Table Of ContentIntroduction to Nonextensive Statistical Mechanics
Constantino Tsallis
Introduction to Nonextensive
Statistical Mechanics
Approaching a Complex World
123
ConstantinoTsallis
CentroBrasileirodePesquisasF´ısicas
RuaXavierSigaud150
22290-180RiodeJaneiro-RJ
Brazil
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ISBN 978-0-387-85358-1 e-ISBN 978-0-387-85359-8
DOI 10.1007/978-0-387-85359-8
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Preface
In1902,afterthreedecadesthatLudwigBoltzmannformulatedthefirstversionof
standardstatisticalmechanics,JosiahWillardGibbsshares,inthePrefaceofhissu-
perbElementaryPrinciplesinStatisticalMechanics[1]:“Certainly,oneisbuilding
onaninsecurefoundation....”AftersuchwordsbyGibbs,itis,stilltoday,uneasy
to feel really comfortable regarding the foundations of statistical mechanics from
firstprinciples.AtthetimethatItakethedecisiontowritethepresentbook,Iwould
certainlysecondhiswords.Severalinterrelatedfactscontributetothisinclination.
First, the verification of the notorious fact that all branches of physics deeply
relatedwiththeoryofprobabilities,suchasstatisticalmechanicsandquantumme-
chanics,haveexhibited,alonghistoryanduptonow,endlessinterpretations,rein-
terpretations, and controversies. All this fullycomplemented by philosophical and
sociologicalconsiderations.Asoneamongmanyevidences,letusmentiontheelo-
quentwordsbyGregoireNicolisandDavidDaems[2]:“Itisthestrangeprivilege
ofstatisticalmechanics tostimulateand nourish passionate discussions related to
itsfoundations,particularlyinconnectionwithirreversibility.Eversincethetimeof
Boltzmannithasbeencustomarytoseethescientificcommunityvacillatingbetween
extreme,mutuallycontradictingpositions.”
Second,Iaminclinedtothinkthat,togetherwiththecentralgeometricalconcept
of symmetry, virtually nothing more basically than energy and entropy deserves
thequalificationofpillarsofmodernphysics.Bothconceptsareamazinglysubtle.
However,energyhastodowithpossibilities,whereasentropywiththeprobabilities
of those possibilities. Consequently, the concept of entropy is, epistemologically
speaking, one step further. One might remember, for instance, the illustrative dia-
log that Claude Elwood Shannon had with John von Neumann [3]: “My greatest
concernwaswhattocallit.Ithoughtofcallingit“information,”butthewordwas
overly used, so I decided to call it “uncertainty.” When I discussed it with John
von Neumann, he had a better idea. Von Neumann told me, “You should call it
entropy,fortworeasons.Inthefirstplaceyouruncertaintyfunctionhasbeenused
in statistical mechanics under that name, so it already has a name. In the second
place,andmoreimportant,nobodyknowswhatentropyreallyis,soinadebateyou
will always have the advantage.” It certainly is frequently that we hear and read
diversified opinions about what shouldand what should not be considered as “the
physicalentropy,”itsconnectionswithheat,information,andsoon.
vii
viii Preface
Third,thedynamicalfoundationsofthestandard,Boltzmann–Gibbs(BG)statis-
ticalmechanicsare,mathematicallyspeaking,notyetfullyestablished.Itisknown
that, for classical systems, exponentially diverging sensitivity to the initial condi-
tions(i.e.,positiveLyapunovexponentsalmosteverywhere,whichtypicallyimply
mixing and ergodicity, properties that are consistent with Boltzmann’s celebrated
“molecularchaoshypothesis”)isasufficientpropertyforhavingameaningfulsta-
tisticaltheory.Moreprecisely,oneexpectsthatthispropertyimplies,formany-body
Hamiltoniansystemsattainingthermalequilibrium,centralfeaturessuchasthecel-
ebratedexponentialweight,introducedanddiscussedinthe1870sbyLudwigBoltz-
mann(veryespeciallyinhis1872[5]and1877[6]papers)inthesocalledμ-space,
thus recovering, as particular instance, the velocity distribution published in 1860
byJamesClerkMaxwell[7].Moregenerally,theexponentialdivergencetypically
leads to the exponential weight in the full phase space, the so-called (cid:2)-space first
proposed by Gibbs. However, are hypothesis such as this exponentially diverging
sensitivitynecessary?Inthefirstplace,arethey,insomeappropriatelogicalchain,
necessaryforhaving BG statisticalmechanics?Iwouldsayyes.Butaretheyalso
necessaryforhavingavalidstatisticalmechanicaldescriptionatallforanytypeof
thermodynamic-like systems?1 I would say no. In any case, it is within this belief
that I write the present book. All in all, if such is today the situation for the suc-
cessful,undoubtedlycorrectforaverywideclassofsystems,universallyused,and
centennial BG statistical mechanics and its associated thermodynamics, what can
wethenexpectforitspossiblegeneralizationonly20yearsafteritsfirstproposal,
in1988?
