c 2010 by Harvey Gould and Jan Tobochnik 25 August 2010

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2 Chapter 1 Preface c 2010 by Harvey Gould and Jan Tobochnik 25 August 2010 This text is about two closely related subjects: thermodynamics and statistical mechanics. Thermodynamics is a general theory of macroscopic systems which provides limits on allowable physical processes involving energy transformations and relations between various measurable quantities. Its power is in its generality. Its limitation is that all quantitative predictions require empirical inputs. Statistical mechanics provides a microscopic foundation for thermodynamics and can be used to make quantitative predictions about macroscopic systems. Thermodynamics has always been important, but is of particular relevance at present because of important policy and engineering issues that require an understanding of thermal systems. These issues include global climate change, the search for new sources of energy, and the need for more efficient uses of energy. Statistical mechanics has become much more important in physics and related areas because its tools can now be implemented on computers and are much more widely applicable. These applications include lattice gauge theories in high energy physics, many problems in astrophysics, biological physics and geophysics, as well as topics traditionally considered outside of physics such as social networks and finance. Although statistical mechanics and thermodynamics are central to many research areas in physics and other sciences, both have had less of a presence in the undergraduate curriculum than classical mechanics, electromagnetism, and quantum mechanics. It wasn t many years ago that statistical mechanics was not even part of the undergraduate physics curriculum at many colleges and universities. Our text is part of an effort to bring some of the recent advances in research into the undergraduate curriculum. Thermodynamics and statistical mechanics are difficult to teach and to learn. The reasons for these difficulties include the following. There is not much emphasis on thermodynamics and statistical mechanics in the introductory physics course sequence, and what is taught in this context is typically not done well. Thermodynamics involves phenomenological reasoning without using microscopic information. This approach is little used in other undergraduate courses. 1

3 CHAPTER 1. PREFACE 2 Students have had little experience making the connection between microscopic and macroscopic phenomena. Many calculations are unfamiliar and involve the use of multivariable calculus. The usual notation is confusing because physicists use the same symbol to denote a physical quantity and a functional form. For example, S represents the entropy whether it is written as a function of the energy E, volume V, and particle number N, or if we replace E by the temperature T. Also the distinction between total and partial derivatives is sometimes confusing. These issues arise in other physics courses, but are more important in thermodynamics because of the many variables involved. A deep understanding of probability theory is important. Probability concepts often seem simple, and we usefully apply them in our everyday experience. However, probability is subtle as evidenced by the frequent misuse of statistics in news stories and the common tendency to attribute causality to events that are random. The solution of a single equation or set of equations such as Newton s laws in mechanics, Maxwell s equations in electrodynamics, and Schrödinger s equation in quantum mechanics is not central to statistical physics. Hence there are no standard procedures that work for a large class of problems, and many of the calculations are unfamiliar to students. There are few exactly solvable problems. There are many diverse applications of statistical and thermal physics, and hence the nature of undergraduate and graduate courses in this area vary more than courses in other subjects. Our text deals with these difficulties in various ways which we discuss in the following. However, we emphasize that learning thermodynamics and statistical mechanics is not easy. Understanding is a personal achievement, won only at the cost of constant intellectual struggle and reflection. No text can replace a good instructor and neither can replace disciplined thinking. One of the features of our text is its use of computer simulations and numerical calculations in a variety of contexts. The simulations and calculations can be used as lecture demonstrations, homework problems, or as a laboratory activity. Instructors and students need not have any background in the use of simulations, and all the simulations are given in a context in which students are asked to think about the results. Our experience is that it is important to discuss various models and algorithms so that students will more easily replace their naive mental models of matter by more explicit ones. For example, many students mental model of a gas is that the molecules act like billiard balls and give off heat when they collide. A discussion of the nature of the interactions between molecules and the algorithms used to simulate their motion according to Newton s second law can show students why the billiard ball model is inadequate and can help them replace their naive model by one in which the total energy is conserved. The simulations also provide useful visualizations of many of the ideas discussed in the text and help make abstract ideas more concrete. For example, Monte Carlo simulations help make the different ensembles more meaningful. We use simulations in the rich context of statistical physics, where discussion of the physics provides motivation for doing simulations, and consideration of various algorithms provides insight into the physics. In addition, the animations provide both motivation and another way of obtaining

