Effec%veFieldTheoryofDissipa%veFluids

HongLiu

MichaelCrossleyarXiv:1511.03646andtoappear

PaoloGlorioso

Conservedquan%%esandhydrodynamics

Considerlongwavelengthdisturbancesofasysteminthermalequilibrium:

conservedquan%%es:cannotrelaxlocally,onlyviatransports

Conservedquan%%esGaplessandlong-livedmodes

Thereshouldexistauniversallowenergyeffec%vetheory.

non-conservedquan%%es:relaxlocally,

Phenomenologicaldescrip%on: Hydrodynamics

(onlyonesinthermalequilibrium)

slowlyvaryingfunc%onsofspace%me

O’Haraetal(2002)

Despitethelongandglorioushistoryofhydrodynamics

Itdoesnotcapturefluctua%ons,whicharealwayspresent,andareimportantinmanyphysicalcontexts:

Transports(long%metail),dynamicalaspectsofphasetransi%ons,non-equilibriumsteadystates,turbulence,finitesizesystems….......

Phenomenologicallevel:stochas%chydro(Landau,Lifshitz)

Goodfornear-equilibriumdisturbances(linearized)

Butnotadequateforfar-from-equilibriumsitua%ons

Rayleigh-Benardproblem

Hydrodynamicfluctua%onshot

cold

Constraints

Cons%tu%verela%ons:notenoughtojustwritedownthemostgeneralderiva%veexpansionconsistentwithsymmetries.

1.Entropycondi%on

2.Onsagerrela%ons:linearresponsematrixmustbesymmetric

Phenomenologicalconstraints:solu%onsshouldsa%sfy:

Arethesecomplete?

Currentformula%onofhydrodynamicsisawkward.

awkward:usesolu%onstoconstrainequa%onsofmo%on

Microscopicorigin?

Clearlysomethingmorefundamentalismissing

Willaddresstheseissuesbydevelopinghydrodynamicsasalowenergyeffec%vefieldtheoryofageneralmany-bodysystematfinitetemperature

Ac%onprincipleforfullfluctua%nghydrodynamicsapplicabletofarfromequilibriumsitua%ons

Closed%mepath

Standardlore:Dissipa%vesystemsdon'thaveanac%onformula%on

Thisissueisnaturallyresolvedbyquantummechanics.

Closed%mepath(CTP)orSchwinger-Keldyshcontour

Interestedindescribingreal-%medynamicsaroundastate,nottransi%onamplitude

Requiresdoublingthenumberofd.o.f.

Closed%mepath

Standardlore:Dissipa%vesystemsdon'thaveanac%onformula%on

Thisissueisnaturallyresolvedbyquantummechanics.

Closed%mepath(CTP)orSchwinger-Keldyshcontour

Interestedindescribingreal-%medynamicsaroundastate,nottransi%onamplitude

Requiresdoublingthenumberofd.o.f.

Effec%vefieldtheoryforfluidsMicroscopicdescrip%on(inthermaldensitymatrix)

Macroscopicphenomena

HydroEFT:

RG

donotwork

3.Integra%onmeasure?

•  thermalstatereflectedincoefficientsofac%on

•  Standardhydro:saddlepointequa%ons

hydrodynamicmodes(2sets)

1.Whatare?

2.Whatarethesymmetriesof?

Searchingforanac%onprinciplefordissipa%vehydrodynamicshasbeenalongstandingproblem,da%ngbackatleasttotheidealfluidac%onofG.Herglotzin1911.

Relativistic: Taub, Salmon, Carter, …...

Includingdissipa%on:GrozdanovandPolonyi;Harder,Kovtun,andRitz;Kovtun,MooreandRomatschke;Endlich,Nicolis,PortoandJ.Wang,…...

Holographic: Nickel, Son; de Boer, Heller, Pinzani-Fokeeva; Crossley, Glorioso., HL, Y. Wang.

Reviews: Jackiw, Nair,PiandPolychronakos;Andersson and Comer

Dubovsky,Gregoire,NicolisandRaiazzihep-th/0512260

The last decade has seen a renewed interest:

Dubovsky,Hui,NicolisandSon,arXiv:1107.0731

Analterna%veapproach:Haehl,LoganayagamandRangamani,arXiv:1510.02494,1511.07809

Hereforconceptualsimplicity,Iwilldescribethestoryforneutralnormalfluids.

