The Entropy current in Hydrodynamics, Superfluid Hydrodymics … · 2011-07-06 · The Theory of charged dissipative hydrodynamics Superﬂuidity from AdS/CFT A Theory of Superﬂuid

The Fluid-Gravity CorrespondenceThe Theory of charged dissipative hydrodynamics

Superfluidity from AdS/CFTA Theory of Superfluid Hydrodynamics

Summary and a crazy conjecture

The Entropy current in Hydrodynamics,Superfluid Hydrodymics and Gravity

Shiraz Minwalla

Department of Theoretical PhysicsTata Institute of Fundamental Research, Mumbai.

Strings 2011, Upsalla, June 2011

Shiraz Minwalla




CollaboratorsJ. Bhattacharya, S. Bhattacharyya and A. YaromBased on arXiv:1101.3332 and arXiv:1105.3733Other Relevant References:arXiv:1004.2707 (Sonner and Withers)arXiv:1101.3330 (Herzog, Lisker, Surowka, Yarom)arXiv:1104.5245 (Shu Lin)arXiv:1106.3576(Neiman and Oz)

Shiraz Minwalla




Introduction

In this talk I will present the most general form of theequations of s wave superfluid hydrodynamics at first orderin the derivative expansion, subject to the constraints ofsymmetry and the second law of thermodynamics.Although superfluid hydrodynamics is a 60 year oldsubject, to my knowledge these equations have nevercorrectly been worked out before in full generalityWe needed to obtain these results in order to make senseof certain computations performed within the framework ofthe so called ‘fluid gravity’ correspondence of AdS/CFT. Istart the talk with a review of this correspondence.

Shiraz Minwalla




The Fluid-Gravity Map

5d negative cosmological constant Einstein Maxwell gravityadmits a 5 parameter set of Reisner Nordstrom black branesolutions. The 5 parameters may be taken to be uµ, T andµ: the 4-velocity, temperature and chemical potential.The Goldstone idea may be used to promote theseparameters to slowly varying fields. Corresponding toevery field configuration of uµ(x), T (x) and µ(x) (subjectto an equation of motion see below) there exists a solutionto the bulk equations of motion. These solutions can beconstructed order by order in ∂

T , and have been explicitlyconstructed to second order in this parameter. The bulksolutions have regular future event horizons.

Shiraz Minwalla




Constitutive Relations

These solutions each have an associated boundary stresstensor Tµν(x) and a charge current Jµ(x). Tµν and Jµ aredetermined in terms of the thermodyanmical fields by socalled constitutive relations

Tµν = (ρ+ P)uµuν + Pηµν + πµν

Jµ = quµ + Jµdiss(1)

The pressure P(T , µ), energy density ρ(T , µ) and chargedensity q(T , µ) are determined by the uniform branesolutions. πµν and Jµdiss represent derivative contributionsto the constitutive relations. They are determined to mth

order by implementation of the fluid-gravitycorrespondence to the same order.

Shiraz Minwalla




Equations of Motion and Entropy Current

Conservation of the boundary stress tensor and current

∂µTµν = FµνJν∂µJµ = cF ∧ F

(2)

yields the equations of fluid dynamics.The event horizon, rH(xµ) of the bulk solutions turns out tobe a local function of thermodynamical fields and theirderivatives. The Hodge dual of the pullback of the horizonarea form to the boundary yields an entropy current for thefluid flow. Hawking’s area increase theorem guaranteesthat

∇µJµS ≥ 0

Shiraz Minwalla




The Theory of constitutive relations

As we have explained, the bulk equations determine theconstitutive relations for πµν and Jµdiss. Relations dependon the precise form of the bulk equations, and change if,e.g. we couple the Maxwell field to neutral scalars.Question: as we vary over all allowed bulk couplings do wescan over all symmetry allowed values πµν and Jµdiss?Answer: No. Generic values for constitutive relations areinconsistent with the positivity of entropy production, as wenow explain. The characterization of the most generalconstitutive relations consistent with non negativity ofentropy production is a question addressed by Landau andLifshitz in their text on Fluid Dynamics. We now repeattheir analysis with certain changes.

