EffectiveFieldTheoryofDissipativeFluids
HongLiu
MichaelCrossley
arXiv:1511.03646
PaoloGlorioso
ConservedquantitiesConsideralong wavelengthdisturbanceofasysteminthermalequilibrium
conserved quantities:cannot relaxlocally,onlyviatransports
Conserved quantitiesGaplessandlong-livedmodes
(only onesinamedium)
Thereshouldexistauniversal lowenergyeffectivetheory.
non-conserved quantities:relaxlocally,
Hydrodynamics
Thermalequilibrium:
Promotethesequantitiestodynamicalvariables:(localequilibrium)
slowlyvaryingfunctionsofspacetime
Expressexpectationvaluesofthestresstensorandconservedcurrent intermsofderivativeexpansionofthesevariables:constitutiverelations.
Equationsofmotion:
d+1variables,d+1equations
Despitethelongandglorioushistoryofhydrodynamics
Itdoesnot capturefluctuations.
Fluctuations
Therearealwaysstatistical fluctuations…..
transports,
Importantinmanycontexts:
Atlowtemperatures,quantum fluctuationscanalsobeimportant.
Longtimetail
dynamicalaspectsofphasetransitions,
non-equilibriumstates,turbulence,
finitesizesystems….
Phenomenologicallevel:stochastic hydro(Landau,Lifshitz)
:noiseswithlocalGaussiandistribution
3.fluctuationsofdynamicalvariablesthemselves
Expect:
1.interactionsamongnoises
2.interactionsbetweendynamicalvariablesandnoises
Untilnownotknownhowtotreatsuchnonlineareffectssystematically.Notevenclearitisagoodquestion.
particularlyimportantfornon-equilibriumsituations.
Constraints
Constitutiverelations:notenough tojustwritedownthemostgeneralderivativeexpansionconsistentwithsymmetries.
1.Entropycondition
2.Onsagerrelations:linearresponsematrixmustbesymmetric
Phenomenologicalconstraints:solutionsshouldsatisfy:
Arethesecomplete?
Currentformulationofhydrodynamicsis awkward.
awkward:usesolutionstoconstrainequationsofmotion
Microscopicderivation?
EffectivetheoryapproachmayalsomakeiteasiertogeneralizehydrodynamicsEOMtolessfamiliarsituations,saywithmomentumdissipations,anomalies.....
develophydrodynamicsasabonafidelowenergyeffectivefieldtheoryofageneralmany-bodysystematfinitetemperature
1.givesafullinteractingtheory ofnoises.
2.Microscopicoriginandcompletenessofphenomenologicalconstraints
3.Newconstraints(nonlinearOnsagerrelations)
Actionprinciplewhichincorporatesbothdissipationsandnoises
Shouldbedistinguished fromanactionwhichjustreproducesstandardeoms (whichmaynotcapturefluctuationscorrectly)
Searchingforanactionprincipleforhydrodynamicshasbeenalongstandingopenproblem,datingbackatleasttoG.Herglotz in1911….....
Allresultsatnon-dissipativelevel….
Manyactivitiessince70’stounderstandhydrodynamicfluctuations….....
Results
1.Hydrodynamicswithclassicalstatisticalfluctuations
isdescribedbyasupersymmetric quantum fieldtheory
2.Hydrodynamicswithquantumfluctuationsalsoincorporated
isdescribedbya“quantum-deformed”(supersymmetric)quantumfieldtheory.
SeealsoHaehl,Loganayagam,Rangamani
Approach:putarelativisticquantummany-bodyssystem inacurvedspacetime
PartII:formulation
Transitionamplitudesv.s.expectationvalues
Weareinterestedinaneffectivetheorydescribingnonlineardynamicsaroundastate.
ShouldbecontrastedwithEFTdescribingtransitionamplitudes,
Closedtimepath(CTP)orSchwinger-Keldysh contour
Shoulddouble alldegreesoffreedom
Hydroeffectivefieldtheory
hydrodynamicmodes
EFTapproach:
1.Whatare? donotwork
2.Whatarethesymmetriesof?
3.Integrationmeasure?
Atlongdistancesandlargetimes:
Allcorrelationfunctionsofthestresstensorandconservedcurrentsinthermalequilibrium
Dynamicalvariables:integratinginToyexample:asingleconservedcurrent
1.Currentconservation:
2.Wmustbenonlocal:Non-localitysolely duetointegratingouthydromodes
Integratein hydromodes:
(a):local (b):Ensure1issatisfied
(c):EOMsmustbeequivalenttocurrentconservations
Proposal:(usetheusualStueckelberger trick)
isalocal action. :hydromodes
Satisfythefollowingconsistencyrequirements:
1.
2.Eoms ofareequivalenttocurrentconservations.