Descrip%onforfullchargedfluidsisgivenin:

Crossley,Glorioso,andHL,arXiv:1511.03646

Canalsobegeneralizedtosuperfluids(S.RajagopalandHL,toappear)

DynamicalvariablesRecallLagrangedescrip%onofhydrodynamics:

:labelfluidelements : describefluidmo%on

An”integra%ng-in”proceduretoiden%fygaplessandlong-livedd.o.f.associatedwithaconservedstresstensor:

:internal%me

AlsonaturalfromHolography:Nickel,Son,arXiv:1009.3094

Standardhydrovariables(whicharenowderivedquan%%es)

Noise:

Asignificantchallenge:ensuretheeomsfromtheac%onofXcanbesolelyexpressedintermsofthesevelocity.(e.g.solidsv.s.fluids)

Symmetries

Requiretheac%ontobeinvariantunder:

labelindividualfluidelements, internal%me

definewhatisafluid!

Itturnsoutthesesymmetriesindeeddomagicforyou:

atthelevelofequa%onsofmo%on,theyensurealldependenceondynamicalvariablescanbeexpressedinandtemperature.

Recoverstandardformula%onofhydrodynamics(modulophenomenologicalconstraints)

Thiswouldbethefullthestoryinausualsitua%on.

Fullnon-linearfluidfluctua%ngdynamicsencodedinnon-trivialdifferen%algeometry:

Consistencycondi%onsandsymmetries

ForasystemdefinedwithCTP:

constraintsoncorrela%onfunc%ons,

Requirefurthersymmetriessa%sfiedbytheoff-shellac%onsothattheseconstraintcondi%onsaresa%sfiedaoerdoingthepathintegral.

g1

g2=

Reflec%vityandKMScondi%ons

Canbesa%sfiedbyimposingaZ2reflec%onsymmetryonbothdynamicalandbackgroundfields

Complexac%on:non-nega%vityofimaginarypart

•  KMScondi%onsplusCPTimplyaZ2symmetryonW:

inversetemperatureatspa%alinfini%escharacterizethermalstateCanbesa%sfiedbyimposingaZ2symmetryontheac%on:localKMScondi%ons

•  Reflec%vitycondi%on:

•  Non-equilibriumfluctua%on-dissipa%onrela%ons

•  Alltheconstraintsfromentropycurrentcondi%onandlinearOnsagerrela%ons

•  Newconstraintsonequa%onsofmo%onfromnonlineargeneraliza%onsofOnsagerrela%ons.

Crossley,Glorioso,HLtoappear

LocalZ2KMScondi%on+equa%onsofmo%on+non-nega%vityofimaginarypartoftheac%on

conservedandentropycurrentatideallevel

Unitarycondi%on

Thismaybeconsideredasacondi%ononpathintegralmeasure.

Todescribehydrodynamicalflucatua%onsproperlyan%-commu%ngobjectsareneeded!

SeealsoHaehl,LoganayagamandRangamaniarXiv:1510.02494

g1=g

g2=g

(BRSTsymmetryhereisaglobalsymmetry)

Resolu%on:Introduceafermionicpartner(“ghost”)foreachdynamicalvariableandrequiretheac%ontohavea“BRST”typesymmetry:

KMScondi%onandSupersymmetry

SeealsoHaehl,LoganayagamandRangamani,arXiv:1510.02494,1511.07809

Strongindica%onsthatBRST+localKMScondi%on

PingGaoandHL,inprogress

emergentfermionicsymmetry:

Classicallimit: Standardsupersymmetryin%medirec%on.

Quantumlevel:indica%onsofa“quantum-deformed”SUSYalgebra

Summary1.Hydrodynamicswithclassicalsta%s%calfluctua%onsisdescribedbyasupersymmetricquantumfieldtheory

2.Hydrodynamicswithquantumfluctua%onsalsoincorporated

isdescribedbya“quantum-deformed”(supersymmetric)quantumfieldtheory.

•  Recoversthestandardhydrodynamics,newconstraints,non-equilibriumfluctua%on-dissipa%onrela%ons

•  Fullnon-linearfluidfluctua%ngdynamicsencodedinnon-trivialdifferen%algeometry.

Bosonicac%onforfullchargedfluidsAdoubleexpansionintermsofnoisesandderiva%ves:

Allcoefficientsarefunc%onsof

Futuredirec%onsFormalism:

Non-rela%vis%c,(toappearwithGlorioso)

Applica%ons:

Transports(long%metails)

Dynamicalaspectsofphasetransi%ons

NovelfixedpointsofhydroEFT,suchasKPZscaling,turbulence….

….........

Non-equilibriumsteadystates,dynamicalflowsofQGP

“quantum-deformed”Supersymmetry…....

superfluids,(toappearwithS.Rajagopal)

DeeperunderstandingofKMS,unitarityandsupersymmetry(inprogresswithPingGao)

Thankyou!