Download Effective Field Theory of Dissipative Fluids

EffectiveFieldTheoryof
DissipativeFluids
HongLiu
PaoloGlorioso
MichaelCrossley
arXiv:1511.03646
Conservedquantities
Consideralong wavelengthdisturbance
ofasysteminthermalequilibrium
non-conserved quantities:relaxlocally,
conserved quantities:cannot relaxlocally,onlyviatransports
Conserved quantities
Gaplessand
(only onesin
long-lived
amedium)
modes
Thereshouldexistauniversal lowenergyeffectivetheory.
Hydrodynamics
Thermalequilibrium:
Promotethesequantitiestodynamicalvariables:(localequilibrium)
slowlyvaryingfunctions
ofspacetime
Expressexpectationvaluesofthestresstensorandconserved
current intermsofderivativeexpansionofthesevariables:
constitutiverelations.
Equationsofmotion:
d+1variables,d+1equations
Despitethelongandglorious
historyofhydrodynamics
Itdoesnot capturefluctuations.
Fluctuations
Therearealwaysstatistical fluctuations…..
Importantinmanycontexts:
Longtimetail
transports,
dynamicalaspectsofphasetransitions,
non-equilibriumstates,
turbulence,
finitesizesystems….
Atlowtemperatures,quantum fluctuationscanalsobeimportant.
Phenomenologicallevel:stochastic hydro(Landau,Lifshitz)
:noiseswithlocalGaussiandistribution
Expect:
1.interactionsamongnoises
2.interactionsbetweendynamicalvariablesandnoises
3.fluctuationsofdynamicalvariablesthemselves
particularlyimportantfornon-equilibriumsituations.
Untilnownotknownhowtotreatsuchnonlinear
effectssystematically.Notevenclearitisagoodquestion.
Constraints
Currentformulationofhydrodynamicsis awkward.
Constitutiverelations:notenough tojustwritedownthemost
generalderivativeexpansionconsistentwithsymmetries.
Phenomenologicalconstraints:solutions shouldsatisfy:
1.Entropycondition
2.Onsagerrelations:linearresponsematrixmustbesymmetric
awkward:usesolutionstoconstrainequationsofmotion
Microscopicderivation?
Arethesecomplete?
develophydrodynamicsasabonafidelowenergyeffectivefield
theoryofageneralmany-bodysystematfinitetemperature
Actionprinciplewhichincorporates
bothdissipationsandnoises
1.givesafullinteractingtheory ofnoises.
2.Microscopicoriginandcompletenessof
phenomenologicalconstraints
3.Newconstraints(nonlinearOnsagerrelations)
Shouldbedistinguished fromanactionwhichjustreproducesstandard
eoms (whichmaynotcapturefluctuationscorrectly)
Effectivetheoryapproachmayalsomakeiteasiertogeneralize
hydrodynamicsEOMtolessfamiliarsituations,saywith
momentumdissipations,anomalies.....
Searchingforanactionprincipleforhydrodynamics
hasbeenalongstandingopenproblem,datingback
atleasttoG.Herglotz in1911….....
Allresultsatnon-dissipativelevel….
Manyactivitiessince70’stounderstand
hydrodynamicfluctuations….....
Results
Approach:putarelativisticquantummany-bodyssystem ina
curvedspacetime
1.Hydrodynamicswithclassicalstatisticalfluctuations
isdescribedbyasupersymmetric quantum fieldtheory
SeealsoHaehl,Loganayagam,Rangamani
2.Hydrodynamicswithquantumfluctuationsalsoincorporated
isdescribedbya“quantum-deformed”(supersymmetric)
quantumfieldtheory.
PartII:formulation
Transitionamplitudesv.s.expectationvalues
Weareinterestedinaneffectivetheorydescribingnonlinear
dynamicsaroundastate.
Closedtimepath(CTP)orSchwinger-Keldysh contour
Shoulddouble alldegreesoffreedom
ShouldbecontrastedwithEFTdescribingtransitionamplitudes,
Hydroeffectivefieldtheory
Atlongdistancesandlargetimes:
Allcorrelationfunctionsof
thestresstensorand
conservedcurrentsin
thermalequilibrium
hydrodynamic
modes
EFTapproach:
1.Whatare?
2.Whatarethesymmetriesof?
3.Integrationmeasure?
donotwork
Dynamicalvariables:integratingin
Toyexample:asingleconservedcurrent
1.Currentconservation:
2.Wmustbenonlocal:Non-localitysolely duetointegrating
outhydromodes
Integratein hydromodes:
(a):local
(b):Ensure1issatisfied
(c):EOMsmustbeequivalenttocurrentconservations
Proposal:(usetheusualStueckelberger trick)
isalocal action.
:hydromodes
Satisfythefollowingconsistencyrequirements:
1.
2.Eoms ofareequivalenttocurrentconservations.
