Introduction to Nonextensive Statistical Mechanics: Approaching a Complex World

575 Pages·2023·25.142 MB·English

by  Constantino Tsallis

Constantino Tsallis
Introduction
to Nonextensive
Statistical
Mechanics
Approaching a Complex World
Second Edition
Introduction to Nonextensive Statistical Mechanics
Constantino Tsallis
Introduction to Nonextensive
Statistical Mechanics
Approaching a Complex World
Second Edition
ConstantinoTsallis
CentroBrasileirodePesquisasFisicas
RiodeJaneiro,Brazil
ISBN 978-3-030-79568-9 ISBN 978-3-030-79569-6 (eBook)
https://doi.org/10.1007/978-3-030-79569-6
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2ndedition:©SpringerNatureSwitzerlandAG2023
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Preface
In 1902, after three decades that Ludwig Eduard Boltzmann formulated the first
versionofstandardstatisticalmechanics,JosiahWillardGibbsshares,inthePreface
of his superb Elementary Principles in Statistical Mechanics [1, 2, 3]: “Certainly,
one is building on an insecure foundation ...”. After such words by Gibbs, it is,
stilltoday,uneasytofeelreallycomfortableregardingthefoundationsofstatistical
mechanicsfromfirstprinciples.SincethetimewhenItookthedecisiontowritethe
presentbook,Iwouldcertainlysecondhiswords.Severalinterrelatedfactscontribute
tothisinclination.
First, the verification of the notorious fact that all branches of physics deeply
related to the theory of probabilities, such as statistical mechanics and quantum
mechanics, have exhibited, along history and up to now, endless interpretations,
reinterpretations,andcontroversies.Allthisisfullycomplementedbyphilosophical
andsociologicalconsiderations.Asoneamongmanyevidences,letusmentionthe
eloquentwordsbyGregoireNicolisandDavidDaems[4]:“Itisthestrangeprivilege
ofstatisticalmechanics tostimulate and nourishpassionate discussions related to
itsfoundations,particularlyinconnectionwithirreversibility.Eversincethetimeof
Boltzmann,ithasbeencustomarytoseethescientificcommunityvacillatingbetween
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,onestepfurther.Onemightremember,forinstance,theillustrativedialog
that Claude Elwood Shannon had with John von Neumann [5, 6]: “My greatest
concernwaswhattocallit.Ithoughtofcallingit‘information’,butthewordwas
overlyused,soIdecidedtocallit‘uncertainty’.WhenIdiscusseditwithJohnvon
Neumann,hehadabetteridea.VonNeumanntoldme,‘Youshouldcallitentropy,for
tworeasons.Inthefirstplace,youruncertaintyfunctionhasbeenusedinstatistical
mechanicsunderthatname,soitalreadyhasaname.Inthesecondplace,andmore
important, nobody knows what entropy really is, so in a debate, you will always
vii
viii Preface
havetheadvantage.’”.Frequentlywehearandreaddiversifiedopinionsaboutwhat
shouldandwhatshouldnotbeconsideredas“thephysicalentropy”,itsconnections
withheat,information,andsoon.
Third,thedynamicalfoundationsofthestandard,Boltzmann-Gibbs(BG)statis-
ticalmechanicsare,mathematicallyspeaking,not yetfullyestablished.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
StoszzahlAnsatz,“molecularchaoshypothesis”)isasufficientpropertyforhavinga
meaningfulstatisticaltheory.Moreprecisely,oneexpectsthatthispropertyimplies,
formany-bodyHamiltoniansystemsattainingthermalequilibrium,centralfeatures
such as the celebrated exponential weight, introduced and discussed in the 1870s
byLudwigBoltzmann(veryespeciallyinhis1872[7]and1877[8]papers)1 inthe
so-called μ-space, thus recovering, as particular instance, the velocity distribution
published in 1860 by James Clerk Maxwell [11]. More generally, the exponential
divergence typically leads to the exponential weight in the full phase space, the
so-called(cid:2)-spacefirstproposedbyGibbs.However,hypothesissuchasthisexpo-
nentially diverging sensitivity, are they necessary? In the first place, are they, in
some appropriate logical chain, necessary for having BG statistical mechanics? I
wouldsayyes.Butaretheyalsonecessaryforhavingavalidstatisticalmechanical
descriptionatallforanytypeofthermodynamic-likesystems?.2 Iwouldsayno.In
any case, it is within this belief that I write the present book. All in all, if such is
todaythesituationforthesuccessful,undoubtedlycorrectforaverywideclassof
systems,universallyused,andcentennialBGstatisticalmechanicsanditsassociated
thermodynamics(“asciencewithsecurefoundations,cleardefinitions,anddistinct
boundaries” as so well characterized by James Clerk Maxwell), what can we then
expect for its possible generalization only a few decades after its first proposal, in
1988?
