Acknowledgments
Ihavebeenthinkingabouttheseideasforanumberofyears nowandalotofpeoplehavebeenkindenoughtolistentomy musings.
FirstoffIwouldliketothankLeoKadanoff,posthumously, forencouragement,support,andespeciallyforhisinspirationalcontributionstothesubjectsthatIfindfascinating. Ifeelveryluckytohavegottentoknowhimalittlesincethe publicationofmyfirstbook.Likewise,Ihavebenefitedgreatly frommanyinteractionswithNigelGoldenfeldoverthelast sixteenyearsorso.
IowealargedebttomyfriendandcolleagueMark Wilson.Wehavebeentalkingaboutphilosophy,science,and mathematicsforthepasttwenty-eightyears.Hehashada majorinfluenceonthewayIthinkaboutproblemsinthe intersectionoftheseareas.
Anumberofmystudents,bothpastandpresent,provided detailedandconstructivecomments.Iammostgratefulto JuliaBursten,KathleenCreel,MichaelMiller,andTravis McKennafortheirvaluablecontributionstothisproject.My co-authoronanumberofpapersrelatedtoissuesdiscussed inthisbook,SaraGreen,gavemesomeofthemosthelpful critiquesbothintheearlystagesofwritingandattheend. Thisbookhasbeenconsiderablyimprovedthankstoher advice.
Severalcolleagueswerealsoveryhelpful,forwhichIam mostgrateful.EricaShumenerhelpedmeunderstandsome aspectsoftherecentmetaphysicalconceptionsoffundamentality.PorterWilliamsgavevaluablecriticismand,alongwith
MikeMiller,helpedmeseetheimportanceforthisproject ofJulianSchwinger’sworkonan“engineeringapproach”to particlephysics.JamesWoodwardofferedmuchbywayof encouragement,support,andvaluablecritiques.ColinAllen readdraftsofallofthechaptersandgavemeexcellent feedback.Likewise,IamhonoredthatmyfriendRoger Jonestookthetimetoreadtheworkandgiveexceptionally constructivecriticism.
IoweagreatdealtoLauraRuetscheandGordonBelot forclubbingmeseveraltimesastheprojectprogressed. Theypushedmeveryhardtoclarifyvariousaspectsofthe argumentsandtodialbacksomeofmymoreoutlandish suggestions.ForthisIamdeeplyappreciative.
Iwouldalsoliketothankseveralanonymousreadersand myeditorfromOxfordUniversityPress,PeterOhlin,for encouragingmeovertheyears.
Finally,IthankDeviandQuinnforkeepingmesane;but notCarolyn,whodidnot.
Preface
Thisbookfocusesonamethodforexploring,explaining,and understandingthebehaviorofmany-bodysystems.Theseare largesystemsconsistingofmanycomponentsthatdisplay distinctbehaviorsatdifferentscales.Theyincludegases, fluids,andcompositematerialssuchaswoodandsteel.In thecontextofcondensed-matterphysicsthismethodwas describedinafamouspaperbyLeoP.KadanoffandPaul C.Martinentitled“HydrodynamicEquationsandCorrelationFunctions.”Itdescribesanapproachtonon-equilibrium behaviorthatfocusesonstructures(representedbycorrelation functions)thatcharacterizemesoscalepropertiesofthesystems.Inotherwords,ratherthanafullybottom-upapproach, startingwiththecomponentsattheatomicormolecularscale, the“hydrodynamicapproach”aimstodescribeandaccount forcontinuumbehaviorsbylargelyignoringdetailsatthe “fundamental”level.
Thismethodologicalapproachactuallyhasitsoriginsin Einstein’sworkonBrownianmotion.ThereEinsteinmade twopioneeringarguments.First,hegavewhatmaybethe firstinstanceofup-scalingorhomogenizationtodetermine aneffective(continuum)valueforamaterialparameter—the viscosity—byconsideringtheheterogeneousmixtureofthe solventandtheBrownianparticles.Itturnsoutthatthis methodisofakindwithmuchworkinthescienceofmaterials. Thisconnectionandthewide-ranginginterdisciplinarynature ofthesemethodsarestressed.
