Statistical and thermal physics: With computer application

497 Pages·2010·3.532 MB·English

by  Gould H., Tobochnik J.

Chapter 1
Preface
(cid:13)c2010 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
empiricalinputs. Statisticalmechanicsprovidesamicroscopicfoundationforthermodynamicsand
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
climatechange,thesearchfornewsourcesofenergy,andtheneedformoreefficientusesofenergy.
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 ap-
plications include lattice gauge theories in high energy physics, many problems in astrophysics,
biologicalphysics andgeophysics,as wellas topics traditionallyconsideredoutside ofphysics such
as social networks and finance.
Although statistical mechanics and thermodynamics are central to many research areas in
physics andother sciences,both havehadless ofa presencein the undergraduatecurriculumthan
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
anduniversities. Our text is partofaneffortto bring someofthe recentadvancesin researchinto
the undergraduate curriculum.
Thermodynamicsandstatisticalmechanicsaredifficulttoteachandto learn. The reasonsfor
these difficulties include the following.
• Thereisnotmuchemphasisonthermodynamicsandstatisticalmechanicsintheintroductory
physics course sequence, and what is taught in this context is typically not done well.
• Thermodynamics involves phenomenological reasoning without using microscopic informa-
tion. This approach is little used in other undergraduate courses.
1
CHAPTER 1. PREFACE 2
• Students have had little experience making the connection between microscopic and macro-
scopic phenomena.
• Manycalculationsareunfamiliarandinvolvetheuseofmultivariablecalculus. Theusualno-
tationisconfusing becausephysicistsusethe samesymboltodenote aphysicalquantityand
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,andweusefullyapplytheminoureverydayexperience. However,probabilityissubtle
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 Schr¨odinger’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.
• Therearemany diverseapplicationsofstatisticalandthermalphysics,andhence the nature
ofundergraduateandgraduatecoursesinthisareavarymorethancoursesinothersubjects.
Our text deals with these difficulties in various ways which we discuss in the following. How-
ever, we emphasize that learning thermodynamics and statistical mechanics is not easy. Under-
standing 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 algorithmsso 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
alsoprovideusefulvisualizationsofmanyoftheideasdiscussedinthetextandhelpmakeabstract
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
providesmotivationfordoingsimulations,andconsiderationofvariousalgorithmsprovidesinsight
intothephysics. Inaddition,theanimationsprovidebothmotivationandanotherwayofobtaining
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 derivedin our text without the use of the computer. However,the simu-
lations 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
overemphasizethe resultsobtainedfromphysicaltheoryandunderemphasizephysicalinsight. We
haveattemptedto providea balancebetweenthesetwogoalsandprovidedifferentresources(sim-
ulations, 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
chapteraredesignedmainlytoencouragestudentstoreadthetextcarefullyandprovideimmediate
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 donotassumeanyspecialbackgroundinthermalphysics. Chapter1discussesthe impor-
tant 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
ofthermalsystems– the needfora quantity(the entropy)inadditionto energyfor explainingthe
unidirectional behavior of macroscopic systems.
It has become fashionable since the 1960s to integrate thermodynamics and statistical me-
chanics. (Before then, thermodynamics was taught separately, and statistical mechanics was not
offeredattheundergraduatelevelatmanycollegesanduniversities.) Theresultofthisintegration
is that many undergraduatesand graduatestudents are notexposedto the phenomenologicalway
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
understandincontext. However,thereismuchthatisunfamiliartostudentswhiletheyarelearning
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.
Becausetheideasofprobabilityarewidelyapplicable,wealsodiscussprobabilityinothercontexts
including Bayes’theorem. These other applications can be skipped without loss of continuity. We
include thembecausea topicsuchasBayes’theoremandtrafficflowcanexcite somestudents and
motivate them to delve deeper into the subject.
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 approachisthat students frequentlyconfuse the singleparticle densityofstates withthe
density of states of the system as a whole and do not graspthe 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
encounteredbyundergraduates. Theconsiderationoflatticesystemsallowsustoapplythecanon-
ical 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 firstpartofChapter 7discussesthe natureofthe chemicalpotentialby consideringsome
simplemodelsandsimulations. Wethendiscusstheroleofthechemicalpotentialinunderstanding
phase transitions with a focus on the van der Waals equation of state.
MostofthematerialinthefirstsixchaptersandChapter7throughSection7.3wouldformthe
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.