Fourth,–lastbutnotleast–nological-deductivemathematicalprocedureexists,
norwillpresumablyeverexist,forproposinganewphysicaltheoryorforgeneraliz-
ingapre-existingone.ItisenoughtothinkaboutNewtonianmechanics,whichhas
already been generalized along at least two completely different (and compatible)
paths, which eventually led to the theory of relativity and to quantum mechanics.
Thisfactisconsistentwiththeevidencethatthereisnouniquewayofgeneralizing
a coherent set of axioms. Indeed, the most obvious manner of generalizing it is to
replace one or more of its axioms by weaker ones. And this can be done in more
than one manner, sometimes in infinite manners. So, if the prescriptions of logics
andmathematicsarehelpfulonlyforanalyzingtheadmissibilityofagivengener-
alization,howgeneralizationsofphysicaltheories,orevenscientificdiscoveriesin
general,occur?Throughalltypesofheuristicprocedures,butmainly–Iwouldsay–
through metaphors [11]. Indeed, theoretical and experimental scientific progress
occurs all the time through all types of logical and heuristic procedures, but the
particularprogressinvolvedinthegeneralizationofaphysicaltheoryimmensely,if
notessentially,reliesonsomekindofmetaphor.2Well-knownexamplesaretheidea
of Erwin Schroedinger of generalizing Newtonian mechanics through a wave-like
1Forexample,wecanreadinarecentpaperbyGiulioCasatiandTomazProsen[9]thefollowing
sentence:“Whileexponentialinstabilityissufficientforameaningfulstatisticaldescription,itis
notknownwhetherornotitisalsonecessary.”
2IwasfirstledtothinkaboutthisbyRoaldHoffmannin1995.
Preface ix
equationinspiredbythephenomenonofopticalinterference,andthediscoveryby
Friedrich August Kekule of the cyclic structure of benzene inspired by the shape
of the mythological Ouroboros. In other words, generalizations not only use the
classical logical procedures of deduction and induction, but also, and overall, the
specifictypeofinferencereferredtoasabduction(orabductivereasoning),which
plays the most central role in Charles Sanders Peirce’s semiotics. The procedures
fortheoreticallyproposingageneralizationofaphysicaltheorysomehowcrucially
rely on the construction of what one may call a plausible scenario. The scientific
valueanduniversalacceptabilityofanysuchaproposalareofcourseultimatelydic-
tatedbyitssuccessfulverifiabilityinnaturaland/orartificialand/orsocialsystems.
Having made all these considerations the best I could, I hope that it must by now
be very transparent for the reader why, in the beginning of this Preface, I evoked
Gibbs’wordsaboutthefragilityofthebasisonwhichwearefounding.
Theword“nonextensive”that–aftersomehesitation–Ieventuallyadopted,in
thetitleofthebookandelsewhere,torefertothepresentspecificgeneralizationof
BGstatisticalmechanicsmay–andoccasionallydoes–causesomeconfusion,and
surelydeservesclarification.Thewholetheoryisbasedonasingleconcept,namely
theentropynoted S which,fortheentropicindexq equaltounity,reproducesthe
q
standard BG entropy,herenoted S .Thetraditionalfunctional S issaidtobe
BG BG
additive.Indeed,forasystemcomposedofanytwo(probabilistically)independent
subsystems, the entropy S of the sum coincides with the sum of the entropies.