Einstein’ssecondargumentledtothefirstexpression ofafundamentaltheoremofstatisticalmechanicscalled
theFluctuation–Dissipationtheorem.Thistheorem,whose importancewasonlyreallyappreciatedmuchlater,provides theprimaryjustificationforthehydrodynamic/correlation functionmethodology.
Thehydrodynamicmethodsexploitthefactthatthere must beheterogeneousstructuresatmesoscalesinbetweenthe atomicandthecontinuum.Suchstructurescanbeemployed todescribeandexplainthebehaviorsofmany-bodysystemsinnearbutnon-equilibriumstates.Inparticular,the methodologyfocusesonunderstandingtransportbehavior— currentsthatappearasaresultofspatialandtemporalnonuniformitieswithrespecttoconservedquantities.Thisallows foranunderstandingofnon-equilibriumbehaviorthatisnot basedonbottom-upderivationsstartingwiththeBoltzmann equation.
Furthermore,thehydrodynamicmethodologyallowsfora novelexplanationoftherelativeautonomyofupper-scalecontinuumtheorieslikefluidmechanics,fromlower-scale,more fundamentaltheories.Wecanunderstandtheremarkablefact thatsuchcontinuumtheoriessurvivedtheatomicrevolution despitebeingontologicallyinaccurate.Thesetheoriestreat systemsashavingnostructurebelowthecontinuumscales, butremainspectacularlysuccessfulintheirengineeringapplicationstoreal-worldproblems.
Reflectionsonthismethodologyledmetoanargument fortreatingthecorrelationalmesoscalestructuresasnatural kinds.Theargumentisneitherbasedonphilosophicalintuitionsaboutthenatureoflawsandlawfulness,noronmetaphysicalconsiderationsoffundamentalityandjointcarving. Iarguethattherearescientificreasonstotreattheparameters characterizingtherelevantaspectsofthemesoscaleasthe rightornaturalvariablesforstudyingheterogeneousmanybodysystems
IarguethattheFluctuation-Dissipationtheoremalong withtheotherconsiderationsinthebookmaybeseen yieldingmetaphysicalconsequences.Theargumentssupporta claimthatmesoscalestructures,andparametersthatdescribe
them,canbe/shouldbeconsidered,inacertainsense,more fundamentalthanthelowestscaleatomicdetails.Iprovide anargumenttotheeffectthatthemesoscaleparametersand variablesarethe right variablesfordoingcondensedmatter physicsbroadlyconstrued.Theyare natural.Inmetaphysical terms,theybettercarvenatureatitsjointsthanlowerscale, presumablymorefundamentalstructures.
Overall,thisbookarguesforamiddlewaybetweencontinuumtheoriesandatomictheories.Exceptforveryspecial cases,thereductionistgoalofprovidingdirectconnections betweenatomictheoryandcontinuummechanicsandthermodynamicsisboundtofail.Aproperunderstandingofthose inter-theoryrelationscanbehadwhenmesoscalesaretaken seriously.
Thereareplacesinthisbookwherethematerialbecomes somewhattechnical.ThisisparticularlythecaseinChapter3 oncorrelationfunctionsandthehydrodynamicdescription ofmany-bodysystems.Ihaveincludedtheequationsand derivationsforreaderswhowishtofollowthedetailedstepsof thearguments.But,Ihopethatthephilosophicalimplications ofthehydrodynamicmethodologyaresufficientlysalientthat readerswhowishtoskimoverthesepartsofthebookmaydo sowithoutlossofunderstanding.
Mostimportantly,Ihopethisbookdemonstratesthe importanceofthemesoscale,middle-out,approachtophysical systems,andthatitoffersanewphilosophicalperspectiveon understandingbehaviorsofsystemsacrossscales.