4 CHAPTER 1. PREFACE 3 understanding. We also discuss some simple numerical programs that calculate various integrals that cannot be expressed in terms of well known functions. It is possible to use the text without ever running a program. The standard results that appear in other texts are derived in our text without the use of the computer. However, the simulations provide a concrete and visual representation of the models that are discussed in statistical mechanics, and thus can lead to a deeper understanding. Texts and physics courses frequently overemphasize the results obtained from physical theory and underemphasize physical insight. We have attempted to provide a balance between these two goals and provide different resources (simulations, various kinds of problems, and detailed discussion in the text) to help students learn to think like physicists. Suggested problems appear both within and at the end of each chapter. Those within the chapter are designed mainly to encourage students to read the text carefully and provide immediate reinforcement of basic ideas and techniques. Problems at the end of the chapters are designed to deepen student understanding and allow students to practice the various tools that they have learned. Some problems are very straightforward and others will likely not be solved by most students. The more difficult problems provide a useful starting point for stimulating student thinking and can be used as the basis of class discussions. We have tried to provide a wide range of problems for students with various backgrounds and abilities. We do not assume any special background in thermal physics. Chapter 1 discusses the important ideas of thermodynamics and statistical physics by appealing to everyday observations and qualitative observations of simple computer simulations. A useful prop during the first several weeks of classes is a cup of hot water. Students are not surprised to hear that the water always cools and are pleased that they are already on their way to understanding one of the basic ideas of thermal systems the need for a quantity (the entropy) in addition to energy for explaining the unidirectional behavior of macroscopic systems. It has become fashionable since the 1960s to integrate thermodynamics and statistical mechanics. (Before then, thermodynamics was taught separately, and statistical mechanics was not offered at the undergraduate level at many colleges and universities.) The result of this integration is that many undergraduates and graduate students are not exposed to the phenomenological way of thinking exemplified by thermodynamics. We devote all of Chapter 2 to this way of reasoning without resorting to statistical mechanics, although we often refer back to what was discussed in Chapter 1. Besides the intrinsic importance of learning thermodynamics, its phenomenological way of reasoning using a few basic principles and empirical observations is as important as ever. Most thermal physics texts discuss the ideas of probability as they are needed to understand the physics. We are sympathetic to this approach because abstract ideas are usually easier to understand in context. However, there is much that is unfamiliar to students while they are learning statistical mechanics, and it is important that students have a firm understanding of probability before tackling problems in statistical mechanics. Our discussion of probability in Chapter 3 includes discussions of random additive processes and the central limit theorem. The latter plays an essential role in statistical mechanics, but it is hardly mentioned in most undergraduate texts. Because the ideas of probability are widely applicable, we also discuss probability in other contexts including Bayes theorem. These other applications can be skipped without loss of continuity. We include them because a topic such as Bayes theorem and traffic flow can excite some students and motivate them to delve deeper into the subject.

5 CHAPTER 1. PREFACE 4 There is a tendency for thermal physics books to look like a long list of disconnected topics. We have organized the subject differently by having just two chapters, Chapter 2 and Chapter 4 on the essential methodology of thermodynamics and statistical mechanics, respectively. These chapters also contain many simple applications because it is difficult to learn a subject in the abstract. In particular, we consider the Einstein solid (a system of harmonic oscillators), discuss the demon algorithm as an ideal thermometer, and use the ideal gas as a model of a dilute gas. After discussing the conceptual foundations of statistical mechanics, most thermal physics texts next discuss the ideal classical gas and then the ideal Fermi and Bose gases. One problem with this approach is that students frequently confuse the single particle density of states with the density of states of the system as a whole and do not grasp the common features of the treatment of ideal quantum gases. We follow Chapter 4 by applying the tools of statistical mechanics in Chapter 5 to the Ising model, a model that is well known to physicists and others, but seldom encountered by undergraduates. The consideration of lattice systems allows us to apply the canonical ensemble to small systems for which students can enumerate the microstates and apply the Metropolis algorithm. In addition, we introduce mean-field theory and emphasize the importance of cooperative effects. Chapter 6 discusses the ideal classical gas using both the canonical and grand canonical ensembles and the usual applications to ideal Fermi and Bose gases. We discuss these topics in a single chapter to emphasize that all treatments of ideal gases involve the single particle density of states. We include programs that calculate numerically several integrals for the ideal Bose and Fermi gases so that the chemical potential as a function of the temperature for fixed density can be determined. In addition we emphasize the limited nature of the equipartition theorem for classical systems and the general applicability of the Maxwell velocity and speed distributions for any classical system, regardless of the nature of the interactions between particles and the phase. The first part of Chapter 7 discusses the nature of the chemical potential by considering some simple models and simulations. We then discuss the role of the chemical potential in understanding phase transitions with a focus on the van der Waals equation of state. Most of the material in the first six chapters and Chapter 7 through Section 7.3 would form the core of a one semester course in statistical and thermal physics. Starred sections in each chapter can be omitted without loss of continuity. It is unlikely that Chapter 8 on classical gases and liquids and Chapter 9 on critical phenomena can be discussed in a one semester course, even in a cursory manner. We hope that their presence will encourage some undergraduates to continue their study of statistical physics. Their main use is likely to be as material for special projects. It is possible to start a course with the development of statistical mechanics in Chapter 4 after a review of the first three chapters. The key results needed from Chapters 1 3 are the fundamental equation de = TdS PdV + µdn, definitions of thermodynamic quantities such as temperature, pressure, and chemical potential in terms of the various thermodynamic potentials, the definitions of the heat capacity and compressibility, and an understanding of how to obtain average quantities from a knowledge of the probability distribution. Of course, knowing these results is not the same as understanding them. Besides including more material than can be covered in a one semester course, our text has some deliberate omissions. Many undergraduate thermal physics texts invoke kinetic theory arguments to derive the ideal gas pressure equation of state. In contrast, our text does not discuss kinetic theory at all. One of the themes of our text is that time is irrelevant in equilibrium statis-