Shiraz Minwalla




Listing Fluid Data

The fluid velocity at any given point breaks the localtangent space SO(3,1) to SO(3). It is useful to label onederivative fluid data by its transformations under SO(3).We start with scalars.One derivative scalarsuµ∂µT ,uµ∂µµ,∂.u.One derivative scalar EOMsuµ∇νTµν = 0∇µJµ = 0Onshell independent scalar dataS1 = ∂µuµ

Shiraz Minwalla




Fluid Data

Similarly onshell independent vector dataV1 = −Pµν∂ν

µT + Fµνuν

TV2 = uν∇νuµ

V3 = FµνuνOnshell independent tensor: σµνOnshell independent 2 derivative scalar dataST

1 = Pµν∂µ∂ν( µ

T

)ST

2 = uµ∂µ∂νuν

ST3 = ∂µ(Fµνuν)

Pµν = uµuν + gµν

σµν = PµαPνβ

(∂αuβ + ∂βuα

2− Pαβ

∂.u3

)Shiraz Minwalla




Constitutive Relations Allowed by Symmetry

The most general constitutive relations allowed by symmetry

PµαPν

βπαβ − Pµν

3Pαβπαβ = −ησµν

(π)abPab

3− ∂P∂ρ

(uµuνπµν) +∂P∂q

(uµJµdiss) = −β∂αuα

Pµα

(Jαdiss +

qρ+ P

(uνπαν))

=3∑

i=1

κiVµi

Note that only those parts of πµν and Jµdiss that are invariantunder field redefinitions of uµ, T and µ are meaningful and maybe specified by constitutive relations.

Shiraz Minwalla




Positivity of Entropy

We must now impose the requirement of non negativity ofentropy production. Landau and Lifshitz proceed asfollows. They define a canonical entropy current

Jµcannon = suµ − µ

TJµdiss −

uνπµν

Twhich is asserted, on intuitive grounds, to be the ‘correct’entropy current for fluid flows.They then use thermodynamics and the equations ofmotion to demonstrate that

∂µJµcannon = −∂µ[uν

T

]πµν − ∂µ

[ µT

]Jµdiss,

and obtain constraints on constitutive coefficients frompositivity of the RHS.

Shiraz Minwalla




It has become clear in other contexts that the intuition thatasserts the ‘correct’ form of the entropy current is notalways infallible. We will not assume it. Instead we simplyallow the entropy current to take the most general onederivative form allowed by symmetries

JµS = Jcanon + s1 S1uµ +3∑

i=1

viVµi

We then proceed to compute ∂µJµS .The result takes the schematic form

∂µJµS =(

independent twoderivative and curvature data

)+(

quadratic form infirst order data

)

Shiraz Minwalla




2 derivative terms in divergence

−v1ST1 + (s1 + v2)ST

2 +(

v3 +v1

T

)ST

3 .

Equating to zerov1 = v3 = 0 and v2 = −s1.Resultant entropy current

JµS = Jcanon + s1(S1uµ − Vµ

2

)Apparent one parameter ambiguity

Shiraz Minwalla




Move to curved space. Additional term in divergence

s1Rαβuαuβ

Thus s1 = 0Resultant entropy current

JµS = Jcanon

Form of entropy current uniquely fixed by requirement ofpositivity. Agrees with the intuition of Landau and Lifshitz

Shiraz Minwalla




∇µJµS =S1

T

[(π)abPab

3− ∂P∂ρ


(uµJµdiss)

]+ V1µ

[Jµdiss +

qρ+ P

(uνπµν)]

−σµνT

[PµαPν

βπαβ − Pµν

3Pαβπαβ

]

PµαPν

βπαβ − Pµν

3Pαβπαβ = −ησµν

Pµα

(Jαdiss +

qρ+ P

(uνπαν))

