real sense, it is closely tied to the nature of experimental observations. In particular, if we conceive of a real experiment that will probe the behavior of an equilibrium system the experiment will , generally consist of observing the response of the system to an imposed perturbation.

With this procedure, we have just proven that according to the second law, the criterion for thermal equilibrium is that the two interacting subsystems have the same temperature. Our proof used the entropy variational principle.

You can verify that the same result follows from the energy variational principle. Next let's change our perspective and consider what happens when the system is initially not at thermal equilibrium-that is initially , j i 2. Eventually, at thermal equilibrium , the two become equal. The question is: how does this equilibrium occur? To answer the question we use the second law again this time , noting that the passage to equilibrium is a natural process so that the change in entropy, AS, is positive i.

Thus, we have just proven that energy flow is from the hot body to the cold body. In summary, therefore, heat is that form of energy flow due to a temperature gradient, and the flow is from hot higher T to cold lower T. Since heat flow and temperature changes are intimately related it , is useful to quantify their connection by introducing heat capacities. The conventional and operational defini- tions are Mi , - - Sl Since 5 is extensive, Cf and Cx are extensive.

The temperature, length per mole and mole , 1 and number for the first piece are r , respectively. Neglect thermal convection to the surroundings and mass flow. However, the ease with which the analyses can be performed is greatly facilitated by introducing certain mathematical concepts and methods. The tricks we learn here and in the remaining sections of this chapter will also significantly add to our computational dexterity when performing statistical mechanical calculations.

The first of these methods is the procedure of Legendre trans- forms. To see why the method is useful, let's use the specific form of the reversible work differential appropriate to systems usually studied in chemistry and biology. In that case, for reversible displacements r f. The chemical potential is defined by the equation above.

Composite system illustrating how mechanical work can be associated with changes in mole numbers. As we will stress in Chapters 2 and 3, the chemical potential is the intensive property that controls mass or particle equilibrium just as temperature controls thermal equilibrium.

Indeed, we will find that gradients in chemical potentials will induce the flow of mass or the rearrangements of atoms and molecules, and the absence of such gradients ensures mass equi- librium. Rearrangements of atoms and molecules are the processes by which equilibria between different phases of matter and between different chemical species are established. Chemical potentials will therefore play a central role in many of our considerations through- out this book. The piston between subsystems I and II is permeable to species 1 but not to species 2.

It is held in place with the application of pressure pA. The other piston, held in place with pressure pB, is permeable to species 2 but not to species 1. Can you think of other devices for which reversible work is connected with changing mole numbers in a system? In view of the form for f. But now suppose an experimentalist insists upon using , T , V, and n to characterize the equilibrium states of a one- component system. The questions are: is that characterization possible, and, if so, what is the thermodynamic function analogous to E that is a natural function of T, V, and n?

The answer comes from considering Legendre transformations. We do this now. It is apparent that this type of construction provides a scheme for introducing a natural function of T, V, and n since T is simply the conjugate variable to S. Exchanges between non-conjugate pairs, however, are not possible for this scheme.

For example, since S, V, n provides enough information to characterize an equilibrium state, so do T, V, n , S,p, n , and r p, n. However p, v, n or , , 5, r, p are not examples of sets of variables that can characterize a state. We will return to this point again when discussing thermo- dynamic degrees of freedom and the Gibbs phase rule. The variational principles associated with the auxiliary functions H A, and G are , Atf 5.

Armed with the auxiliary functions, many types of different measure- ments can be interrelated. As an example, consider dSldV T,n. Two more examples further demonstrate these methods and serve to show how thermal experiments are intimately linked with equation of state measurements. With this meaning in mind, consider the internal energy E, which is extensive, and how it depends upon S and X, which are also extensive.

Differentiate this equation with respect to A: a But we also know that n dfiuu. Let us now specialize as in the previous two sections: r f. These are zeroth-order homogeneous functions of the extensive variables. This equation is true for any A.

Volume per mole of water at 1 atm pressure. Hence, a variable indicating this size, required for extensive pro- perties, is not needed to characterize intensive properties. Having said all these things, here is a simple puzzle, the solution of which provides a helpful illustration: If an intensive property of a one-component equilibrium system can be characterized by two other intensive properties, how do you explain the experimental behavior of liquid water illustrated in Fig.