Dynamicalvariables(II)Forstresstensor,weputthesysteminacurvedspacetime
Conservationofstresstensor:
Integrateinhydromodes: Promotespacetime coordinatestodynamicalfields
1.2.Xeoms areequivalenttoconservationofstresstensor
anemergent spacetime withcoordinates
Interpretationof: labelindividualfluidelements, internaltime
:motionofafluidelementinphysicalspacetime
SowejustrecoveredtheLagrangedescriptionofafluid!
Asastartingpoint,wecouldsimplydoublethedegreesoffreedomintheLagrangedescription.
Abithistory:
NickelandSonshowedthecovariantversionarisesnaturallyfromholography(arXiv:1103.2137).
DoubledcopiesappearedinHaehl,Loganayagam,RangamaniarXiv:1502.00636, andCrossley,Glorioso,HL,WangarXiv:1504.07611.
UsingasinglecopyofasdynamicalvariableforanidealfluidactiondatedbacktoG.Herglotz in1911.
CovariantwasusedbyTaub in1954.
Rediscoveredin2005byDubovsky,Gregoire,Nicolis andRattazziinhep-th/0512260andfurtherdevelopedbyDubovsky,Hui,Nicolis and Son inarXiv:1107.0731 ,......
Standardhydrovariables(whicharenowderivedquantities)
Asignificantchallenge: ensuretheeoms fromtheactionofXandcanbesolelyexpressedintermsofthesevelocitytypeofvariables.(e.g.solids v.s.fluids)
Symmetries(I)Nowneedtospecifythesymmetriesof
Notethatitisdefinedinfluidspacetime
Requiretheactiontobeinvariantunder:
Interpretationof: labelindividualfluidelements, internaltime
definewhatisafluid!
Itturnsoutthesesymmetriesindeeddomagicforyou:
atthelevelofequationsofmotion,theyensurealldependenceondynamicalvariablescanbeexpressedin
Recoverstandardformulationofhydrodynamics(modulo phenomenological constraints)
Thiswouldbethefullthestoryinausualsituation.
Fullnon-linearfluidfluctuatingdynamicsencodedinnon-trivialdifferentialgeometry:
Symmetries(II)WeareconsideringEFTforasystemdefinedwithCTP:
Thegeneratingfunctionalhasthefollowingproperties:
• KMSconditionplusPTimplyaZ2 symmetryonW:
• Reflectivitycondition:
• Unitaritycondition:
Fullbosonic theoryReflectivityconditioncanbeeasilyimposed,leadingtoacomplex action.
ImposingKMSconditionisverytricky.
AlltheconstraintsfromentropycurrentconditionandlinearOnsagerrelations
NewconstraintsonequationsofmotionfromnonlinearOnsagerrelations.
proposal:localKMScondition,aZ2symmetryontheaction
Imaginarypartoftheactionnon-negative
FermionsandSupersymmetry
isa“topological”conditiononthemeasureofpathintegrals
Introducefermionic partners(“ghost”fieds)fordynamicalvariablesandrequiretheactiontohaveaBRSTtypesymmetry.
SeealsoHaehl etalarXiv:1510.024941511.07809Unitarity condition:
Ataquadratic levelindynamicalfields,onefindsthatlocalKMSconditionleadstoanemergentfermionic symmetry.
Butnotclearhowtowritedownanonlinearactionwithsuchanalgebra.
Requiresa“quantum-deformed”SUSY
Classicallimit:
becomestandardsupersymmetry intimedirection.
Inthislimitonecanwritedownasupersymmetric completionofthefullbosonic hydrodynamicaction.
Notethatintheclassicallimit,pathintegralremains,capturingstatisticalfluctuations.
Example:nonlinearstochasticdiffusion
Considerthetheoryforasingleconservedcurrent,wheretherelevantphysicsisdiffusion.
Dynamicalvariables: (or)
Roughly,:standarddiffusionmode,:thenoise.
Ifignoringinteractionsofnoise
AvariationofKardar-Parisi-Zhang equation
Summary
Fermionic excitationsandEmergentsupersymmetry.
AnEFTforgeneraldissipativefluids.
Recoversthestandardhydrodynamicsasequationsofmotion,constitutiverelations,constraints.
Encodesquantumandthermalfluctuationssystematicallyinapathintegralexpansion.
Fullnon-linearfluidfluctuatingdynamicsencodedinnon-trivialdifferentialgeometry.
FuturedirectionsFormalism:
Non-relativisticlimit,superfluids,Anisotropic,inhomogeneous,“quantum-deformed”Supersymmetry
…....Applications:
Longtimetails,runningofviscosities,
Dynamicalaspectsofclassicalandquantumphasetransitions
ScalingbehaviorinhydrobehaviorviafixedpointsofQFTs,suchasKPZscaling,turbulence….
….........
Non-equilibriumsteadystates,dynamicalflowsofQGP
ThankYou