BG
Theentropy S (q (cid:2)= 1)violatesthisproperty,andisthereforenonadditive.Aswe
q
see,entropicadditivitydepends,fromitsverydefinition,onlyonthefunctionalform
oftheentropyintermsofprobabilities.Thesituationisgenericallyquitedifferent
forthethermodynamicconceptofextensivity.Anentropyofasystemorofasub-
system is said extensive if, for a large number N of its elements (probabilistically
independentornot),theentropyis(asymptotically)proportionaltoN.Otherwise,it
isnonextensive.Thisistosay,extensivitydependsonboththemathematicalformof
theentropicfunctionalandthecorrelationspossiblyexistingwithintheelementsof
thesystem.Consequently,fora(sub)systemwhoseelementsareeitherindependent
or weakly correlated, the additive entropy S is extensive, whereas the nonaddi-
BG
tive entropy S (q (cid:2)= 1) is nonextensive. In contrast, however, for a (sub)system
q
whoseelementsaregenericallystronglycorrelated,theadditiveentropyS canbe
BG
nonextensive, whereas the nonadditive entropy S (q (cid:2)= 1) can be extensive for a
q
specialvalueofq.Probabilisticsystemsexistsuchthat S isnotextensiveforany
q
valueofq,eitherq =1orq (cid:2)=1.Allthesestatementsareillustratedinthebodyof
the book.3 We shall also see that, consistently, the indexq appears to characterize
3Duringmorethanonecentury,physicistshaveprimarilyaddressedweaklyinteractingsystems,
andthereforetheentropicformwhichsatisfiesthethermodynamicalrequirementofextensivity
is S .Aregretfulconsequenceofthisfactisthatentropicadditivityandextensivityhavebeen
BG
practically considered as synonyms in many communities, thus generating all kinds of confu-
sions and inadvertences. For example, our own book Nonextensive Entropy—Interdisciplinary
Applications[69]shoulddefinitivelyhavebeenmoreappropriatelyentitledNonadditiveEntropy—
InterdisciplinaryApplications!Indeed,alreadyinitsfirstchapter,anexampleisshownwherethe
nonadditiveentropyS (q(cid:2)=1)isextensive.
q
x Preface
universality classes of nonadditivity, by phrasing this concept similarly to what is
doneinthestandardtheoryofcriticalphenomena.Withineachclass,oneexpectsto
findinfinitelymanydynamicalsystems.
Coming back to the name nonextensive statistical mechanics, would it not be
moreappropriatetocallitnonadditivestatisticalmechanics? Certainlyyes,ifone
focusesontheentropythatisbeingused.However,thereis,ononehand,thefact
that the expression nonextensive statistical mechanics is by now spread in thou-
sandsofpapers.Thereis,ontheotherhand,thefactthatimportantsystemswhose
approachisexpectedtobenefitfromthepresentgeneralizationoftheBGtheoryare
long-range-interactingmany-bodyHamiltoniansystems.Forsuchsystems,thetotal
energyiswellknowntobenonextensive,eveniftheextensivityoftheentropycan
bepreservedbyconvenientlychoosingthevalueoftheindexq.
Stillatthelinguisticandsemanticlevels,shouldwereferto S asanentropyor
q
justasanentropicfunctionalorentropicform?And,evenbeforethat,whyshould
suchaminor-lookingpointhaveanyrelevanceinthefirstplace?Thepointisthat,
inphysics,sincemorethanonecentury,onlyoneentropicfunctionalisconsidered
“physical”inthethermodynamicalsense,namelythe BG one.Inotherareas,such
as cybernetics, control theory, nonlinear dynamical systems, information theory,
many other (well over 20!) entropic functionals have been studied and/or used as
well. In the physical community only the BG form is undoubtfully admitted as
physically meaningful because of its deep connections with thermodynamics. So,
whatabout S inthisspecificcontext?Avarietyofthermodynamicalarguments–
q
extensivity,Clausiusinequality,firstprincipleofthermodynamics,andothers–that
arepresentedlateron,definitivelypointS asaphysicalentropyinaquiteanalogous
q
sensethat S surelyis.Letusfurtherelaboratethispoint.