The answer to this apparent paradox is that p and V are conjugate variables. Hence p, V, n do not uniquely characterize an equilibrium state. The above experimental curve does not contradict this expectation. This puzzle illustrates the importance of using non-conjugate variables to characterize equilibrium states. Additional Exercises 1. Derive the analog of the Gibbs-Duhem equation for a rubber band.

Compute the change in entropy per unit mass for this process assuming the rubber band obeys the equation of state given in Exercise 1. It plays an important role in the theory of phase transitions when deriving what are known as "scaling" equa- tions of state for systems near critical points. Bibliography Herbert Callen pioneered the teaching of thermodynamics through a ' sequence of postulates, relying on the student s anticipation of a statistical microscopic basis to the principles.

Our discussion in this chapter is strongly influenced by Callen's first edition: B. Callen, Thermodynamics John Wiley, N. In traditional treatments of thermodynamics, not a word is spoken about molecules, and the concept of temperature and the second law are motivated entirely on the basis of macroscopic observations. Here are two texts that embrace the traditional approach: J G. Kirk wood and 1. Oppenheim, Chemical Thermodynamics. McGraw-Hill, N. Oppenheim, Thermodynamics Elsevier Scien-.

Elementary texts are always useful. Cambridge University, Cambridge, The derivations are based on the procedure already introduced in Chapter 1. In particular, we first assume the system under investigation is stable and at equilibrium, and we then examine the thermodynamic changes or response produced by perturbing the system away from equi- librium. The perturbations are produced by applying so-called " internal constraints.

According to the second law of thermodynamics such processes lead , to lower entropy or higher free energy provided the system was initially at a stable equilibrium point. Thus, by analyzing the signs of thermodynamic changes for the processes, we arrive at inequalities consistent with stability and equilibrium.

These conditions are known as equilibrium and stability criteria. We first discuss equilibrium criteria and show, for example, that having T, p9 and ju constant throughout a system is equivalent to assuring that entropy or the thermodynamic energies are extrema with respect to the partitioning of extensive variables.

To distinguish between maxima or minima, one must continue the analysis to consider the sign of the curvature at the extrema. This second step in the development yields stability criteria. When developing the theory of statistical mechanics in Chapter 3, we will find that these " " criteria, often referred to as convexity properties, can be viewed as statistical principles that equate thermodynamic derivatives to the value of mean square fluctuations of dynamical quantities.

After examining several thermodynamic consequences of equi- librium and stability, we apply these criteria to the phenomena of phase transitions and phase equilibria. The Clausius-Clapeyron equation and the Maxwell construction are derived. These are the thermodynamic relationships describing how, for example, the boil- ing temperature is determined by the latent heat and molar volume change associated with the liquid-gas phase transition.

While results of this type are insightful, it should be appreciated that a full understanding of phase transitions requires a statistical mechanical treatment.

Phase transformations are the results of microscopic fluctuations that, under certain circumstances, conspire in concert to bring about a macroscopic change. The explanation of this cooperati- vity, which is one of the great successes of statistical mechanics, is discussed in Chapter 5. Each phase comprises a different subsystem. Repartitionings of extensive variables can be accom- plished by shufling portions of the extensive variables between the different phases. Here, one might note that this formula neglects energies as- sociated with surfaces i.

The neglect of surface energy is in fact an approximation. It produces a negligible error, however, when considering large i. The reason is that the energy of a bulk phase is proportional to N, the number of molecules in the phase, while the surface energy goes as Af Hence,. A two-phase system. But for now , we focus on the bulk phases.

Note that if the fluctuations were to be constrained to be of only one sign, then the equilibrium conditions would be inequalities rather than equalities.

The argument for unconstrained variations or l f uctuations is easily extended to any number of phases. Exercise 2. One may also demonstrate that these criteria are both necessary and sufficient conditions for equilibrium.

At this point, let us pause to consider a bit more about the meaning and significance of chemical potentials. To do this, consider the composite system in Fig. That is, matter flows from high jU to low jU. As a final remark concerning chemical potentials, note that we have only imagined the repartitioning of species among different phases or regions of space in the system.

Another possibility is the rearrangements accomplished via chemical reactions. The analysis of these processes leads to the criteria of chemical equilibria. For now, we leave this analysis as an exercise for the reader; it will be performed in the text, however, when we treat the statistical mechanics of gas phase chemical equilibria in Chapter 4. Conditions derived from this relation are called stability criteria. That is, after the occurrence of a small fluctuation, the system will return to the equilibrium state.

