The eightfold way to dissipation
DDCPT-14/65
The eightfold way to hydrodynamic dissipation
Felix M. Haehl, ∗ R. Loganayagam, † and Mukund Rangamani ‡ Centre for Particle Theory & Department of Mathematical Sciences,Durham University, South Road, Durham DH1 3LE, United Kingdom Institute for Advanced Study, Einstein Drive, Princeton, NJ 08540, USA.
We provide a complete characterization of hydrodynamic transport consistent with the second lawof thermodynamics at arbitrary orders in the gradient expansion. A key ingredient in facilitatingthis analysis is the notion of adiabatic hydrodynamics, which enables isolation of the genuinelydissipative parts of transport. We demonstrate that most transport is adiabatic. Furthermore, ofthe dissipative part, only terms at the leading order in gradient expansion are constrained to besign-definite by the second law (as has been derived before).
I. INTRODUCTION
Hydrodynamics is the universal low energy descrip-tion at sufficiently high temperatures of quantum sys-tems near thermal equilibrium. The dynamical fields arethe intensive parameters that describe the near thermaldensity matrix viz., temperature T , chemical potential µ along with the fluid velocity ( u µ , u µ u µ = −
1) whichsets the local frame in which the state appears thermal.The background sources are the metric g µν and the flavorsources A µ . The hydrodynamic state in a given back-ground is then completely characterized by a ‘thermalvector’ β µ and ‘thermal twist’ Λ β defined via B ≡ (cid:26) β µ = u µ T , Λ β = µT − β σ A σ (cid:27) . (1)The response to the background sources are encoded inthe energy momentum tensor ( T µν ) and charge current( J µ ) of the theory given in terms of the hydrodynamicfields. The dynamical equations are the statements ofconservation. In the presence of external sources andquantum anomalies (incorporated by the inflow Hall cur-rents T µ ⊥ H and J ⊥ H ) one has with D µ = ∇ µ + [ A µ , · ] ∇ ν T µν = J ν · F µν + T µ ⊥ H D ν J ν = J ⊥ H . (2)Phenomenologically, a hydrodynamicist findsconstitutive relations that express the currents interms of the fields. The operators are tensors builtout of B , the background sources { g µν , A µ } , and theirgradients, multiplied by transport coefficients which arearbitrary scalar functions of T, µ . A-priori this ‘currentalgebra’ formulation appears simple, since classifyingsuch unrestricted tensors is a straightforward exercise inrepresentation theory.However, hydrodynamic currents should satisfy a fur-ther constraint [1] – the second law of thermodynamicshas to hold for arbitrary configurations of the low energy ∗ [email protected] † [email protected] ‡ [email protected] dynamics. In practice, one demands the existence of anentropy current J µS with non-negative definite divergence ∇ µ J µS ≥ η, ζ, σ ≥
0, which is physically intuitive. To dateno complete classification has been obtained at higherorders, though the impressive analyses of [2–4] come quiteclose.From a (Wilsonian) effective field theorist’s perspec-tive this phenomenological current algebra-like approachis unsatisfactory. Not only is the entropy current notassociated with any underlying microscopic principle,but also the origin of dynamics as conservation is ob-scure. A-priori a Wilsonian description for density ma-trices should involve working with doubled microscopicdegrees of freedom, a la Schwinger-Keldysh or Martin-Siggia-Rose-Janssen-deDominicis. But one has yet to un-derstand the couplings between the two copies (influencefunctionals) allowed by the second law, which ought toencode information about dissipation (and curiously alsoanomalies [5]).In this letter we describe a new framework for hydro-dynamic effective field theories and provide a completeclassification of transport. In particular, hydrodynamictransport admits a natural decomposition into adiabaticand dissipative components: the latter contribute to en-tropy production, while the former don’t. At low ordersterms such as viscosities are dissipative; a major surpriseis that most higher order transport is adiabatic!Adiabatic transport can be captured by an effectiveaction with not only Schwinger-Keldysh doubling of thesources, but also a new gauge principle, U (1) T KMSgauge invariance, with a gauge field A ( T ) . This symmetryimplies adiabaticity i.e., off-shell entropy conservation,providing thereby a rationale for J µS (dissipative dynam-ics arises in the Higgs phase). We use this to prove aneightfold classification of adiabatic transport. Togetherwith a key theorem from [3], we further argue that dis-sipative hydrodynamic transport is constrained by thesecond law only at leading order in gradients. In thefollowing we will sketch the essential features of our con- a r X i v : . [ h e p - t h ] M a y struction; details will appear in companion papers [6]. II. ADIABATIC HYDRODYNAMICS AND THEEIGHTFOLD WAY