ItispossibletostartacoursewiththedevelopmentofstatisticalmechanicsinChapter4after
areviewofthefirstthreechapters. ThekeyresultsneededfromChapters1–3arethefundamental
equation dE =TdS−PdV +µdN, definitions of thermodynamic quantities such as temperature,
pressure,andchemicalpotential interms ofthe various thermodynamic potentials,the definitions
oftheheatcapacityandcompressibility,andanunderstandingofhowtoobtainaveragequantities
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 ar-
guments 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-
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
thattheidealgaspressureequationofstateisderivedinstatisticalmechanicsbyignoringtheinter-
actionsbetweenparticles,butthe kinetictheoryderivationofthesameequationofstateexplicitly
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.
InadditiontothechaptersavailableviathePrincetonUniversityPresswebsite,<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 (<www.compadre.org/stp>). 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
networkoffreeonlineresourcecollectionssupportingfaculty,students,andteachersinphysicsand
astronomyeducation. We encourageothers to submit materials on statistical and thermalphysics
to <www.compadre.org/stp>.
The software associated with our text is available in several formats and can be downloaded
from <www.compadre.org/stp> or <press.princeton.edu/titles/9375.html>. All the pro-
gramsareopensource. Thecompiledprogramsarejarfilesandcanbe runlikeanyotherprogram
on your computer by double-clicking on the file. The applications can also be run in applet mode
whenembeddedintoanhtmlpageandrunwithinabrowser. (Directionsfordoingsoareavailable
on both websites.) Java 1.5 or greater must be installed on your computer to run the programs.
Javacanbe downloadedatnocostfrom<java.com> forWindows andLinux andis includedwith
Mac OS X.
Alloftheprogramsmentionedinthetextandafewothersarepackagedtogetherwithrelated
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
bemodifiedwithoutdownloadingthesourcecodeandrecompilingit. TheEJS(EasyJavaSimula-
tions)programsusually havea simpler interface,but the sourcecode is partofthe jarfile andcan
beeasilymodified. TodothelatteritisnecessarytodownloadEJSfrom<www.um.es/fem/Ejs/>.
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 originallyencouragedus to write a text onstatisticalandthermal physicsand to
apply for a National Science Foundation grant. The two NSF grants, PHY-98-01878 and DUE-
0442481, 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
beenstimulatedby discussionswith manycolleagues. We wouldliketo thankJillLarkinandFred
Reiffortheir usefuladviceandencouragementatthe beginningofthe STPproject. We usedFred
Reif’s classic thermal physics text for many years and our text shows its strong influence.
We owea special debt to Wolfgang Christian, Anne Cox, Doug Brown,Francisco Esquembre,
and Mario Belloni of the Open Source Physics Project, <http://www.compadre.org/osp/> for
CHAPTER 1. PREFACE 6
their many direct and indirect contributions to the STP project. The software would not have
beenpossiblewithoutWolfgangChristian’syeomanworkdevelopingandmanagingtheJava-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 Op-
penheim,SidRedner,andRoyceZiaforreadingindividualchaptersandmakingusefulsuggestions.
Our students havebeen patient (not that they had much choice), andmany of them gaveus valu-
able 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 un-
clear 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
<www.compadre.org/stp>. (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 workedwith 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 encourage-
ment. 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 emails 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, <[email protected]>
Jan Tobochnik, <[email protected]>
Contents
1 From Microscopic to Macroscopic Behavior 1
1.1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Some Qualitative Observations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
1.3 Doing Work and the Quality of Energy. . . . . . . . . . . . . . . . . . . . . . . . . 4
1.4 Some Simple Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.5 Measuring the Pressure and Temperature . . . . . . . . . . . . . . . . . . . . . . . 15
1.6 Work, Heating, and the First Law of Thermodynamics . . . . . . . . . . . . . . . . 19
1.7 *The Fundamental Need for a Statistical Approach . . . . . . . . . . . . . . . . . . 19
1.8 *Time and Ensemble Averages . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
1.9 Models of Matter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
1.9.1 The ideal gas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
1.9.2 Interparticle potentials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
1.9.3 Lattice models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
1.10 Importance of Simulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
1.11 Dimensionless Quantities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
1.12 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
1.13 Supplementary Notes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
1.13.1 Approach to equilibrium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
1.13.2 Mathematics refresher . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
Vocabulary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
Additional Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
Suggestions for Further Reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
2 Thermodynamic Concepts 31
2.1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
2.2 The System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
2.3 Thermodynamic Equilibrium . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
i
CONTENTS ii
2.4 Temperature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