Fourth,—lastbutnotleast—nological-deductivemathematicalprocedureexists,
norwillpresumablyeverexist,forproposinganewphysicaltheoryorforgeneral-
izingapre-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
replaceoneormoreofitsaxiomswithweakerones.Andthiscanbedoneinmore
than one manner, sometimes in infinite manners. So, if the prescriptions of logics
andmathematicsarehelpfulonlyforanalyzingtheadmissibilityofagivengeneral-
ization,howdogeneralizationsofphysicaltheories,orevenscientificdiscoveriesin
1Englishtranslationin[9];seealso[10].
2Forexample,wecanreadinarecentpaperbyGiulioCasatiandTomazProsen[12]thefollowing
sentence:“Whileexponentialinstabilityissufficientforameaningfulstatisticaldescription,itis
notknownwhetherornotitisalsonecessary.”
Preface ix
general,occur?Throughalltypesofheuristicprocedures,butmainly—Iwouldsay—
through methaphors [13]. Indeed, theoretical and experimental scientific progress
occurs all the time through all types of logical and heuristic procedures, but the
particularprogressinvolvedinthegeneralizationofaphysicaltheoryimmensely,if
notessentially,reliesonsomekindofmetaphor.3Well-knownexamplesaretheidea
of Erwin Schroedinger of generalizing Newtonian mechanics through a wave-like
equationinspiredbythephenomenonofopticalinterference,andthediscoveryby
FriedrichAugustKekuleofthecyclicstructureofbenzeneinspiredbytheshapeof
themythologicalOuroboros.Inotherwords,generalizationsnotonlyusetheclas-
sicallogicalproceduresofdeductionandinduction,butalso,andoverall,thespecific
typeofinferencereferredtoasabduction(orabductivereasoning),whichplaysthe
most central role in Charles Sanders Peirce’s semiotics. The procedures for theo-
reticallyproposingageneralizationofaphysicaltheorysomehowcruciallyrelyon
theconstructionofwhatonemaycallaplausiblescenario.Thescientificvalueand
universal acceptability of any such a proposal are of course ultimately dictated by
itssuccessfulverifiabilityinnaturaland/orartificialand/orsocialsystems.Having
made all these considerations, I hope that it must by now be very transparent for
the reader why, in the beginning of this Preface, I evoked Gibbs’ words about the
fragilityofthebasisonwhichwearefounding.
Newton’sdecompositionofwhitelightintorainbowcolors,notonlyprovideda
deeperinsightintothenatureofwhatweknowtodaytoclassicallybeelectromag-
netic waves, but also opened the door to the discovery of infrared and ultraviolet.
Whiletryingtofollowthemethodsofthisgreatmaster,itismycherishedhopethat
thepresent,nonextensivegeneralizationofBoltzmann–Gibbsstatisticalmechanics,
mayprovideadeeperunderstandingofthestandardtheory,inadditiontoproposing
someextensionofthedomainofapplicabilityofthepowerfulmethodsofstatistical
mechanics. The book is written at a graduate course level, and some basic knowl-
edgeofquantumandstandardstatisticalmechanics,aswellasthermodynamics,is
assumed.Thestyleis,however,slightlydifferentfromaconventionaltextbook,in
the sense that not all the results that are presented are proved. The quick ongoing
developmentofthefielddoesnotyetallowforsuchanambitioustask.Variouspoints
ofthetheoryarepresentlyonlypartiallyknownandunderstood.So,hereandthere
weareobligedtoproceedwithheuristicarguments.Thebookisunconventionalalso
inthesensethathereandtherehistoricalandothersideremarksareincludedaswell.
Some sections of the book, the most basic ones, are presented with all details and
intermediatesteps;someothers,moreadvancedorquitelengthy,arepresentedonly
throughtheirmainresults,andthereaderisreferredtotheoriginalpublicationsto
knowmore.Wehope,however,thataunifiedperceptionofstatisticalmechanics,its
background,anditsbasicconceptsdoesemerge.