6 CHAPTER 1. PREFACE 5 tical mechanics. We suspect that kinetic theory arguments confuse students if presented early in their learning of statistical mechanics. Part of the confusion is probably associated with the fact that the ideal gas pressure equation of state is derived in statistical mechanics by ignoring the interactions between particles, but the kinetic theory derivation of the same equation of state explicitly includes these interactions. Similarly, many derivations of the Maxwell velocity distribution give the misleading impression that it is applicable only to a dilute gas. A solutions manual for instructors is available from Princeton University Press. In addition to the chapters available via the Princeton University Press website, <press.princeton.edu/titles/9375 we plan to add more chapters on topics in statistical physics to the STP (Statistical and Thermal Physics) collection of the ComPADRE digital library (< In particular, we plan to add a chapter on kinetic theory and other dynamical phenomena and a chapter on applications of statistical physics to nonthermal systems. The emphasis of the latter will be on applications to current research problems. In addition, there are resources for the teaching of statistical and thermal physics from other teachers on the ComPADRE website. The latter is a network of free online resource collections supporting faculty, students, and teachers in physics and astronomy education. We encourage others to submit materials on statistical and thermal physics to < The software associated with our text is available in several formats and can be downloaded from < or <press.princeton.edu/titles/9375.html>. All the programs are open source. The compiled programs are jar files and can be run like any other program on your computer by double-clicking on the file. The applications can also be run in applet mode when embedded into an html page and run within a browser. (Directions for doing so are available on both websites.) Java 1.5 or greater must be installed on your computer to run the programs. Java can be downloaded at no cost from <java.com> for Windows and Linux and is included with Mac OS X. All of the programs mentioned in the text and a few others are packaged together with related curricular material in the STP Launcher. This convenient way of organizing and running other Java programs was developed by Doug Brown. Alternatively, you can download the programs individually. The STP programs generally have a wider range of inputs and outputs, but cannot be modified without downloading the source code and recompiling it. The EJS (Easy Java Simulations) programs usually have a simpler interface, but the source code is part of the jar file and can be easily modified. To do the latter it is necessary to download EJS from < We have many people to thank. In particular, we are especially grateful to Louis Colonna- Romano for drawing almost all of the figures. Lou writes programs in postscript the way others write programs in Java or Fortran. Milton Cole originally encouraged us to write a text on statistical and thermal physics and to apply for a National Science Foundation grant. The two NSF grants, PHY and DUE , we have received have given us and our students the resources to write the software associated with the text and the opportunity to attend conferences and workshops where we have been stimulated by discussions with many colleagues. We would like to thank Jill Larkin and Fred Reif for their useful advice and encouragement at the beginning of the STP project. We used Fred Reif s classic thermal physics text for many years and our text shows its strong influence. We owe a special debt to Wolfgang Christian, Anne Cox, Doug Brown, Francisco Esquembre, and Mario Belloni of the Open Source Physics Project, < for

7 CHAPTER 1. PREFACE 6 their many direct and indirect contributions to the STP project. The software would not have been possible without Wolfgang Christian s yeoman work developing and managing the Java-based Open Source Physics Library and its many APIs. Anne Cox helped convert most of the STP curricular material to EJS, set up the STP collection on ComPADRE, and provided many of the solutions to the first three chapters with the help of her student Chris Bauer. We thank Anne Cox, Bill Klein, and Beate Schmittman for using drafts of our text as their course textbook. We also would like to thank Milton Cole, Don Lemons, Jon Machta, Irwin Oppenheim, Sid Redner, and Royce Zia for reading individual chapters and making useful suggestions. Our students have been patient (not that they had much choice), and many of them gave us valuable feedback. In particular, we would like to thank Kipton Barros, Sarah Brent, Ranjit Chacko, Andy Coniglio, Jake Ellowitz, Michael Robitaille, Chris Sataline, Hui Wang, Laurel Winter, and Junchao Xia for their questions and suggestions. Jake Ellowitz also wrote many of the solutions. Drafts of the text have been available online for several years. We have benefited from the feedback that we have received from people who have written to us to correct typos and unclear passages. These people include Bernard Chasan, Pedro Cruz, Oscar R. Enriquez, Jim Fox, Rob Kooijman, Matus Medo, Nedialko M. Nedeltchev, Pouria Pedram, Jan Ryckebusch, and Oruganti Shanker. We also appreciate the general comments that we have received about the usefulness of the online text. Individual chapters of the text will remain freely available online at < (The page numbers of the online version will differ from the printed copy.) We remember Lynna Spornick who worked for many years at The Johns Hopkins University Applied Physics Laboratory, and who first worked with us on incorporating computer simulations into the teaching of statistical mechanics and thermodynamics. It has been a pleasure to work with Ingrid Gnerlich of Princeton University Press. Her encouragement and flexibility have made it possible for us to complete this project and see it in print. We would also like to thank Brigitte Pelner for her patience in correcting the many errors we found after submitting our manuscript. We thank our colleagues at Clark University and Kalamazoo College for their encouragement. In particular, we would like to thank Sujata Davis and Peggy Cauchy without whom our departments would cease to function. We are grateful to our friends and especially our families, Patti Orbuch Gould, Andrea Moll Tobochnik, Steven Tobochnik, Howard Tobochnik, Joshua and Rachel Zucker Gould, Emily and Ventura Rodriguez, and Evan Gould, for their understanding and patience during the completion of the text. In this day of easy internet access, it is still difficult to get feedback other than general comments. We would be grateful for s regarding corrections (typos and otherwise) and suggestions for improvements. Despite the comments and suggestions we have received, there are probably still more typos, errors, and unclear explanations. Blame them on entropy. Harvey Gould, <hgould@clarku.edu> Jan Tobochnik, <jant@kzoo.edu>