5FromBrownianMotiontoBendingBeams85
Chapter1 Introduction
Philosophersofscience,particularlyphilosophersofphysics, havefocusedtheirattentionon foundationalproblems in physicaltheories.Theyhavealsoexaminedtherelations between non-fundamental theoriesandthoseconsideredto be fundamental ormore fundamental thanothers.Theterms “foundational”and“fundamental”getbandiedabout,used sometimesinterchangeably,andsometimesindifferentways. Asaroughcharacterization,Ishalltake“fundamental”primarilytorefertotheoriesreflectingsomekindofhierarchy witha(the)fundamentaltheorybeingthebottomorperhaps the“ground”uponwhichhigher,lessfundamental,theories maydepend.1 Ishalltake“foundational”torefertotypesof deepproblemsthatlieatthecoreofanindividualtheoryorat theinterfacebetweenapairoftheories.Theseproblemsare mostoftenofaconceptualorbroadlylogicalnature.(More aboutthisinthenextsection.)Metaphysiciansofscience want,tosomeextent,todiscussthefundamental.Philosophers ofspecificphysicaltheoriesoftenwriteaboutfoundational problemsorpuzzlesinthosetheories.
1Theexistenceofdependencerelations(reductiverelations,for example)andtheexistenceofsomeonefundamentaltheory(atheory ofeverything)are,ofcourse,deeplydisputedclaims.Ijustnotethatfact fornow.
AMiddleWay:ANon-FundamentalApproachtoMany-BodyPhysics.RobertW.Batterman, OxfordUniversityPress. c ⃝ OxfordUniversityPress2021. DOI:10.1093/oso/9780197568613.003.0001
Ontheassumptionthatthedistinctionbetween“foundationalproblems”and“fundamentalvs.non-fundamental theories”isreasonablycogent,thisbookprimarilyaddresses thelatter.Ibelievethatthefocusonfoundationalproblemsin physicaltheorieshasledmanyphilosophersofphysics(myself, attimes,included)tofailtoappreciatesomeimportant aspectsofphysicalmethodologythathavebeendevelopedin thestudyofmany-bodysystemsconstruedmostbroadly.This isprimarilybecauseattentiontofoundationalproblemsofa conceptualnaturedirectsourgazetowardtheproperlogical structureofthetheoryanditsproperaxiomaticformulation, ratherthantowardtheprocessesbywhichscientistsactually developmodelsandtheories.Ibelievethatthisisresiduefrom thedaysoflogicalpositivismandlogicalempiricism.
Thestudyofmany-bodysystemsismuchmorethanthe studyofthecorrectaxiomaticstructureofclassicaltheoriesor quantumtheories.Someofthemostsuccessfulandprofound methodsforgainingunderstandingofcollectiveproperties ofmany-bodysystemstreatsuchsystemsfroma fieldtheoretic perspective.Thisperspectivefocusesonstructuresat scalesintermediatebetweentheatomicscaleandthecontinuumscaleoffluidmechanicsandthermodynamics.Following condensed-mattertheorists,wecanrefertothemesoscale descriptionsas“hydrodynamicdescriptions.”Myhopeisthat byfocusingonthesemethodswecangainconsiderableinsight intorelationsbetweenfundamentalandnon-fundamental theories.Specifically,Iwillarguethatmesoscalequantities andparametersare,forthepurposeofunderstandingthebulk behaviorofmany-bodysystems,muchsuperiortoquantities andparametersatfundamentalatomicscales.Theyare,Ihope toargue,the natural variablesorparametersforthatpurpose.
1.1PhilosophyandFoundational Problems Philosophyofphysicshasinpracticelargelyconcerneditself withfoundationalproblemsinphysicaltheories.Examplesof
foundationalproblemsfromstatisticalmechanicsandthermodynamicsincludetheproblemofirreversibilityandthe justificationofappealstoergodicityinthecontextofequilibriumstatisticalmechanics.Thefirstconcernshowtoaccount forthemanifesttemporalasymmetriesintheworldatthe scaleofeverydayobjects,giventhecompletetemporal symmetry ofthedynamicsgoverningthebehaviorsoftheatomic andmolecularconstituentsofsucheverydayobjects.2 The secondinvolvesthejustificationofidentifyingtimeaverages withensembleaveragesinvariousstatisticalcalculationsof equilibriumproperties.3 Quantummechanicsprovidesmany suchintractablefoundationalpuzzles:Whatistheproper waytounderstandmeasurementinthetheory?Whatisthe natureofthe(in)famousnon-localityinthattheory?Isthe theoryreallyincompatiblewiththeexistenceofhiddenvariables?Relativistictheoriesalsoraisefoundationalproblems concerningthecausalstructureofspacetime.Ofcurrent(and deepinterest)aretheconnectionsbetweengeneralrelativistic theoriesofgravityandquantumfieldtheories.Thereare extremedifficultiesinreconcilingthesetwotheoriesathigh energiesorshortdistances(Burgess,2004;Donoghue,2012).