Thebookisorganizedintofourparts,namelyPartI—BasicsorHowtheTheory
Works, Part II—Foundations or Why the Theory Works, Part III—Applications or
WhatfortheTheoryWorks,andPartIV—Last(ButNotLeast).Thefirstpartconsti-
tutesapedagogicalintroductiontothetheoryanditsbackground(Chaps.1–3).The
3IwasfirstledtothinkaboutthisbyRoaldHoffmannin1995.
x Preface
secondpartcontainsthestateoftheartinitsdynamicalfoundations,inparticularhow
theindex(indices)qcanbeobtained,insomeparadigmaticcases,frommicroscopic
firstprinciplesor,alternatively,frommesoscopicprinciples(Chaps.4–6).Thethird
partisdedicatedtolistingbriefpresentationsoftypicalapplicationsofthetheoryand
itsconcepts,oratleastofitsfunctionalforms,aswellaspossibleextensionsexisting
intheliterature(Chap.7).Finally,thefourthpartconstitutesanattempttoplacethe
present—intensively evolving, open to further contributions and insights4—theory
into contemporary science, by addressing some frequently asked or still unsolved
currentissues(Chap.8).AnAppendixwithusefulformulaehasbeenaddedatthe
end,aswellasanotheronediscussingescortdistributionsandq-expectationvalues.
It may be useful to point out at this stage that this book can be quite conve-
nientlyreadalongtwopossibletracks.Thefirsttrackconcernsreadersatagraduate
student level. It basically consists of reading Chaps. 1, 2, 3, 5, and 8. The second
trackconcernsreadersattheresearchlevelwithsomepracticeinstandardstatistical
mechanicalmethods.ItbasicallyconsistsofreadingChaps.3–8.Forbothtracks,let
usemphasizethatChap.7containsalargeamountofdiversifiedapplications.The
reader may focus on those of his/her main preference. In the present new edition,
we have refined some concepts, included some recent analytical discussions, and
addedaconsiderablenumberofapplicationsaswellasexperimentalandnumerical
verifications.Letusmentionalsothatithasbeenunfortunatelyimpossibletounify
thenotationsalongallthechaptersofthebook,duetothefactthatverymanyfigures
havebeenreproducedfromalargenumberofpapersintheliterature.
Towardthisend,itisagenuinepleasuretowarmlyacknowledgethecontributions
ofM.Gell-Mann,maîtreàpenser,withwhomIhavehadfrequentanddelightfully
deepconversationsonthesubjectofnonextensivestatisticalmechanics...aswellas
onmanyothers.Manyotherfriendsandcolleagueshavesubstantiallycontributedto
theideas,results,andfigurespresentedinthisbook.Thosecontributionsrangefrom
insightfulquestionsorremarks—sometimesfairlycritical—toentiremathematical
developments and seminal ideas. Their natures are so diverse that it becomes an
impossible task to duly recognize them all. So, faute de mieux, I decided to name
theminalphabeticalorder,beingcertainthatIambynomeansdoingjusticetotheir
enormousanddiversifiedintellectualimportance.Inallcases,mygratitudecouldnot
bedeeper.TheyareS.Abe,F.C.Alcaraz,R.F.Alvarez-Estrada,S.Amari,G.F.J.
Ananos,C.Anteneodo,V.Aquilanti,N.Ay,G.BakerJr.,F.Baldovin,M.Baranger,
G.G.Barnafoldi,C.Beck,I.Bediaga,G.Bemski,T.S.Biro,A.R.Bishop,H.Blom,
B.M.Boghosian,E.Bonderup,J.P.Boon,E.P.Borges,L.Borland,T.Bountis,E.
Brezin,A.Bunde,L.F.Burlaga,B.J.C.Cabral,M.O.Caceres,S.A.Cannas,A.
Carati,F.Caruso,M.Casas,G.Casati,N.Caticha,A.Chame,P.-H.Chavanis,C.E.
Cedeño,L.J.L.Cirto,J.Cleymans,E.G.D.Cohen,A.Coniglio,M.CoutinhoFilho,
E.M.F.Curado,S.Curilef,J.S.Dehesa,S.A.Dias,A.Deppman,A.Erzan,L.R.