The key ingredient of our analysis which enables theclassification scheme is the notion of adiabaticity. Themain complications in hydrodynamics arise from at-tempting to implement the second law of thermodynam-ics on-shell. Significant simplification can be achieved bytaking the constraints off-shell. One natural way to dothis is to extend the inequality ∇ µ J µS ≥ ∇ µ J µS + β µ (cid:16) ∇ ν T µν − J ν · F µν − T µ ⊥ H (cid:17) + (Λ β + β λ A λ ) · (cid:0) D ν J ν − J ⊥ H (cid:1) = ∆ ≥ , (3)with ∆ capturing the dissipation and “ · ” denotes flavourindex contraction.While taking the second-law inequality off-shell allowsus to ignore on-shell dynamics, one can obtain the moststringent conditions by examining the boundary of thedomain where we marginally satisfy the constraint. Wedefine an adiabatic fluid as one where the off-shell en-tropy production is compensated for precisely by energy-momentum and charge transport. We thus motivate thestudy of the adiabaticity equation obtained from (3) bysetting ∆ = 0. We will refer to the set of functionals { J µS , T µν , J µ } that satisfy ∆ = 0 as the adiabatic consti-tutive relations.Implications of adiabaticity were first studied in thecontext of anomalous transport in [8] and are exploredin greater detail in [6]. In the following we will quotesome of the salient results of our analysis and explainhow it helps with the taxonomy.Intuitively, the notion of adiabaticity is an off-shell gen-eralization of non-dissipativeness; imposing (2) we learnthat the entropy current has to be conserved on-shell.Moreover, apart from quantum anomalies encoded by theHall currents, the contributions at each order in the gra-dient expansion can be decoupled. It is quite remark-able that this corner of the hydrodynamic constitutiverelations is sufficient to delineate all the constraints ontransport. We will first outline different classes of solu-tions to the adiabaticity equation (3) and then in § IIIexplain how it can be utilized for taxonomic purposes.The adiabatic transport finds a natural classificationinto eight primary classes – see Fig. 1. We emphasizethat adiabatic constitutive relations encode those trans-port coefficients which never appear in the expression forentropy production. Together with Class D (dissipative)we exhaust all transport along this eightfold path.To understand the nomenclature and taxonomy, let usstart with Class A which comprises of transport fixed
FIG. 1. The eightfold way of hydrodynamic transport. by the quantum anomalies of the QFT. Such anomaloustransport gives a particular solution to the adiabaticityequation (3), cf., [8] – the anomalous Hall currents can beviewed as inhomogeneous source terms. This allows usto dispense with them once and for all and focus thenceon the non-anomalous adiabaticity equation.The simplest solutions to (3) can be obtained byrestricting to hydrostatic equilibrium (Class H). Onesubjects the fluid to arbitrary slowly varying, time-independent external sources { g µν , A µ } . The backgroundtime-independence implies the existence of Killing vec-tor and gauge transformation, K = { K µ , Λ K } , with δ K g µν = δ K A µ = 0. Identifying the hydrodynamic fieldswith these background isometries β µ = K µ , Λ β = Λ K solves (3). This information can equivalently be encodedin a hydrostatic partition function [9, 10] which is thegenerating functional of (Euclidean) current correlators.Varying this partition function, we can then obtain aclass of constitutive relations that solve (3).The partition function has two distinct components:hydrostatic scalars H S and vectors H V . The transforma-tion properties refer to the transverse spatial manifoldobtained by reducing along the (timelike) isometry di-rection. The scalars H S are terms one is most familiarwith; e.g., the pressure p as a functional of intensive pa-rameters (which now are determined by the backgroundKilling fields). The vectors P σ in H V are both transverseto the Killing field and conserved on the co-dimension oneachronal slice, i.e., K σ P σ = ∇ σ P σ = 0. These scalars and vectors in the generating function generate thetensor operators for the currents (including part of dissipativetransport) upon variation with respect to the background metricand gauge field.