Dynamicalvariables(II)
Forstresstensor,weputthesysteminacurvedspacetime
Conservationofstresstensor:
Promotespacetime coordinatesto
Integrateinhydromodes:
dynamicalfields
1.
2.Xeoms areequivalentto
conservationofstresstensor
anemergent spacetime withcoordinates
Interpretationof:
labelindividualfluidelements,
internaltime
:motionofafluidelementinphysicalspacetime
SowejustrecoveredtheLagrangedescriptionofafluid!
Asastartingpoint,wecouldsimplydoublethedegreesoffreedom
intheLagrangedescription.
Abithistory:
Usingasinglecopyofasdynamicalvariableforan
idealfluidactiondatedbacktoG.Herglotz in1911.
CovariantwasusedbyTaub in1954.
Rediscoveredin2005byDubovsky,Gregoire,Nicolis andRattazzi
inhep-th/0512260andfurtherdevelopedbyDubovsky,Hui,
Nicolis and Son inarXiv:1107.0731 ,......
NickelandSonshowedthecovariantversionarisesnaturally
fromholography(arXiv:1103.2137).
DoubledcopiesappearedinHaehl,Loganayagam,Rangamani
arXiv:1502.00636, andCrossley,Glorioso,HL,Wang
arXiv:1504.07611.
Standardhydrovariables(whicharenowderivedquantities)
Asignificantchallenge: ensuretheeoms fromtheaction
ofXandcanbesolelyexpressedintermsofthesevelocity
typeofvariables.(e.g.solids v.s.fluids)
Symmetries(I)
Nowneedtospecifythesymmetriesof
Notethatitisdefinedinfluidspacetime
Interpretationof:
labelindividualfluidelements,
Requiretheactiontobeinvariantunder:
definewhatisafluid!
internaltime
Itturnsoutthesesymmetriesindeeddomagicforyou:
atthelevelofequationsofmotion,theyensurealldependence
ondynamicalvariablescanbeexpressedin
Recoverstandardformulationofhydrodynamics
(modulo phenomenological constraints)
Fullnon-linearfluidfluctuatingdynamicsencoded
innon-trivialdifferentialgeometry:
Thiswouldbethefullthestoryinausualsituation.
Symmetries(II)
WeareconsideringEFTfora
systemdefinedwithCTP:
Thegeneratingfunctionalhasthefollowingproperties:
• Reflectivitycondition:
• KMSconditionplusPTimplyaZ2 symmetryonW:
• Unitaritycondition:
Fullbosonic theory
Reflectivityconditioncanbeeasilyimposed,leadingtoa
complex action.
Imaginarypartoftheactionnon-negative
ImposingKMSconditionisverytricky.
proposal:localKMScondition,aZ2symmetryontheaction
Alltheconstraintsfromentropycurrent
conditionandlinearOnsagerrelations
Newconstraintsonequationsofmotionfromnonlinear
Onsagerrelations.
FermionsandSupersymmetry
Unitarity condition:
SeealsoHaehl etal
arXiv:1510.02494
1511.07809
isa“topological”conditiononthemeasureofpathintegrals
Introducefermionic partners(“ghost”fieds)fordynamical
variablesandrequiretheactiontohaveaBRSTtypesymmetry.
Ataquadratic levelindynamicalfields,onefindsthatlocalKMS
conditionleadstoanemergentfermionic symmetry.
Butnotclearhowtowritedownanonlinearactionwithsuchanalgebra.
Requiresa“quantum-deformed”SUSY
Classicallimit:
becomestandardsupersymmetry intimedirection.
Inthislimitonecanwritedownasupersymmetric completion
ofthefullbosonic hydrodynamicaction.
Notethatintheclassicallimit,pathintegralremains,
capturingstatisticalfluctuations.
Example:nonlinearstochasticdiffusion
Considerthetheoryforasingleconservedcurrent,where
therelevantphysicsisdiffusion.
Dynamicalvariables:
(or)
Roughly,:standarddiffusionmode,:thenoise.
Ifignoringinteractionsofnoise
AvariationofKardar-Parisi-Zhang equation
Summary
AnEFTforgeneraldissipativefluids.
Recoversthestandardhydrodynamicsasequations
ofmotion,constitutiverelations,constraints.
Encodesquantumandthermalfluctuations
systematicallyinapathintegralexpansion.
Fullnon-linearfluidfluctuatingdynamicsencoded
innon-trivialdifferentialgeometry.
Fermionic excitationsandEmergentsupersymmetry.
Futuredirections
Formalism:
Non-relativisticlimit,
superfluids,
Anisotropic,inhomogeneous,
“quantum-deformed”Supersymmetry
…....
Applications:
Longtimetails,runningofviscosities,
Non-equilibriumsteadystates,dynamicalflowsofQGP
Dynamicalaspectsofclassicaland
quantumphasetransitions
Scalingbehaviorinhydrobehaviorviafixedpoints
ofQFTs,suchasKPZscaling,turbulence….
….........
ThankYou