Evangelista,J.D.Farmer,R.Ferreira,M.A.Fuentes,L.Galgani,J.P.Gazeau,P.-G.
deGennes,A.Giansanti,A.Greco,P.Grigolini,D.H.E.Gross,G.R.Guerberoff,
E.Guyon,M.Hameeda,R.Hanel,H.J.Haubold,R.Hersh,H.J.Herrmann,H.J.
4Foraregularlyupdatedbibliographyofthesubjectseehttp://tsallis.cat.cbpf.br/biblio.htm.
Preface xi
Hilhorst,R.Hoffmann,G.‘tHooft,K.Huang,M.Jauregui,H.J.Jensen,P.Jizba,L.
P.Kadanoff,G.Kaniadakis,T.A.Kaplan,S.Kawasaki,J.Korbel,D.Krakauer,P.T.
Landsberg,V.Latora,C.M.Lattes,E.K.Lenzi,S.V.F.Levy,H.S.Lima,J.A.S.
Lima,M.L.Lyra,S.D.Mahanti,A.M.Mariz,J.Marsh,A.M.Mathai,R.Maynard,
G.F.Mazenko,E.Megias,R.S.Mendes,L.C.Mihalcea,L.G.Moyano,J.Naudts,
K.P.Nelson,G.Nicolis,F.D.Nobre,J.Nogales,F.A.Oliveira,P.M.C.Oliveira,I.
Oppenheim,A.W.Overhauser,G.Parisi,R.Pasechnik,G.P.Pavlos,R.Piasecki,A.
Plastino,A.R.Plastino,A.Pluchino,D.Prato,P.Quarati,S.M.D.Queiros,A.K.
Rajagopal,A.Rapisarda,M.A.Rego-Monteiro,M.S.Ribeiro,A.Robledo,M.C.
Rocca,A.Rodriguez,S.Ruffo,G.Ruiz,M.Rybczynski,S.R.A.Salinas,Y.Sato,V.
Schwammle,L.Silva,L.R.daSilva,R.N.Silver,I.D.Soares,A.M.C.Souza,H.
E.Stanley,D.A.Stariolo,D.Stauffer,S.Steinberg,R.Stinchcombe,H.Suyari,F.
Vallianatos,H.L.Swinney,F.A.Tamarit,P.Tempesta,W.J.Thistleton,S.Thurner,
U.Tirnakli,L.Tisza,F.Topsoe,R.Toral,A.C.Tsallis,A.F.Tsallis,E.L.P.Tsallis,F.
TsallisViegas,S.Umarov,P.Van,M.E.Vares,M.C.S.Vieira,C.Vignat,J.Villain,
A.F.Vinas,S.Vinciguerra,G.Viswanathan,R.S.Wedemann,B.Widom,G.Wilk,
H.O.Wio,Z.Wlodarczyk,D.J.Zamora,andI.I.Zovko.Unavoidably,Imusthave
forgottentomentionsome—thisideastarteddevelopingoverthreedecadesago!—to
themmymostgenuineapologiesandgratitude.Finally,asinvirtuallyallthefieldsof
scienceandveryespeciallyduringthefirststagesofanynewdevelopment,thereare
alsoafewcolleagueswhoseintentionshavenotbeen—Iconfess—verytransparent
to me. But they have nevertheless—perhaps even unwillingly—contributed to the
progressoftheideasthatarepresentedinthisbook.Theysurelyknowwhotheyare.
My gratitude goes to them as well: it belongs to human nature to generate fruitful
ideasthroughalltypesofpaths.
Along the years, I have relevantly benefited from the partial financial support
ofvariousAgencies,especiallytheBrazilianCNPq,FAPERJ,PRONEX/MCT,and
CAPES,theUSANSF,SFI,SIInternational,AFRL,andJohnTempletonFoundation,
theItalianINFNandINFM,amongothers.Iamindebtedtoallofthem.
Finally,someofthefiguresthatarepresentedinthepresentbookhavebeenrepro-
ducedfromvariouspublicationsindicatedcasebycase.Igratefullyacknowledgethe
graciousauthorizationtodosofromtheirauthors.
RiodeJaneiro,Brazilthroughtheperiods2004–2009and2018–2023
RiodeJaneiro,Brazil ConstantinoTsallis