Table Of ContentWhen Things Grow Many
When Things Grow Many
Complexity, Universality and Emergence in Nature
L. S. Schulman
GreatClarendonStreet,Oxford,OX26DP,
UnitedKingdom
OxfordUniversityPressisadepartmentoftheUniversityofOxford.
ItfurtherstheUniversity’sobjectiveofexcellenceinresearch,scholarship,
andeducationbypublishingworldwide.Oxfordisaregisteredtrademarkof
OxfordUniversityPressintheUKandincertainothercountries
©LawrenceS.Schulman2022
Themoralrightsoftheauthor[s]havebeenasserted
Impression:1
Allrightsreserved.Nopartofthispublicationmaybereproduced,storedin
aretrievalsystem,ortransmitted,inanyformorbyanymeans,withoutthe
priorpermissioninwritingofOxfordUniversityPress,orasexpresslypermitted
bylaw,bylicenceorundertermsagreedwiththeappropriatereprographics
rightsorganization.Enquiriesconcerningreproductionoutsidethescopeofthe
aboveshouldbesenttotheRightsDepartment,OxfordUniversityPress,atthe
addressabove
Youmustnotcirculatethisworkinanyotherform
andyoumustimposethissameconditiononanyacquirer
PublishedintheUnitedStatesofAmericabyOxfordUniversityPress
198MadisonAvenue,NewYork,NY10016,UnitedStatesofAmerica
BritishLibraryCataloguinginPublicationData
Dataavailable
LibraryofCongressControlNumber:2021936303
ISBN978-0-19-886188-1(hbk.)
DOI:10.1093/oso/9780198861881.001.0001
Printedandboundby
CPIGroup(UK)Ltd,Croydon,CR04YY
LinkstothirdpartywebsitesareprovidedbyOxfordingoodfaithand
forinformationonly.Oxforddisclaimsanyresponsibilityforthematerials
containedinanythirdpartywebsitereferencedinthiswork.
Dedicated to
my wife, Claire
Preface
This is a book about statistical mechanics, but not so much about statistical physics.
The distinction is in the applications, not the methods. The central idea is that when
dealing with objects that have a large number of constituents qualitatively new phe-
nomenaemerge.Thisisamessageonecantakefromphysicsandapplytoamyriadof
topics.Much,butnotall,ofthematerialisbasedoncoursesgivenatClarksonUniver-
sity and the Georgia Institute of Technology. The presentation is aimed at advanced
undergraduates and graduate students, including—in fact, especially—those not di-
rectly involved in traditional areas of physics. The general theme is that statistical
mechanics is useful in many areas of science, both those where the subject developed,
suchaschemistry,aswellasmorerecentapplicationsinbiological,financial,linguistic,
social and other fields.
The organization of the book is chaotic—perhaps as befits the subject matter. At
one point I thought it best to sort things out along topics, biology here, sociology in
its own chapter, and so forth. But as I wrote and thought, the areas became so inter-
twinedthatIabandonedthisplan.Thereareapplicationsofpowerlawstolinguistics,
sociology, economics, and more. Ditto for maximum entropy methods; ditto for self-
organizedcriticalityandsoon.Sointheend,Ijumparoundabit,sometimesfocusing
on the application, sometimes on the method. This means that if you’re interested in
economics—and although you might care about firefly synchronization, it’s not your
priority—you’ll have to hunt around a bit. I have tried to make the index compensate
for the lack of clear boundaries.
The first eight chapters are an introduction to the methods of statistical physics,
althoughmanyexamplesfromNatureareprovidedalongtheway(inAppendixBmore
formalresultsinstatisticalphysicsaregiven).Theproblemsdealtwitharesimplified,
butthemethodsarethoseusedinrichersituations.Andrichersituationsfollow;quite
a few individual applications are given in the final chapters although they are only a
smallfractionofthosetreatedintheliterature.(AppendixApresentsthenotationwe
use.)
As to the mathematical level, I have been free in the use of equations, although
in many cases footnotes or appendices are provided with some of the necessary back-
ground. The goal is to make this readable to as large an audience as possible. The
footnotes, which are plentiful, serve three purposes. They fill in material that may
be unfamiliar to the reader, they cover advanced material whose details often can be
ignored, and finally there are ordinary footnotes, guides to citations or extra material
whose insertion in the main text would be a distraction.