8 Contents 1 FromMicroscopictoMacroscopicBehavior Introduction Some Qualitative Observations Doing Work and the Quality of Energy Some Simple Simulations Measuring the Pressure and Temperature Work, Heating, and the First Law of Thermodynamics *The Fundamental Need for a Statistical Approach *Time and Ensemble Averages Models of Matter The ideal gas Interparticle potentials Lattice models Importance of Simulations Dimensionless Quantities Summary Supplementary Notes Approach to equilibrium Mathematics refresher Vocabulary Additional Problems Suggestions for Further Reading ThermodynamicConcepts Introduction The System Thermodynamic Equilibrium i

9 CONTENTS ii 2.4 Temperature Pressure Equation of State Some Thermodynamic Processes Work The First Law of Thermodynamics Energy Equation of State Heat Capacities and Enthalpy Quasistatic Adiabatic Processes The Second Law of Thermodynamics The Thermodynamic Temperature The Second Law and Heat Engines Entropy Changes Equivalence of Thermodynamic and Ideal Gas Scale Temperatures The Thermodynamic Pressure The Fundamental Thermodynamic Relation The Entropy of an Ideal Classical Gas The Third Law of Thermodynamics Free Energies Thermodynamic Derivatives *Applications to Irreversible Processes Joule or free expansion process Joule-Thomson process Supplementary Notes The mathematics of thermodynamics Thermodynamic potentials and Legendre transforms Vocabulary Additional Problems Suggestions for Further Reading Conceptsof Probability Probability in Everyday Life The Rules of Probability Mean Values The Meaning of Probability Information and uncertainty *Bayesian inference Bernoulli Processes and the Binomial Distribution

10 CONTENTS iii 3.6 Continuous Probability Distributions The Central Limit Theorem (or Why Thermodynamics Is Possible) *The Poisson Distribution or Should You Fly? *Traffic Flow and the Exponential Distribution *Are All Probability Distributions Gaussian? *Supplementary Notes Method of undetermined multipliers Derivation of the central limit theorem Vocabulary Additional Problems Suggestions for Further Reading StatisticalMechanics Introduction A Simple Example of a Thermal Interaction Counting Microstates Noninteracting spins A particle in a one-dimensional box One-dimensional harmonic oscillator One particle in a two-dimensional box One particle in a three-dimensional box Two noninteracting identical particles and the semiclassical limit The Number of States of Many Noninteracting Particles: Semiclassical Limit The Microcanonical Ensemble (Fixed E, V,andN) The Canonical Ensemble (Fixed T, V,andN) Connection Between Thermodynamics and Statistical Mechanics in the Canonical Ensemble Simple Applications of the Canonical Ensemble An Ideal Thermometer Simulation of the Microcanonical Ensemble Simulation of the Canonical Ensemble Grand Canonical Ensemble (Fixed T, V,andµ) *Entropy is not a Measure of Disorder Supplementary Notes The volume of a hypersphere Fluctuations in the canonical ensemble Vocabulary Additional Problems

11 CONTENTS iv Suggestions for Further Reading MagneticSystems Paramagnetism Noninteracting Magnetic Moments Thermodynamics of Magnetism The Ising Model The Ising Chain Exact enumeration Spin-spin correlation function Simulations of the Ising chain *Transfer matrix Absence of a phase transition in one dimension The Two-Dimensional Ising Model Onsager solution Computer simulation of the two-dimensional Ising model Mean-Field Theory *Phase diagram of the Ising model *Simulation of the Density of States *Lattice Gas Supplementary Notes The Heisenberg model of magnetism Low temperature expansion High temperature expansion *Bethe approximation Fully connected Ising model Metastability and nucleation Vocabulary Additional Problems Suggestions for Further Reading Many-Particle Systems The Ideal Gas in the Semiclassical Limit Classical Statistical Mechanics The equipartition theorem The Maxwell velocity distribution The Maxwell speed distribution