= κVµ1

(π)abPab

3− ∂P∂ρ


(uµJµdiss) = −β∂αuα

Shiraz Minwalla




Summary- Part 1

The positivity of entropy production require that 2 out of the5 possible constitutive parameters vanish. The entropycurrent is fixed to take the canonical form. Explicitcomputations in Einstein Maxwell theory bear out boththese predictions.In order to completely constrain the entropy current weimposed the requirement that the fluid obey the 2nd laweven when formulated on a weakly curved backgroundmetric.Of the 5 first order pieces of data, σµν , ∂.u and Vµ

1 aredissipative (result in entropy production). The remaining 2pieces of data, Vµ

2 and Vµ3 are nondissipative to this order.

Shiraz Minwalla




Hairy black branes

It was pointed out by Gubser that asymptotically AdScharged black branes are sometimes unstable to thecondensation of charged bulk scalars.The end point of this instability is a ‘hairy’ black brane; abrane in equilibrium with a charged scalar condensateoutside its horizon.Under the AdS/CFT duality, a hairy black brane describesa superfluid at rest. The stuff inside the horizon of theblack brane is roughly dual the ‘normal’ part of the fluid,while the scalar (Bose) condensate outide the horizon isroughly dual to the superfluid.

Shiraz Minwalla




It turns out that hairy black brane solutions admit a naturald − 1 parameter generalization. It is possible to constructstationary hairy black brane solutions in which which Ai isa nontrivial and reduces to a constant at the boundary , (allfields are functions only of the radial coordinate r in thesesolutions).This solution is dual to a configuration in which the normalfluid is at rest, while the superfluid moves with 4 velocityproportional to ξµ = ∂µφ

b −Abµ (Ab

µ is the boundary value ofthe gauge field and φb is the boundary value of thecharged scalar). We can of course produce furthersolutions by boosting these configurations.This is satisfying as superfluids are well known to admitexactly stationary solutions in which the normal fluidmoves relative to the superfluid.

Shiraz Minwalla




In net, we have an 8 parameter set of stationary solutionslabeled by T , uµ, ξµ. T is the temperature of the blackbrane. uµ, the boost parameter, may be thought of as thevelocity of the normal fluid. Aµb encodes the chemicalpotential of the configuration (via the equation µ = A.u)and the 4 velocity of the superfluid.Natural question: what is the value of the stress tensor andcharge current in equilibrium configuration labeled by the 8parameters above?In a beautiful paper a year or so ago, Sonner and Withersused abstract (black hole thermodynamics type) reasoningto demonstrate that the answer is given in terms of a singlethermodynamical function, P(T , µ, ξ) by the equations

Shiraz Minwalla




Tµν = (ρn + P)uµuν + Pηµν +ρs

ξ2 ξµξν

Jµ = qnuµ − qsξµ

ξ

uµξµ = µ

(3)

where

ρn + P = qnµ+ Tsρs = µsqs

µs = ξ = ξµuµsdP = sdT + qsdµs + qndµn

(4)

Precisely the Landau-Tisza formShiraz Minwalla




Once again we should be able to generate bulk solutions forarbitrarily slowly varying fields uµ(x), T (x) and ξµ(x), providedthese configurations obey the equations of motion

∂µTµν = FµνJν∂µJµ = cF ∧ F

∂µξν − ∂νξµ = Fµν

(5)

The fluid -gravity map, implemented order by order, generatesconstitutive relations at the same order. These relations - whencombined with constitutive relations - yield equations of motionfor the boundary ‘superfluid’

Shiraz Minwalla




As before, the constitutive relations take the form

Tµν = (ρn + P)uµuν + Pηµν +ρs

ξ2 ξµξν + πµν

Jµ = qnuµ − qsξµ

ξ+ Jµdiss

(6)

Here the stress tensor is written as a function of the 8fields ξµ(x), T (x) and uµ(x). πµν and Jµdiss represent thecontributions of terms that are of atleast first or higherorder in the derivatives of fluid fields.In order to close those equations we need expressions forπµν and Jµdiss. For every configuration that solves theseequations we have a solution of the bulk equations, thathas a regular event horizon, and so an entropy current withpositive divergence.