And now for the mea culpa: First, I confess to a certain amount of myopia. I did
not read every paper on every subject. I didn’t even read a significant fraction of
Preface vii
them.SoinmanycasesI’moutofdate,inmanycasesIreportwhatI’mfamiliarwith,
omitting significant other work. I apologize to those I slight. There are also value
judgements: The data are inadequate for the conclusion, or some article misses the
point,andsoforth.IthinkI’mright,butmaybenot.Sotakethose“conclusions”witha
grainofsalt.Italsomustbesaidthatthisbookhasanemphasisdifferentfrombookson
aparticularsubjectmatter.I’mnotgoingtoteachanyoneecology,linguistics,finance
or any specific field. Rather this should teach approaches to a problem. (Witness, the
number of times I express ignorance. And ignorance can take two forms: my own or
that of the human race.) There’s also an apology due over the selection of matlab™
for the sample programs. I once knew Fortran and a smattering of other computer
languages, but in the last few years I’ve focused on matlab™. Yes, I know that (for
example) Python is (1) free and (2) superior in some respects (Airy functions as of
2015).True,somebooksgivegenericprograms,butI’llhavetoleavesuchtranslations
to readers. There is also the issue of classification. Does linguistics (Zipf’s law) belong
in the social sciences? Does an electronic application of the Kuramoto model belong
in biological sciences? Maybe yes, maybe no, but please don’t worry about it.
And finally it’s likely that some instructors will develop new topics that I don’t
treat.Tellmeaboutit!1 Theremightbeasecondedition.Thisalsogoesforinteresting
applications that don’t appear in the present volume. I’m sure there are topics that
are omitted through my own ignorance.
1My email address is schulman (at) clarkson.edu and variations are given on both of my (out of
date)Clarksonwebsites.Icanalsobereachedatschulman137(at)gmail.com.
Acknowledgments
I am grateful to people at all institutions where this course has been taught and
especially to the students who participated in the presentation of these ideas.
Those students who contributed were Rachel E. Barker, Jonathan Brassard, Pablo
Bravo, Isabel Dengos, Lawson Glasby, Nick Gravish, Conner J. Herndon, Shane
J. Jacobeen, Matthew W. Jensen-Bukovinsky, Shengkai Li, Michael Lovall, Megan
G. Matthews, Christopher R. McBryde, James (Cordy) McCord, James McInerny,
KaranP.Mehta,CollinReff,KaranShah,JamesTaylor,JacobTroupe,andMackenna
Wood.Inadditionmanycolleagueshavecontributedtomyownunderstandingaswell
asposingquestionstowhichIoftenhadnoanswer.Thosewhohavedirectlyinfluenced
this work are John Beggs, Bruce Boghosian, Marcos da Luz, Bernard Gaveau, Dan
Goldman, Nicholas Gotelli, John Harte, Ady Mann, Eva Mihóková, David Storch,
Lucas Wetzel, and Marina Wosniack
Contents
1 Introduction 1
1.1 Building 2
2 Ideal gas 6
2.1 Fluctuations of the ideal gas 8
3 Rubber bands 11
3.1 The game 11
3.2 Analysis 11
3.3 Simulation 13
3.4 Independent folk 15
3.5 How often does the prediction go wrong? 17
4 Percolitis 19
4.1 An epidemic model 19
4.2 Discussion 22
4.3 Behavior of the order parameter near the critical point 24
4.4 Approach to equilibrium 25
4.5 Discreteness and fluctuations* 30
4.6 Self-organized criticality (SOC): Applications to galaxies
and mean field theory 34
4.7 The truth about percolitis 38
4.8 Abstract percolation 42
4.9 Percolation applications 44
4.10 True epidemiology 45
5 Ferromagnetism 50
5.1 Curie–Weiss ferromagnets 55
5.2 Magnetization √ 57
5.3 Fluctuations greater than N 60
6 Maximum entropy methods 63
6.1 Information 63
6.2 Maximum entropy 66
6.3 Using maximum entropy to study Supreme Court voting 71