2.5 Pressure Equation of State . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
2.6 Some Thermodynamic Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
2.7 Work. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
2.8 The First Law of Thermodynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
2.9 Energy Equation of State . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
2.10 Heat Capacities and Enthalpy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
2.11 Quasistatic Adiabatic Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
2.12 The Second Law of Thermodynamics . . . . . . . . . . . . . . . . . . . . . . . . . . 53
2.13 The Thermodynamic Temperature . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
2.14 The Second Law and Heat Engines . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
2.15 Entropy Changes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
2.16 Equivalence of Thermodynamic and Ideal Gas Scale Temperatures . . . . . . . . . 70
2.17 The Thermodynamic Pressure. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
2.18 The Fundamental Thermodynamic Relation . . . . . . . . . . . . . . . . . . . . . . 72
2.19 The Entropy of an Ideal Classical Gas . . . . . . . . . . . . . . . . . . . . . . . . . 74
2.20 The Third Law of Thermodynamics . . . . . . . . . . . . . . . . . . . . . . . . . . 74
2.21 Free Energies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
2.22 Thermodynamic Derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
2.23 *Applications to Irreversible Processes . . . . . . . . . . . . . . . . . . . . . . . . . 84
2.23.1 Joule or free expansion process . . . . . . . . . . . . . . . . . . . . . . . . . 85
2.23.2 Joule-Thomson process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86
2.24 Supplementary Notes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88
2.24.1 The mathematics of thermodynamics. . . . . . . . . . . . . . . . . . . . . . 88
2.24.2 Thermodynamic potentials and Legendre transforms . . . . . . . . . . . . . 91
Vocabulary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
Additional Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94
Suggestions for Further Reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
3 Concepts of Probability 106
3.1 Probability in Everyday Life. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106
3.2 The Rules of Probability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
3.3 Mean Values . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
3.4 The Meaning of Probability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116
3.4.1 Information and uncertainty. . . . . . . . . . . . . . . . . . . . . . . . . . . 118
3.4.2 *Bayesianinference. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122
3.5 Bernoulli Processes and the Binomial Distribution . . . . . . . . . . . . . . . . . . 127
CONTENTS iii
3.6 Continuous Probability Distributions . . . . . . . . . . . . . . . . . . . . . . . . . . 140
3.7 The Central Limit Theorem (or Why Thermodynamics Is Possible) . . . . . . . . . 144
3.8 *The PoissonDistribution or Should You Fly? . . . . . . . . . . . . . . . . . . . . 147
3.9 *Traffic Flow and the Exponential Distribution . . . . . . . . . . . . . . . . . . . . 148
3.10 *Are All Probability Distributions Gaussian? . . . . . . . . . . . . . . . . . . . . . 151
3.11 *Supplementary Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153
3.11.1 Method of undetermined multipliers . . . . . . . . . . . . . . . . . . . . . . 153
3.11.2 Derivation of the central limit theorem . . . . . . . . . . . . . . . . . . . . . 155
Vocabulary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159
Additional Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159
Suggestions for Further Reading . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169
4 Statistical Mechanics 172
4.1 Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172
4.2 A Simple Example of a Thermal Interaction . . . . . . . . . . . . . . . . . . . . . . 174
4.3 Counting Microstates. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184
4.3.1 Noninteracting spins . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184
4.3.2 A particle in a one-dimensional box . . . . . . . . . . . . . . . . . . . . . . 185
4.3.3 One-dimensional harmonic oscillator . . . . . . . . . . . . . . . . . . . . . . 188
4.3.4 One particle in a two-dimensional box . . . . . . . . . . . . . . . . . . . . . 188
4.3.5 One particle in a three-dimensional box . . . . . . . . . . . . . . . . . . . . 190
4.3.6 Two noninteracting identical particles and the semiclassical limit . . . . . . 191
4.4 The Number of States of Many Noninteracting Particles: Semiclassical Limit . . . 193
4.5 The Microcanonical Ensemble (Fixed E, V, and N) . . . . . . . . . . . . . . . . . 194
4.6 The Canonical Ensemble (Fixed T, V, and N) . . . . . . . . . . . . . . . . . . . . 200
4.7 Connection Between Thermodynamics and Statistical Mechanics in the Canonical Ensemble207
4.8 Simple Applications of the Canonical Ensemble . . . . . . . . . . . . . . . . . . . . 208
4.9 An Ideal Thermometer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211
4.10 Simulation of the MicrocanonicalEnsemble . . . . . . . . . . . . . . . . . . . . . . 215
4.11 Simulation of the Canonical Ensemble . . . . . . . . . . . . . . . . . . . . . . . . . 216
4.12 Grand Canonical Ensemble (Fixed T, V, and µ) . . . . . . . . . . . . . . . . . . . 217
4.13 *Entropy is not a Measure of Disorder . . . . . . . . . . . . . . . . . . . . . . . . . 219
4.14 Supplementary Notes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 221
4.14.1 The volume of a hypersphere . . . . . . . . . . . . . . . . . . . . . . . . . . 221
4.14.2 Fluctuations in the canonical ensemble . . . . . . . . . . . . . . . . . . . . . 222
Vocabulary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223
Additional Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223