Hydrostatics fixes a part of the constitutive rela-tions by imposing relations between a-priori independenttransport coefficients [9]. These relations (Class H F ) cap-ture the fact that non-vanishing hydrostatic currents ex-pressed as independent tensor structures in equilibrium,arise from a single partition function. More importantly,dangerous terms which can produce sign-indefinite diver-gence of entropy current are eliminated in Class H F .The second set of solutions of (3) are generated by gen-eralizing the scalar part of the partition function to time-dependent configurations, a la Landau-Ginzburg. We callthese Lagrangian (Class L) solutions, since one can finda local Lagrangian (or Landau-Ginzburg free-energy) ofthe hydrodynamic fields and sources L [ β µ , Λ β , g µν , A µ ].The currents are defined through standard variationalcalculus which can be expressed after suitable integra-tions by parts as1 √− g δ (cid:0) √− g L (cid:1) = 12 T µν δg µν + J µ · δA µ + T h σ δ β σ + T n · ( δ Λ β + A σ δ β σ ) + bdy. terms (4)while the entropy density is defined as (nb: J µS = s u µ ) s ≡ (cid:18) √− g δδT ˆ √− g L [ Ψ ] (cid:19) (cid:12)(cid:12)(cid:12)(cid:12) { u σ , µ, g αβ , A α } =fixed with Ψ ≡ { β µ , Λ β , g µν , A µ } . Diffeomorphism and gaugeinvariance of L together imply a set of Bianchi identi-ties, which together with the definition of J µS suffices toshow that (3) is satisfied. In the above equation, onecan interpret { h σ , n } as characterizing the adiabatic heatcurrent and adiabatic charge density which satisfy a re-lation of the form T s + µ · n = − u σ h σ .It is intuitively clear that by restricting Class L so-lutions to hydrostatics we recover the partition functionscalars H S . As a result one can write L = H S ∪ H S withH S denoting scalar invariants that vanish identically inhydrostatics; hence hydrostatic scalars take values in acoset manifold L / H S .There are two other adiabatic constitutive relationswhich are non-hydrostatic but non-dissipative. Oneclass of adiabatic constitutive relations describe Berry-like transport (Class B) which can be parameterized as( T µν ) B ≡ − N ( µν )( αβ ) δ B g αβ + X ( µν ) α · δ B A α ( J α ) B ≡ − X ( µν ) α δ B g µν − S [ αβ ] · δ B A β . (5) Obtaining the dynamical equations of motion i.e., conservationin Class L requires a constrained variational principle whereinone only considers variations in the Lie orbit of a reference con-figuration, cf., [6]. Class L is equivalent up to a Legendre trans-form to the non-dissipative effective action formalism developedin [11, 12]. Thus the effective action describes a proper subset ofadiabatic constitutive relations.
Here N ( µν )( αβ ) = −N ( αβ )( µν ) , X µνα , and S αβ are arbitrary local functionals of Ψ with indicated(anti)symmetry properties, such that along with J µS = − β ν T µν − µT J µ , the adiabaticity equation is satisfied[6]. A prime example for structures of the type (5) arethe parity odd shear tensor in 3 dimensions which con-tributes to Hall viscosity (Class B). Thus, the tensors N µναβ , X µνα , and S αβ can be thought of as a general-ization of the notion of odd viscosities and conductivities.We will denote the other class as Class H V which canbe parameterized as:( T µν ) H V ≡ (cid:104) D ρ C ρ ( µν )( αβ ) N δ B g αβ + 2 C ρ ( µν )( αβ ) N D ρ δ B g αβ (cid:105) + D ρ C ρ ( µν ) α X · δ B A α + 2 C ρ ( µν ) α X · D ρ δ B A α ( J α ) H V ≡ (cid:104) D ρ C ρ ( µν ) α X δ B g µν + 2 C ρ ( µν ) α X D ρ δ B g µν (cid:105) + D ρ C ρ ( αβ ) S · δ B A β + 2 C ρ ( αβ ) S · D ρ δ B A β (6)where C ρ ( µν )( αβ ) N = C ρ ( αβ )( µν ) N . The entropy current has asimilar form as in Class B along with an additional con-tribution which is quadratic in δ B g µν and δ B A µ . Finally,we have exactly conserved vectors (Class C) which can beadded to the entropy current without modification of theconstitutive relations. They describe possible topologicalstates which transport entropy but no charge or energy.We claim that the above classification is exhaustive: Theorem:
The eightfold classes of adiabatic hydro-dynamic transport can be obtained from a scalar La-grangian density L T (cid:2) β µ , Λ β , g µν , A µ , ¯ g µν , ¯ A µ , A ( T ) µ (cid:3) : L T = 12 T µν ¯ g µν + J µ · ¯ A µ + ( J σS + β ν T νσ + (Λ β + β ν A ν ) · J σ ) A ( T ) σ . (7)As indicated the Lagrangian density depends not onlyon the hydrodynamic fields and the background sources,but also the ‘Schwinger-Keldysh’ partners of the sources { ¯ g µν , ¯ A µ } and a new KMS gauge field A ( T ) µ . This La-grangian is invariant under diffeomorphisms and gaugetransformations and under U (1) T which acts only onthe sources as a diffeomorphism or gauge transformationalong B . The U (1) T gauge invariance implies a Bianchiidentity, which is nothing but the adiabaticity equation(3). Furthermore, a constrained variational principle forthe fields { β µ , Λ β } ensures that the dynamics of thetheory is simply given by conservation. We anticipate Here δ B denotes Lie derivatives implementing diffeomorphismsand flavour gauge transformations by B , i.e., δ B g µν = 2 ∇ ( µ β ν ) and δ B A µ = D µ (Λ β + β σ A σ ) + β ν F νµ . Anomalies if present are dealt with using the inflow mechanism[13]. L T then includes a topological theory in d + 1 dimen-sions coupled to the physical d -dimensional QFT (at the bound-ary/edge). that the KMS gauge field plays a crucial role in imple-menting non-equilibrium fluctuation-dissipation relationswhich follow from the KMS condition; its significanceboth in hydrodynamic effective field theories as well asin holography will be discussed in a future work [14]. III. THE ROUTE TO DISSIPATION
Having classified solutions to the adiabaticity equationlet us now turn to the characterization of hydrodynamictransport including dissipative terms (Class D). We willdo so by first systematically eliminating all of the adia-batic transport by the following algorithm:1. Enumerate the total number of transport coeffi-cients, Tot k∂ , at the k th order in the derivative ex-pansion. This can be done by either working in apreferred fluid frame, or more generally by classify-ing frame-invariant scalar, vector and tensor data.2. Find the particular solution to the anomaly inducedtransport (if any); this fixes all terms in Class A.3. Restrict to hydrostatic equilibrium. The (inde-pendent) non-vanishing scalar fields and transverseconserved vectors determine H S and H V respec-tively (after factoring out terms which are relatedup to total derivatives), which parameterize the(Euclidean) partition function [9, 10].4. Classify the number of tensor structures enteringconstitutive relations that survive the hydrostaticlimit. Since they are to be determined from H S andH V respectively, we should have a number of hy-drostatic relations H F . In general the hydrostaticconstrained transport coefficients are given as lin-ear differential combinations of unconstrained ones.5. Determine the Class L scalars that vanish in hy-drostatic equilibrium H S from the list of frame in-variant scalars after throwing out terms in H S (andthose related by total derivatives).6. Find all solutions to Class B and H V terms at thedesired order in the gradient expansion by clas-sifying potential tensor structures {N , X , S} and { C N , C X , C S } respectively. We have now solved forthe adiabatic part of hydrodynamics.7. The remainder of transport is dissipative and con-tributes to ∆ (cid:54) = 0. Class D is subdivided intotwo classes: terms constrained by the second lawlie in Class D v , while those in Class D s contributesub-dominantly to entropy production and are ar-bitrary. The goal at this stage is to isolate the D v terms; fortunately they only show up only at theleading order in the gradient expansion ( k = 1);for k ≥ s (cf.,[3, 4]). 8. Finally, Class D s can be written in terms of dissi-pative tensor structures using the same formalismemployed for Class B, except now we pick a differ-ent symmetry structure to ensure ∆ (cid:54) = 0.Steps 1-6 can be implemented straightforwardly in the U (1) T invariant L T , but we will exemplify this algorithmby a more pedestrian approach below. In Table I weprovide a classification of transport for few hydrodynamicsystems up to second order in gradient expansion. Fluid Type Tot H S H S H F H V A B H V D Neutral 1 ∂ ∂
15 3 2 5 0 0 2 0 3Weyl neutral 2 ∂ ∂ ∂