Asnoted,theseproblemsoftenhintatpotential“logical flaws”attheverycoreofscientifictheories.Myconceptionof “foundationalproblems”dependspartlyuponthepresenceof suchlogicalissues.4 Philosophersandmathematicalphysicists tendtoapproachsomeoftheseproblemsbytryingtofirm upthelogicalfoundationsofthetheories.Sometimesthese attemptsinvolvetryingtofindtheproperaxiomatization
2See(Sklar,1993).
3See(Moore,2015;MalamentandZabell,1980;EarmanandRédei, 1996;Batterman,1998;Sklar,1993;WerndlandFrigg,2015a,b).
4Iambeingpurposivelyvagueaboutthenatureoftheselogicalflaws andissues.(Hence,theuseofscarequotes.)Theideaisthatthereare conceptualconflictsthatappearhiddenuntilphilosophicalspotlightsare focusedoncoretenetsofthetheories.Ibelievethefocusonlogical inconsistencieshasitsrootsinthepositivistandempiricistapproachesto sciencedevelopedintheearlypartofthetwentiethcentury.Anhistorical analysisintothisisbeyondthescopeofthisinvestigation.
forthetheory—onethateliminatesanyinconsistenciesor incompatibilities.Attemptsliketheseconnecttheconception ofa foundational problemtotheproperarticulationor formulationofthe/arelevant fundamental theory.
Theaxiomatic/algebraicprograminquantumfieldtheory providesaveryniceexampleofthisconnection.Thatapproach aimstofindtheproperaxiomssoastoplacequantumfield theoryonfirmlogicalgrounds5 (HalvorsonandMueger,2006).
Ithinkitisfairtosaythatalargepartofphilosophyof physicshasbeendirectedtowardinvestigatingrealandpresumed“logical”inconsistenciesinvarioustheories.Additionally,philosophersofphysicshaveaimedtofindnecessaryand sufficientconditionsthatjustifyappealstocertainproperties (e.g.ergodicity,locality,etc.).Ialsothinkitisfairtosay thatconsiderableprogresshasbeenmadeonmanyofthese foundationalproblems.
Itisalsoclear,astheexampleofaxiomaticquantumfield theorydemonstrates,thatatleastoneroutetoaddressing foundational problemsistoproperlyaxiomatizethe fundamental theory.Inthissense,thereisarelativelydirect connectionbetweenattemptingtosolvefoundationalproblems andHilbert’s 6th problemofaxiomatizingphysicaltheories (specificallymechanicaltheories).HereisHilbert:
Theinvestigationsonthefoundationsofgeometry suggesttheproblem: Totreatinthesamemanner, bymeansofaxioms,thosephysicalsciencesin whichmathematicsplaysanimportantpart;in thefirstrankarethetheoryofprobabilitiesand mechanics. (Hilbert,1902,p.454)
HilbertnotesthatBoltzmann’swork Lecturesonthe PrinciplesofMechanics (VorlesungenüberdiePrincipeder Mechanik)“suggeststheproblemofdevelopingmathemat-
5Insodoing,oneaimistoavoidthe“inconsistencies”accusedof leadingtodivergencesandunwantedinfinitiesinperturbativequantum fieldtheories.
icallythelimitingprocesses,theremerelyindicated,whichlead fromtheatomisticviewtothelawsofmotionofcontinuua.”