12 CONTENTS v 6.3 Occupation Numbers and Bose and Fermi Statistics Distribution Functions of Ideal Bose and Fermi Gases Single Particle Density of States Photons Nonrelativistic particles The Equation of State of an Ideal Classical Gas: Application of the Grand Canonical Ensemble Blackbody Radiation The Ideal Fermi Gas Ground state properties Low temperature properties The Heat Capacity of a Crystalline Solid The Einstein model Debye theory The Ideal Bose Gas and Bose Condensation Supplementary Notes Fluctuations in the number of particles Low temperature expansion of an ideal Fermi gas Vocabulary Additional Problems Suggestions for Further Reading The ChemicalPotential and Phase Equilibria Meaning of the chemical potential Measuring the chemical potential in simulations The Widom insertion method The chemical demon algorithm Phase Equilibria Equilibrium conditions Simple phase diagrams Clausius-Clapeyron equation The van der Waals Equation of State Maxwell construction *The van der Waals critical point *Chemical Reactions Vocabulary Additional Problems Suggestions for Further Reading

13 CONTENTS vi 8 ClassicalGasesand Liquids Introduction Density Expansion The Second Virial Coefficient *Diagrammatic Expansions Cumulants High temperature expansion Density expansion Higher order virial coefficients for hard spheres The Radial Distribution Function Perturbation Theory of Liquids The van der Waals equation *The Ornstein-Zernicke Equation and Integral Equationsforg(r) *One-Component Plasma Supplementary Notes The third virial coefficient for hard spheres Definition of g(r) in terms of the local particle density X-ray scattering and the static structure function Vocabulary Additional Problems Suggestions for Further Reading CriticalPhenomena Landau Theory of Phase Transitions Universality and Scaling Relations A Geometrical Phase Transition Renormalization Group Method for Percolation The Renormalization Group Method and the One-Dimensional Ising Model The Renormalization Group Method and the Two-Dimensional Ising Model Vocabulary Additional Problems Suggestions for Further Reading A.1 Physical Constants and Conversion Factors A.2 Hyperbolic Functions A.3 Approximations A.4 Euler-Maclaurin Formula A.5 Gaussian Integrals

14 CONTENTS vii A.6 Stirling s Approximation A.7 Bernoulli Numbers A.8 Probability Distributions A.9 Fourier Transforms A.10 The Delta Function A.11 Convolution Integrals A.12 Fermi and Bose Integrals

15 Chapter 1 From Microscopic to Macroscopic Behavior c 2010 by Harvey Gould and Jan Tobochnik 4 May 2010 We explore the fundamental differences between microscopic and macroscopic systems, note that bouncing balls come to rest and hot objects cool, and discuss how the behavior of macroscopic systems is related to the behavior of their microscopic constituents. Computer simulations are introduced to demonstrate the general qualitative behavior of macroscopic systems. 1.1 Introduction Our goal is to understand the properties of macroscopic systems, that is, systems of many electrons, atoms, molecules, photons, or other constituents. Examples of familiar macroscopic objects include systems such as the air in your room, a glass of water, a coin, and a rubber band examples of a gas, liquid, solid, and polymer, respectively. Less familiar macroscopic systems include superconductors, cell membranes, the brain, the stock market, and neutron stars. We will find that the type of questions we ask about macroscopic systems differ in important ways from the questions we ask about systems that we treat microscopically. For example, consider the air in your room. Have you ever wondered about the trajectory of a particular molecule in the air? Would knowing that trajectory be helpful in understanding the properties of air? Instead of questions such as these, examples of questions that we do ask about macroscopic systems include the following: 1. How does the pressure of a gas depend on the temperature and the volume of its container? 2. How does a refrigerator work? How can we make it more efficient? 3. How much energy do we need to add to a kettle of water to change it to steam? 1