51 7 5 17 0 0 11 2 9TABLE I. Transport taxonomy for some simple (parity-even)fluid systems in d ≥
4. The fluid type refers to whetherwe describe pure energy-momentum transport (neutral) ortransport with a single global symmetry (charged). We haveindicated the derivative order at which we are working by k∂ . IV. AN EXAMPLE: WEYL INVARIANTNEUTRAL FLUID
To illustrate our construction consider a (parity-even)Weyl invariant neutral fluid which has been studied ex-tensively in the holographic context [15–17]. Weyl invari-ance implies that the stress tensor must be traceless andbuilt out of Weyl covariant tensors. Our classificationsuggests the following constitutive relation written in abasis adapted to the eightfold way: T µν = p ( d u µ u ν + g µν ) − η σ µν + ( λ − κ ) σ <µα σ ν>α + ( λ + 2 τ − κ ) σ <µα ω ν>α + τ (cid:0) u α D W α σ µν − σ <µα ω ν>α (cid:1) + λ ω <µα ω ν>α + κ (cid:0) C µανβ u α u β + σ <µα σ ν>α + 2 σ <µα ω ν>α (cid:1) . (8)To obtain this note that for a neutral fluid there are noanomalies so A = 0. At first order there is only a ClassD term η σ µν which contributes to ∆ = 2 η σ , leadingto η ≥ ω µν ω νµ and W R ;hence H S = 2 corresponding to λ and κ terms. As σ µν vanishes in hydrostatics only two tensors survive the The fluid tensors are defined via the decomposition ∇ µ u ν = σ ( µν ) + ω [ µν ] + d − Θ ( g µν + u µ u ν ) − a ν u µ and <> denotesthe symmetric, transverse (to u µ ) traceless projection. TheWeyl covariant derivative [18] (and associated curvatures) pre-serve homogeneity under conformal rescaling. In particular, W R = R + 2( d − (cid:16) ∇ α W α − d − W (cid:17) , with Weyl connection W µ = a µ − Θ d − u µ appears in Eq. (9). limit; thus there are no constraints, H F = 0. There areno transverse vectors and so H V = H V = 0. Surprisingly( λ + 2 τ − κ ) σ <µα ω ν>α is a Class B term – it can beobtained from N [( µν ) | ( αβ )] ∼ ( λ + 2 τ − κ ) ( ω µα P νβ +perms . ). There is one non-hydrostatic scalar σ whichis in H S corresponding to τ term above. This leavesus with one Class D term which can be inferred to be( λ − κ ) σ <µα σ ν>α . Its contribution to entropy productionis ∇ µ J µS ∼ ( λ − κ ) σ αν σ νβ σ αβ . This being sub-dominantto the leading order η σ entropy production, it followsthat ( λ − κ ) belongs to Class D s .While this completes the classification, we note onerather intriguing fact. For holographic fluids dual to twoderivative gravity, the second order constitutive relations(cf., [17]) can be derived from a Class L Lagrangian: L W = − πG d +1 (cid:18) πTd (cid:19) d − × (cid:20) W R ( d −
2) + 12 ω + 1 d Har (cid:18) d − (cid:19) σ (cid:21) (9)where Har( x ) = γ e + Γ (cid:48) ( x )Γ( x ) is the Harmonic number func-tion ( γ e is Euler’s constant). The first two terms are inH S while σ ∈ H S and they give contributions to each ofthe five second order transport coefficients. We thereforehave two relations: λ + 2 τ − κ = 0 , λ − κ = 0 . (10)Eliminating κ we have τ = λ − λ which was ar-gued to in fact be a universal property of two deriva-tive gravity theories [19]. Curiously, the first relationis also obeyed in kinetic theory to the orders in whichcomputations are available [20]. We advance this as theevidence that our eightfold classification explains varioushitherto unexplained coincidences in both perturbativetransport calculations and non-perturbative results fromAdS/CFT.The second relation in (10) suggests that the sub-leading entropy production from ( λ − κ ) is absent inAdS black holes. Inspired by earlier observations re-garding lower bound of shear viscosity η/s ≥ π [21],we conjecture that holographic fluids obtained in thelong-wavelength limit of strongly interacting quantumsystems obey a principle of minimal dissipation. Thefluid/gravity correspondence provides the shortest pathin the eightfold way: AdS black holes scramble fast tothermalize, but are slow to dissipate! The relation between { τ, κ, λ } appears not to hold in higherderivative gravitation theories. It however must be borne in mindthat generic higher derivative gravity theories are unlikely to bedual to unitary QFTs. We thank S. Grozdanov, E. Shaverin, A.Starinets and A.Yarom for sharing their results. ACKNOWLEDGMENTS
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