Hilbertposedasecondaxiomatizationproblem:
Converselyonemighttrytoderivethelawsofthe motionofrigidbodiesbyalimitingprocessfrom asystemofaxiomsdependingupontheideaof continuouslyvaryingconditionsofamaterialfillingallspacecontinuously,theseconditionsbeing definedbyparameters.(Hilbert,1902,p.454)
Thisproblemhasreceivedlittleattentionbyphilosophers. However,thehomogenizationmethodsinvestigatedinthis book,whilenotstartingfromaxioms,doattempttoshow howmaterialbehaviorsofcontinuumbodiescanarisefrom limitingconsiderationsstartingwithcompositesofdifferent materialsexhibitingdiverseproperties.Furthermore,these differentpropertiesaredefinedintermsofparameters,aswe shallsee.
Considertheproblemofconnectingtheatomistictheories tothecontinuumtheoriesviaaxiomatizationandlimitingtheorems.Iftherightaxiomatizationcanbefoundalongwiththe properlimittheorems,wewouldbeabletoderive directly the lawsofcontinuummechanicsfromtheatomisticfundamental theory.Presumably,havingdonethiswewouldalsohavean understandingofhow(forexample)irreversiblecontinuumscalebehaviorsaregroundedinthetime-symmetriclawsof theatomictheory.Ibelieveitisfairtosaythatthisproject hasnotbeencompletelysuccessful.
Ontheotherhand,wewillseeinthecourseofthis bookthattherearegenuinereasonstoholdthatpartof theconnectionoflower-scaletocontinuum-scalebehaviors canberealizedbyafocusonintermediate(meso-)scales.In fact,Ibelievethatphilosophicaldiscussionsconcerningintertheoryandinter-levelrelationshavefocusedonalimitedsetof possibilities(Nagel,1961;Batterman,2002;Dizadji-Bahamani etal.,2010;Schaffner,2013;Fodor,1974;Kim,1992).For example,thereareNagel-liketype-typereductionswherea
non-fundamentaltheoryispresumablyreducedto(derived from)amorefundamentalone.Thereareweakerproposals whereanon-fundamentaltheoryissaidtosuperveneonamore fundamentalone,presupposingtoken-tokenreductions.In generalthough,discussionsofinter-theory/inter-levelrelations haveprimarilysought directrelations betweenthereducing, morefundamentaltheoryandthereduced,lessfundamental theory.
Iintendtoarguethatthesesupposeddirectrelationsare rarelypossible.Ialsointendtopresentafield-theoretically motivatedmethodology—a hydrodynamicmethodology—for making indirect,mesoscalemediatedconnectionsbetweencontinuumtheoriesandmorefundamentalatomicandmolecular theories.Thismethodologyiswidespreadandguideswork inphysics,materialsscience,andevenbiology.Furthermore, itworksdespitethefactthatitalmostcompletelyignores fundamentalatomicandmoleculardetails.Itmaybeunderstood,asnoted,tobeapartialrealizationofHilbert’sconverse axiomatizationproblem.
1.2AutonomyandFundamentality
Thereisafeatureofmanytheoreticalhierarchiesthatstand intherelation“theory X ismorefundamentalthantheory Y ” thatstandardreductivestrategiesgenerallyignoreorhave missedaltogether.Thisisthefactthatmanylessfundamental theoriesaretocertainextents autonomous fromtheirmore fundamentalpartners.Thereductivestrategiesignorethis autonomy,astheirmaingoalistoshowhowtheupperscale,lessfundamentaltheories,are derivablefrom,andhence dependentupon theirmorefundamentalcounterparts.