16 CHAPTER 1. FROM MICROSCOPIC TO MACROSCOPIC BEHAVIOR 2 4. Why are the properties of water different from those of steam, even though water and steam consist of the same type of molecules? 5. How and why does a liquid freeze into a particular crystalline structure? 6. Why does helium have a superfluid phase at very low temperatures? Why do some materials exhibit zero resistance to electrical current at sufficiently low temperatures? 7. In general, how do the properties of a system emerge from its constituents? 8. How fast does the current in a river have to be before its flow changes from laminar to turbulent? 9. What will the weather be tomorrow? These questions can be roughly classified into three groups. Questions 1 3 are concerned with macroscopic properties such as pressure, volume, and temperature and processes related to heating and work. These questions are relevant to, thermodynamics which provides a framework for relating the macroscopic properties of a system to one another. Thermodynamics is concerned only with macroscopic quantities and ignores the microscopic variables that characterize individual molecules. For example, we will find that understanding the maximum efficiency of a refrigerator does not require a knowledge of the particular liquid used as the coolant. Many of the applications of thermodynamics are to engines, for example, the internal combustion engine and the steam turbine. Questions 4 7 relate to understanding the behavior of macroscopic systems starting from the atomic nature of matter. For example, we know that water consists of molecules of hydrogen and oxygen. We also know that the laws of classical and quantum mechanics determine the behavior of molecules at the microscopic level. The goal of statistical mechanics is to begin with the microscopic laws of physics that govern the behavior of the constituents of the system and deduce the properties of the system as a whole. Statistical mechanics is a bridge between the microscopic and macroscopic worlds. Question 8 also relates to a macroscopic system, but temperature is not relevant in this case. Moreover, turbulent flow continually changes in time. Question 9 concerns macroscopic phenomena that change with time. Although there has been progress in our understanding of time-dependent phenomena such as turbulent flow and hurricanes, our understanding of such phenomena is much less advanced than our understanding of time-independent systems. For this reason we will focus our attention on systems whose macroscopic properties are independent of time and consider questions such as those in Questions Some Qualitative Observations We begin our discussion of macroscopic systems by considering a glass of hot water. We know that if we place a glass of hot water into a large cold room, the hot water cools until its temperature equals that of the room. This simple observation illustrates two important properties associated with macroscopic systems the importance of temperature and the arrow of time. Temperature

17 CHAPTER 1. FROM MICROSCOPIC TO MACROSCOPIC BEHAVIOR 3 is familiar because it is associated with the physiological sensations of hot and cold and is important in our everyday experience. The direction or arrow of time raises many questions. Have you ever observed a glass of water at room temperature spontaneously become hotter? Why not? What other phenomena exhibit a direction of time? The direction of time is expressed by the nursery rhyme: Humpty Dumpty sat on a wall Humpty Dumpty had a great fall All the king s horses and all the king s men Couldn t put Humpty Dumpty back together again. Is there a direction of time for a single particle? Newton s second law for a single particle, F = dp/dt, implies that the motion of particles is time reversal invariant; that is, Newton s second law looks the same if the time t is replaced by t and the momentum p by p. There is no direction of time at the microscopic level. Yet if we drop a basketball onto a floor, we know that it will bounce and eventually come to rest. Nobody has observed a ball at rest spontaneously begin to bounce, and then bounce higher and higher. So based on simple everyday observations, we can conclude that the behaviors of macroscopic bodies and single particles are very different. Unlike scientists of about a century or so ago, we know that macroscopic systems such as a glass of water and a basketball consist of many molecules. Although the intermolecular forces in water produce a complicated trajectory for each molecule, the observable properties of water are easy to describe. If we prepare two glasses of water under similar conditions, we know that the observable properties of the water in each glass are indistinguishable, even though the motion of the individual particles in the two glasses is very different. If we take into account that the bouncing ball and the floor consist of molecules, then we know that the total energy of the ball and the floor is conserved as the ball bounces and eventually comes to rest. Why does the ball eventually come to rest? You might be tempted to say the cause is friction, but friction is just a name for an effective or phenomenological force. At the microscopic level we know that the fundamental forces associated with mass, charge, and the nucleus conserve total energy. Hence, if we include the energy of the molecules of the ball and the floor, the total energy is conserved. Conservation of energy does not explain why the inverse process, where the ball rises higher and higher with each bounce, does not occur. Such a process also would conserve the total energy. So a more fundamental explanation is that the ball comes to rest consistent with conservation of the total energy and with some other principle of physics. We will learn that this principle is associated with an increase in the entropy of the system. For now, entropy is just a name, and it is important only to understand that energy conservation is not sufficient to understand the behavior of macroscopic systems. 1 By thinking about the constituent molecules, we can gain some insight into the nature of entropy. Let us consider the ball bouncing on the floor again. Initially, the energy of the ball is associated with the motion of its center of mass, and we say that the energy is associated with one degree of freedom. After some time the energy becomes associated with the individual molecules near the surface of the ball and the floor, and we say that the energy is now distributed over many degrees of freedom. If we were to bounce the ball on the floor many times, the ball and the 1 We will learn that as for most concepts in physics, the meaning of entropy in the context of thermodynamics and statistical mechanics is very different from its meaning as used by nonscientists.