Arbitrary composite system with two compartments. As an example, consider a composite system with two compart- ments, as pictured in Fig. Thus, a stable system will have a positive Cv. The gradient in T will cause heat flow, and the flow will be from high Tto low T.

This illustrates the physical content of stability criteria. If they are obeyed, the spon- taneous process induced by a deviation from equilibrium will be in a direction to restore equilibrium. One should note that it is not permissible to consider fluctuations in T since these variational theorems refer to experiments in which internal constraints vary internal extensive variables keeping the total extensive variables fixed.

T is intensive, however, and it makes no sense to consider a repartitioning of an intensive variable. Thus if the pressure of a stable , system is increased isothermally its volume will decrease.

A general rule for stability criteria should now be apparent. Let stand for the internal energy or a Legendre transform of it which is a natural function of the extensive variables Xl,. However, the second law i. Triple point": a-p-7 phase equilibria " T Fig. A hypothetical phase diagram. Here, jicja is the mole fraction of species i in phase or. This formula is the Gibbs phase rule. As an illustration, consider a one-component system.

Without coexisting phases, there are two degrees of freedom; p and T are a convenient set. The system can exist anywhere in the p-T plane. Three phases coexist at a point, and it is impossible for more than three phases to coexist in a one-component system. Thus, a possible phase diagram is illustrated in Fig.

For example, the content of the first of these equations is illustrated in Fig. The second law says that at constant T p, and n, the stable , equilibrium state is the one with the lowest Gibbs free energy which is «jU for a one-component system.

This condition determines which of the two surfaces corresponds to the stable phase on a particular side of the oc-fi coexistence line. According to this picture, a phase transition is associated with the intersection of Gibbs surfaces. Chemical potential surfaces for two phases.

If the two surfaces happen to join smoothly to one another, then v and 5 are continuous during the phase change. When that happens, the transition is called second order or higher order. A first-order transition is one in which, for example, v T, p is discontinuous. For a one-component system, a second-order transition can occur at only one point-a critical point.

In a two-component system, one can find lines of second-order phase transitions, which are called critical lines. An isotherm in the p-v plane. Notice that the right-hand side is ill-defined at a second- order phase transition.

Another way to view phase equilibria is to look at a thermo- dynamic plane in which one axis is an intensive field and the other axis is the conjugate variable to this field. For example, consider the p-v plane for a one-component system In Fig. Notice how v T,p is discontinuous i. Here is a puzzle to think about: For water near 1 atm pressure and 0oC temperature, the solid phase, ice I, has a larger volume per mole than the liquid.

Which one might be an isotherm for water? Wouldn t this behavior violate stability? Perhaps Fig. Isobar in the v-Tplane. For many systems the , picture looks like that shown in Fig. At times this representation is very informative. But sometimes it becomes difficult to use because v and T are not conjugate, and as a result, v T,p is not necessarily a monotonic function of T.

The a vs. A questionable Helmholz free energy per mole on an isotherm. Finally, since p is fixed by T when two phases are in equilibrium, the double tangent line drawn between u a and is the free energy per mole in the two-phase region v a!

Phase diagram for a simple material. One assumes for these theories that the instability is bridged by a phase transition located by a Maxwell construction the dashed line. Show that below a certain temperature the van der Waals equation of state implies a free energy that is unstable for some densities.

For example, in a one- component simple fluid like argon the phase diagram looks like the diagram pictured in Fig. If an approximate theory is used in which an instability is associated with a phase transition, the locus of points surrounding the unstable region is called the spinodal.

The spinodal must be enveloped by the coexistence curve. For example the van der Waals , equation yields a diagram like the one pictured in Fig.

If two phases are in equilibrium, there is a surface or interface of material between them. Let us now focus attention on this interface. See Fig. I refrain from treating advanced theoretical techniques e. I also treat only briefly the traditional statistical thermodynamics subjects of ideal gases and gas phase chemical equilibria. In the former case, this book should provide necessary background for the interested student to pursue advanced courses or readings in many-body theory.

In the latter case, since there are already so many excellent texts devoted to these topics, it is wastefully redundant to spend much time on them here.

Furthermore, these subjects have now become rather standard in the materials presented in undergraduate physical chemistry courses. Do you like this book? Within the context of that model, I discuss both mean field approximations and the renormalization group theory. In the latter case, I know of no other introductory text presenting a self-contained picture of this important subject.