AsanexampleofthekindofautonomytowhichIrefer, considertherelationbetweenfluidmechanicsandmolecular dynamics:Thereisadeepsenseinwhichfluidmechanics—a theorythattreatsfluidsascontinuousblobswithnostructure atall—isautonomousfrommorefundamentaltheoriesthat
recognizethatfluidsareactuallycomposedofvariouskinds ofmolecules.Theontologicallyincorrecttheoryofcontinuous blobsworksamazinglywell,despitethefactthatfluidsare actuallycomposedofcollectionsofdiscretemolecules.Thisis afact,adatum,thatwewanttounderstandandexplain.The generalstructureofthisexplanationisdiscussedinChapter2. Muchofthisbookisdevotedtovariousstrategiesand methodologiesthatplayaroleinrealizingthisunderstanding. Thereisanironyherewhenonecomparestheprevious talkoffundamentalityandthisnotionofautonomywithsome contemporarytheorizinginthemetaphysicsoffundamentality.Itseemsthatmanymetaphysicaldiscussionsconnect fundamentalitywithsomenotionof(in)dependence.Thus, Tahko,notingthatthereisahostof(in)dependencerelations Di,formulatesvariousconceptionsoffundamentalitybased uponthefollowingintuition:
Acommonwaytothinkaboutfundamentalityis intermsofindependence,wherebyforanynotion ofdependence D,anentityis D-fundamentalif andonlyifitdoesnotdependD onanythingelse (oronanythingelsethatdoesnotdependonit).
(Tahko,2018,p.3)
Ofcourse,wearefocusingontheorieshere,butgiventhatthe theoriesaredescriptiveoftheentities, primafacie,wemight trytoorderourhierarchyoftheoriesusing(in)dependence relationsaccordingtothepreviousschema.Ifwedothis, asameansforelucidatingtherelation“theory X ismore fundamentalthantheory Y ,”werundirectlyintothefollowing situation.Manyofourscientifictheories(continuumtheories likethermodynamicsandfluidmechanics)areconsideredto bephenomenologicalandthereforenon-fundamental.Aims todemonstratethisarethefocusofthevariousproposals fortheoryreductionmentionedpreviously.But,asnoted, thesenon-fundamentaltheoriesoftenexhibitakindofrelative autonomyor independence fromthelower-scaletheoriesthat
areoftenconsideredtobemorefundamental.Thus,thiskind ofautonomyorrelative/conditional independence6 doesnot (atleastnoteasily)mapontosomemetaphysicians’discussionsoffundamentalityintermsofindependence.Upper-scale phenomenaofthesortweareconsideringoftendisplaya remarkableinsensitivitytochangesinlower-scaledetails.
Infact,ifwepayattentiontoactualmethodsfrom condensed-matterphysicsandfrommaterialsscience,wewill seethatconnectionsbetweenmorefundamentalandless fundamentaltheoriesaremediatedbyintermediatemesoscale ormesoleveltheoriesandmodels.Often,onedoesnotneed (norcanoneobtain)manydetailsfromthefundamental theorytomakeupscalingconnections.Thisisaverygood thingasmuchofphysicswouldnotbepossibleifallthe detailswererequired.Thisisnotjustamatterofpractical convenience,asignoringlower-scaledetailsandfocusingon intermediatemesoscales(asmuchofcondensed-matterphysics does)actuallyallowsustounderstandtheautonomyjust mentioned.Furthermore,suchafocuswillsuggestthata betteraccountofwhatmakesonetheorymorefundamental thananothercanbehadbygivingupondirectreductive connectionsinfavorofindirectorderingsbasedonrelations ofrelativeautonomy.Theselatterconnectionsgiveusa differentwayofunderstandingwhatmakesonetheorymore fundamentalthananother.Itisneitherareductiverelation norametaphysicalrelationoffundamentalityintermsof independence.
1.3Two-ishSensesofFundamental
ItisworthnotingthatCliffordTruesdell7 makesadistinction between“structuretheories”and“continuumtheories”that
6SeethediscussionofWoodwardon“theproblemofvariablechoice” inChapter7.