18 CHAPTER 1. FROM MICROSCOPIC TO MACROSCOPIC BEHAVIOR 4 floor would each feel warm to our hands. So we can hypothesize that energy has been transferred from one degree of freedom to many degrees of freedom while the total energy has been conserved. Hence, we conclude that the entropy is a measure of how the energy is distributed. What other quantities are associated with macroscopic systems besides temperature, energy, and entropy? We are already familiar with some of these quantities. For example, we can measure the air pressure in a basketball and its volume. More complicated quantities are the thermal conductivity of a solid and the viscosity of oil. How are these macroscopic quantities related to each other and to the motion of the individual constituent molecules? The answers to questions such as these and the meaning of temperature and entropy will take us through many chapters Doing Work and the Quality of Energy We already have observed that hot objects cool, and cool objects do not spontaneously become hot; bouncing balls come to rest, and a stationary ball does not spontaneously begin to bounce. And although the total energy is conserved in these processes, the distribution of energy changes in an irreversible manner. We also have concluded that a new concept, the entropy, needs to be introduced to explain the direction of change of the distribution of energy. Now let us take a purely macroscopic viewpoint and discuss how we can arrive at a similar qualitative conclusion about the asymmetry of nature. This viewpoint was especially important historically because of the lack of a microscopic theory of matter in the 19th century when the laws of thermodynamics were being developed. Consider the conversion of stored energy into heating a house or a glass of water. The stored energy could be in the form of wood, coal, or animal and vegetable oils for example. We know that this conversion is easy to do using simple methods, for example, an open flame. We also know that if we rub our hands together, they will become warmer. There is no theoretical limit to the efficiency at which we can convert stored energy to energy used for heating an object. What about the process of converting stored energy into work? Work, like many of the other concepts that we have mentioned, is difficult to define. For now let us say that doing work is equivalent to the raising of a weight. To be useful, we need to do this conversion in a controlled manner and indefinitely. A single conversion of stored energy into work such as the explosion of dynamite might demolish an unwanted building, but this process cannot be done repeatedly with the same materials. It is much more difficult to convert stored energy into work and the discovery of ways to do this conversion led to the industrial revolution. In contrast to the primitiveness of an open flame, we have to build an engine to do this conversion. Can we convert stored energy into useful work with 100% efficiency? To answer this question we have to appeal to observation. We know that some forms of stored energy are more useful than others. For example, why do we burn coal and oil in power plants even though the atmosphere and the oceans are vast reservoirs of energy? Can we mitigate global climate change by extracting energy from the atmosphere to run a power plant? From the work of Kelvin, Clausius, Carnot and others, we know that we cannot convert stored energy into work with 100% efficiency, and we must necessarily waste some of the energy. At this point, it is easier to understand the reason 2 Properties such as the thermal conductivity and viscosity are treated in the online supplement (see < or <press.princeton.edu/titles/9375.html>).

19 CHAPTER 1. FROM MICROSCOPIC TO MACROSCOPIC BEHAVIOR 5 for this necessary inefficiency by microscopic arguments. For example, the energy in the gasoline of the fuel tank of an automobile is associated with many molecules. The job of the automobile engine is to transform this (potential) energy so that it is associated with only a few degrees of freedom, that is, the rolling tires and gears. It is plausible that it is inefficient to transfer energy from many degrees of freedom to only a few. In contrast, the transfer of energy from a few degrees of freedom (the firewood) to many degrees of freedom (the air in your room) is relatively easy. The importance of entropy, the direction of time, and the inefficiency of converting stored energy into work are summarized in the various statements of the second law of thermodynamics. It is interesting that the second law of thermodynamics was conceived before the first law of thermodynamics. As we will learn, the first law is a statement of conservation of energy. Suppose that we take some firewood and use it to heat a sealed room. Because of energy conservation, the energy in the room plus the firewood is the same before and after the firewood has been converted to ash. Which form of the energy is more capable of doing work? You probably realize that the firewood is a more useful form of energy than the hot air and ash that exists after the firewood is burned. Originally the energy was stored in the form of chemical (potential) energy. Afterward the energy is mostly associated with the motion of the molecules in the air. What has changed is not the total energy, but its ability to do work. We will learn that an increase in entropy is associated with a loss of ability to do work. We have an entropy problem, not an energy problem. 1.4 Some Simple Simulations So far we have discussed the behavior of macroscopic systems by appealing to everyday experience and simple observations. We now discuss some simple ways of simulating the behavior of macroscopic systems. Although we cannot simulate a macroscopic system of particles on a computer, we will find that even small systems of the order of a 100 particles are sufficient to illustrate the qualitative behavior of macroscopic systems. We first discuss how we can simulate a simple model of a gas consisting of molecules whose internal structure can be ignored. In particular, imagine a system of N particles in a closed container of volume V and suppose that the container is far from the influence of external forces such as gravity. We will usually consider two-dimensional systems so that we can easily visualize the motion of the particles. For simplicity, we assume that the motion of the particles is given by classical mechanics, and hence we need to solve Newton s second law for each particle. To compute the total force on each particle we have to specify the nature of the interaction between the particles. We will assume that the force between any pair of particles depends only on the distance between them. This simplifying assumption is applicable to simple liquids such as liquid argon, but not to water. We will also assume that the particles are not charged. The force between any two particles is repulsive when their separation is small and weakly attractive when they are far apart. For convenience we