BG
Complexityisnowadaysafrequentlyusedyetpoorlydefined–atleastquantita-
tivelyspeaking–concept.Ittriestoembraceagreatvarietyofscientificandtech-
nologicalapproachesofalltypesofnatural,artificial,andsocialsystems.Aname,
plectics, has been coined by Murray Gell-Mann to refer to this emerging science
[12].Oneofthemain–necessarybutbynomeanssufficient–featuresofcomplex-
ityhastodowiththefactthatbothveryorderedandverydisorderedsystemsare,in
thesenseofplectics,consideredtobesimple,notcomplex.Ubiquitousphenomena,
suchastheoriginoflifeandlanguages,thegrowthofcitiesandcomputernetworks,
citations of scientific papers, co-authorships and co-actorships, displacements of
living beings, financial fluctuations, turbulence, are frequently considered to be
complex phenomena. They all seem to occur close, in some sense, to the frontier
between order and disorder. Most of their basic quantities exhibit nonexponential
behaviors, very frequently power-laws. It happens that the distributions and other
relevantquantitiesthatemergenaturallywithintheframeofnonextensivestatistical
mechanicsarepreciselyofthistype,becomingoftheexponentialtypeintheq =1
limit.Oneofthemosttypicaldynamicalsituationshastodowiththeedgeofchaos,
occurring in the frontier between regular motion and standard chaos. Since these
twotypicalregimeswouldclearlybeconsidered“simple”inthesenseofplectics,
oneisstronglytemptedtoconsideras“complex”theregimeinbetween,whichhas
some aspects of the disorder of strong chaos but also some of the order lurking
Preface xi
nearby.4 Nonextensive statistical mechanics turns out to be appropriate precisely
for that intermediate region, thus suggesting that the entropic index q could be a
convenientmannerforquantifyingsomerelevantaspectsofcomplexity,surelynot
in all cases but probably so far vast classes of systems. Regular motion and chaos
aretimeanalogsforthespaceconfigurationsoccurringrespectivelyincrystalsand
fluids.Inthissense,theedgeofchaoswouldbetheanalogofquasi-crystals,glasses,
spin-glasses, and other amorphous, typically metastable structures. One does not
expectstatisticalconceptstobeintrinsicallyusefulforregularmotionsandregular
structures. On the contrary, one naturally tends to use probabilistic concepts for
chaos and fluids. These probabilistic concepts and their associated entropy, S ,
BG
would typically be the realm of BG statistical mechanics and standard thermody-
namics. It appears that, in the marginal cases, or at least in very many of them,
between great order and great disorder,the statisticalprocedures can stillbe used.
However, the associated natural entropy would not anymore be the BG one, but
S with q (cid:2)= 1. It then appears quite naturally the scenario within which BG sta-
q
tistical mechanics is the microscopic thermodynamical description properly asso-
ciatedwithEuclideangeometry,whereasnonextensivestatisticalmechanicswould
bethepropercounterpartwhichhasprivilegedconnectionswith(multi)fractaland
similar,hierarchical,statisticallyscale-invariant,structures(atleastasymptotically
speaking). As already mentioned, a paradigmatic case would be those nonlinear
dynamical systems whose largest Lyapunov exponent is neither negative (easily
predictablesystems)norpositive(strongchaos)butvanishinginstead,e.g.,theedge
of chaos (weak chaos5). Standard, equilibrium critical phenomena also deserve a
specialcomment.Indeed,Ihavealwayslikedtothinkandsaythat“criticalityisa
littlewindowthroughwhichonecanseethenonextensiveworld.”Manypeoplehave
certainlyhadsimilarinsights.AlbertoRobledo,FilippoCaruso,andIhaverecently
exhibitedsomerigorousevidences–tobediscussedlateron–alongthisline.Not
that there is anything wrong with the usual and successful use of BG concepts to
discuss the neighborhood of criticality in cooperative systems at thermal equilib-
rium! But, if one wants to make a delicate quantification of some of the physical
concepts precisely at the critical point, the nonextensive language appears to be a
privilegedoneforthistask.Itmaybesoformanyanomaloussystems.Paraphrasing
AngelPlastino’s(A.PlastinoSr.)laststatementinhislectureatthe2003Villasimius
meeting,“fordifferentsizesofscrewsonemustusedifferentscrewdrivers”!
Aproposalofageneralizationofthe BG entropyasthephysicalbasisfordeal-
ing,instatisticalmechanicalterms,withsomeclassesofcomplexsystemsmight–
4Itisfrequentlyencounterednowadaysthebeliefthatcomplexityemergestypicallyattheedge
ofchaos.Forinstance,thefinalwordsoftheAbstractofalecturedeliveredinSeptember2005
byLeonO.ChuaatthePolitecnicodiMilanowere“Explicitmathematicalcriteriaaregivento
identifyarelativelysmallsubsetofthelocally-activeparameterregion,calledtheedgeofchaos,
wheremostcomplexphenomenaemerge.”[14].
5Inthepresentbook,theexpression“weakchaos”isusedinthesenseofasensitivitytotheinitial
conditionsdivergingwithtimeslowerthanexponentially,andnotinothersensesusedcurrentlyin
thetheoryofnonlineardynamicalsystems.