7Truesdellwasanidiosyncratic,sometimesirasciblemathematician, naturalphilosopher,andphysicist.Hemadegreatstridesindeveloping continuummechanicsandputtingitonreasonablyfirmfoundations.He
alsomapsontoadistinction,respectively,betweenfundamentalandlessfundamentaltheories(TruesdellandNoll,1992, pp.5–8).Wecangetasenseofthedistinctionbyfocusing onhisscornfor“realphysicists”—thosewhoexclusivelystudy structuretheories:
Thetrainingofprofessionalphysiciststodayputs themunderheavydisadvantagewhenitcomes tounderstandingphysicalphenomena,muchas didatrainingintheologysomecenturiesago. Ignorancecommonlyventsitselfinexpressionsof contempt.Thus“physics”,bydefinition,isbecome exclusivelythestudyofthestructureofmatter, whileanyonewhoconsidersphysicalphenomena onasupermolecularscaleiskickedasideasnot beinga“real”physicist.Often“real”physicists letitbeknownthatallgrossphenomenaeasily couldbedescribedandpredictedperfectlywell bystructuraltheories;thatasidefromthelackof “fundamental”(i.e.structural)interestinallthings concerningordinarymaterialssuchaswater,air, andwood,theblockstoatruly“physical”(i.e. structural)treatmentare“onlymathematical”. (Truesdell,1984,p.47)
Theideathatthefundamentalstructuraltheoriesare in principle abletodescribeandpredictallgross(continuum) propertiesis,ofcourse,ahallmarkofreductionistthinking. Ibelieveitreflectsmuchofthemotivationfortalkingabout fundamentalvs.non-fundamentaltheoriesinthefirstplace. Overthecourseofthisbook,wewillfindanumberofreasons tobeskepticalaboutthis inprinciple claim.
Stillanotherimportant(thoughrelated)senseof“fundamental”reflectsthefactthatsometheoriesandsomemethods areepistemicallymoretrustworthythanothers.Theyleadto
wasthefounderandeditorofthejournals ArchiveforRationalMechanics andAnalysis andthe ArchiveforHistoryofExactSciences.
true,universalizable,andempiricallyverifiableclaimsabout theworld.Itisoftenassumed,Ibelieve,thatthemost fundamentaltheoryinthefirstsensewillautomaticallybethe onethatisfundamentalinitstrustworthiness.8 Onelesson tobelearnedfromthediscussionstofollowisthatthis assumptionisalmostalwaysmistaken.
Itisoftenthecasethatderivingpredictable,observable consequencesfromfundamentaltheoriesis,ifnotimpossible, thenextremelydifficult.Onecandismissthisdifficultyby appealto“inprinciple”derivabilityasjustnoted,butsuch dismissalsarerarely,ifever,accompaniedbyasuggestionas tohowthismightbedone.Muchofsciencesoldiersonanyway andstrivestomakeempiricallytestableconnectionswiththe world.Many-bodysystemsposeparticularlydifficultproblems formakingdirectconnectionswithobservablephenomena. Thisisobvious,anditisclearlythereasonwhystatistical methodsinmechanicsweredevelopedbyBoltzmannand Maxwell etal. inthefirstplace.
Asnoted,Iwouldliketofocusonaparticularsetof methodsdevelopedforthetheoryofsolid-stateorcondensedmatterphysicsthataimtomakeconnectionswithmeasurementparticularlydirectandtransparent.Thatistosay, thesemethodsrepresentourbest,mostefficient,andmost trustworthymeansforobtaininginformationaboutmanybodysystems.Studyingthese hydrodynamicmethods isthe primaryaimofthisbook.Ibelievethattherearesome ratherdeepphilosophicallessonstobelearnedbyexamining thegeneralmethodologyoftreatingmany-bodysystemsby usingthesetechniques.Specifically,Iwillarguethatamiddleoutormesoscaleapproachtostudyingmany-bodysystems issuperiorinmanycontextstomethodsthatstartwith fundamentaltheories.Inaddition,Ihopetoshowthatthere aregenuinescientificreasonsfortreatingmesoscalequantities orparametersas natural variableswithwhichtocharacterize
8Itisprettyclearthatthisassumptionisonething(therearemany!) thatTruesdellfindsobjectionable.
suchsystems.9 Thus,theconclusionisthatinimportant instances,fundamentalvariablesare not thenaturalvariables. Thenextsectionbeginstolayouttheingredientsrequiredto supporttheseclaims.