Chapter 6 presents another very important subject not treated in other texts of this level—the Monte Carlo method. Here, I again use the Ising model as a concrete basis for discussion. Maris and L. Kadanoff, Am. The one-dimensional case serves to illustrate principles of quantum Monte Carlo.

The Metropolis algorithm is described, and programs are provided for the student to experiment with a microcomputer and explore the power and limitations of the method. In Chapter 7, we consider the equilibrium statistical mechanics of classical fluids. In chemistry, this is a very important topic since it provides the basis for understanding solvation. Some of the topics, such as the Maxwell-Boltzmann velocity distribution, are rather standard.

But others are less so. In particular, definitions and descriptions of pair correlation functions for both simple and molecular fluids are presented, the connection between these func- tions and X-ray scattering cross sections is derived, and their relationship to chemical equilibria in solutions is discussed. Finally, an illustration of Monte Carlo for a two-dimensional classical fluid of hard disks is presented which the student can run on a microcomputer.

The last chapter concerns dynamics—relaxation and molecular motion in macroscopic systems that are close to or at equilibrium. In particular, I discuss time correlation functions, the fluctuation-dis- sipation theorem and its consequences for understanding chemical kinetics, self-diffusion, absorption, and friction. Once again, in the context of modern science, these are very important and basic topics. But in terms of teaching the principles of non-equilibrium statistical mechanics, the subject has too often been considered as an advanced or special topic.

I am not sure why this is the case. A glance at Chapter 8 shows that one may derive the principal results such as the fluctuation-dissipation theorem with only a few lines of algebra, and without recourse to sophisticated mathematical methods e.

In all the chapters, I assume the reader has mastered the mathematical methods of a typical three-semester undergraduate calculus course. With that training, the student may find some of the mathematics challenging yet manageable. In this context, the most difficult portion of the book is Chapters 3 and 4 where the concepts of probability statistics are first encountered.

But since the material in these chapters is rather standard, even students with a weak background but access to a library have been able to rise to the occasion. Students who have taken the course based on this text have been advanced undergraduates or beginning graduates majoring in biochemistry, chemistry, chemical engineering, or physics. They usually master the material well enough to answer most of the numerous Exercise questions.

After their study of this book, I do hope a significant number of students will pursue more advanced treatments of the subjects than those I present here. The Bibliography at the end of each chapter suggests places for the students to start this reading. In this sense, this book serves as both an introduction and a guide to a discipline too broad and powerful for any one text to adequately describe.

In creating this book, I have benefited from the help of many people. John Wheeler has given his time unstintingly to help weed out logical errors and points of confusion. Encouragement and advice from Attila Szabo are greatly appreciated. I am also grateful to John Light for his helpful review of an earlier version of the text. Several students and my wife, Elaine, wrote and tested the computer programs included in the book.

Elaine provided a great deal of advice on the content of the book as well. Finally, I am indebted to Evelyn Carlier and Sandy Smith, respectively, for their expert manuscript preparations of the first and final versions of the text.

Philadelphia and Berkeley D. Thermodynamics, Fundamentals 1. First Law of Thermodynamics and Equilibrium 1. Second Law 1. Variational Statement of Second Law 1. Application: Thermal Equilibrium and Temperature 1. Auxiliary Functions and Legendre Transforms 1. Maxwell Relations 1. Extensive Functions and the Gibbs-Duhem Equation 1. Conditions for Equilibrium and Stability 2. Multiphase Equilibrium 2. Stability 2. Application to Phase Equilibria 2. Statistical Mechanics 3. The Statistical Method and Ensembles 3.

Microcanonical Ensemble and the Rational Foundation of Thermodynamics 3. A Simple Example 3. Generalized Ensembles and the Gibbs Entropy Formula 3.

Fluctuations Involving Uncorrelated Particles 3. Non-Interacting Ideal Systems 4. Occupation Numbers 4. Photon Gas 4. Ideal Gases of Real Particles 4. Electrons in Metals 4. Classical Ideal Gases, the Classical Limit 4. A Dilute Gas of Atoms 4. Dilute Gas of Diatomic Molecules 4. Statistical Mechanical Theory of Phase Transitions 5.

Ising Model 5. Lattice Gas 5. Broken Symmetry and Range of Correlations 5.



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