20 CHAPTER 1. FROM MICROSCOPIC TO MACROSCOPIC BEHAVIOR 6 u LJ (r) ε 1.0 σ 2.0 σ 3.0 σ r Figure 1.1: Plot of the Lennard-Jones potential u LJ (r), where r is the distance between the particles. Note that the potential is characterized by a length σ and an energy ǫ. will usually assume that the interaction is given by the Lennard-Jones potential given by 3 [ (σ ) 12 ( σ ) ] 6 u LJ (r) = 4ǫ, (1.1) r r where r is the distance between two particles. A plot of the Lennard-Jones potential is shown in Figure 1.1. The r 12 form of the repulsive part of the interaction was chosen for convenience only and has no fundamental significance. The attractive 1/r 6 behavior at large r is the van der Waals interaction. 4 The force between any two particles is given by f(r) = du/dr. In macroscopic systems the fraction of particles near the walls of the container is negligibly small. However, the number of particles that can be studied in a simulation is typically For these small systems the fraction of particles near the walls of the container would be significant, and hence the behavior of such a system would be dominated by surface effects. The most common way of minimizing surface effects and to simulate more closely the properties of a macroscopic system is to use what are known as toroidal boundary conditions. These boundary conditions are familiar to computer game players. For example, a particle that exits the right edge of the box, reenters the box from the left side. In one dimension this boundary condition is equivalent to taking a piece of string and making it into a loop. In this way a particle moving on the wire never reaches the end (the surface). Given the form of the interparticle potential, we can determine the total force on each particle due to all the other particles in the system. We then use Newton s second law of motion to find 3 This potential is named after John Lennard-Jones, , a theoretical chemist and physicist at Cambridge University. The Lennard-Jones potential is appropriate for closed-shell systems, that is, rare gases such as Ar or Kr. 4 The van der Waals interaction arises from an induced dipole-dipole effect. It is present in all molecules, but is important only for the heavier noble gas atoms. See for example, Brehm and Mullin.

21 CHAPTER 1. FROM MICROSCOPIC TO MACROSCOPIC BEHAVIOR 7 the acceleration of each particle. Because the acceleration is the second derivative of the position, we need to solve a second-order differential equation for each particle (in each direction) given the initial position and velocity of each particle. (For a two-dimensional system of N particles, we would have to solve 2N differential equations.) These differential equations are coupled because the acceleration of a particular particle depends on the positions of all the other particles. Although we cannot solve the resultant set of coupled differential equations analytically, we can use straightforward numerical methods to solve these equations to a good approximation. This way of simulating dense gases, liquids, solids, and biomolecules is called molecular dynamics. 5 In the following two problems we will explore some of the qualitative properties of macroscopic systems by doing some simple simulations. Before you actually do the simulations, think about what you believe the results will be. In many cases the most valuable part of the simulation is not the simulation itself, but the act of thinking about a concrete model and its behavior. The simulations are written in Java and can be run on any operating system that supports Java You may download all the programs used in this text from <press.princeton.edu/titles/9375.html> or < as a single file (the STP Launcher) or as individual programs. Alternatively, you can run each simulation as an applet using a browser. Problem 1.1. Approach to equilibrium Suppose that we divide a box into three equal parts and place N particles in the middle third of the box. 6 The particles are placed at random with the constraint that no two particles can be closer than the length parameter σ. This constraint prevents the initial force between any two particles from being too big, which would lead to the breakdown of the numerical method used to solve the differential equations. The velocity of each particle is assigned at random and then the velocity of the center of mass is set to zero. At t = 0, we remove the barriers between the three parts and watch the particles move according to Newton s equations of motion. We say that the removal of the barrier corresponds to the removal of an internal constraint. What do you think will happen? Program ApproachToEquilibriumThreePartitions implements this simulation. 7 Double click on the jar file to open the program, and click the Start button to begin the simulation. The program shows the motion of the particles in the box and plots the number of particles in the left (n 1 ), center (n 2 ), and right (n 3 ) part of the box as a function of time. The input parameter is N, the number of particles initially in the center cell, so that n 1 = 0, n 2 = N, and n 3 = 0 at t = 0. Give your answers to the following questions before you do the simulation. (a) Does the system appear to show a direction of time for N = 6? (b) What is the nature of the time dependence of n 1, n 2, and n 3 as a function of the time t for N = 27? Does the system appear to show a direction of time? Choose various values of N that are multiples of 3 up to N = 270. Is the direction of time better defined for larger N? 5 The nature of molecular dynamics is discussed in Gould, Tobochnik, and Christian, Chap We have divided the box into three parts so that the effects of the toroidal boundary conditions will not be as apparent as if we had initially confined the particles to one half of the box. 7 We will omit the prefix stp whenever it will not cause confusion.