Shiraz Minwalla




In a corner of parameter regime of a specific model(identified by Herzog) we were able to practicallyimplement the fluid gravity map described abstractlybehind to compute πµν and Jµdiss explicitly to first order in aderivative expansionFor technical reasons we worked with a theory of aminimally coupled scalar field of large charge e andm2 = −2. The bulk equations we used wereEinstein-Maxwell theory, plus Chern Simons A ∧ F ∧ F .Our results did not fit within the 13 parameterClark-Putterman constitutive relations listed in the textbook on superfluid hydrodynamics by Putterman. For thisreason we were forced to address the question similar tothe one answered above: what is the most general formallowed for πµν and Jµdiss ?

Shiraz Minwalla




The method: imitates the presentation above with somedifferences. At any given point we have two vectors, uµ(x),ξµ(x). Residual symmetry group generically, SO(3). Weclassify data in representations of SO(2).We then proceed to ennumerate onshell indpendent data.For instance, assuming parity invariance we find 7 scalars,7 vectors and 2 tensors at first order in the derivativeexpansion.We write down the most general constitutive relationallowed by symmetry and then impose non negativity ofthe divergence of the entropy current. We also impose theso called ‘Onsager Reciprocity Relation’ to ensure CPTinvariance of our results.

Shiraz Minwalla




Landau and Lifshitz and Clark and Putterman undertake asimilar analysis assuming the entropy current take acanonical form very similar to that described above.As in our previous analysis, we proceed without makingany assumptions about the form of the entropy current (21parameters assuming parity, 32 parameters without), butconstrain the latter, along with constitutive parameters, viathe requirement of positivity, even for superfluid flows in aweakly curved background spacetime.

Shiraz Minwalla




Results: Parity Even

If we assume that the superfluid preserves parity, it turnsout that (upto a physically inessential ambiguity) theentropy current is uniquely fixed to take the canonical form.Dissipative terms are constrained to lie in a 14 parameterfamily. (Clark and Putterman claimed this was a 13parameter family, but there was a mistake in their analysisand they missed a parameter). It turns out that the gravityresult with zero bulk Chern Simons term does indeed liewithin this corrected Clark Putterman familyIn sum the ‘Landau Lifshitz’ intuition for the structure of theentropy current is impressively borne out in the parity evensector. As a bonus we were able to correct a conceptuallyminor error in the literature.

Shiraz Minwalla




Results: Parity Odd

In this case it turns out that there are 4 additionaldissipative coefficients - ignored by Clark and Putterman-even if we assume that the entropy current takes thecanonical form.More improtantly, however, in this case the entropy currentis not uniquely fixed to take the canonical form, but itselfhas a two parameter ambiguity. It turns out that these twonew paramters manifest themselves as two newparameters in the space of allowed dissipative terms,taking the total number of such terms to 20. All this is trueboth in the presence and in the absence of a U(1)3

anomaly for the current. Our gravitational computationsconfirm that in specific examples the entropy current doesdeviate from the canonical form.

Shiraz Minwalla




Results: Comparison with Gravity

Although we are still in the process of performing athorough check, it so far appears that all our gravitationalresults in the general case (with nonzero bulk ChernSimons term) fit with the 20 parameter predictiondescribed above.It is a simple matter to take the non relativistic limit of our20 parameter set of dissipative terms. No parametersvanish in the process.

Shiraz Minwalla




Practitioners of superfluidity are often most interested inthe special case in which the normal and superfluidvelocities are taken not to deviate much from one another.Formally, one sets these two velocities equal but allowstheir derivatives to be independent.In this special limit Landau-Lifshitz report a 5 parameterset of dissipative terms. Our results agree with theirs in theparity even sector, but include 2 new transport coefficientsupon allowing for parity violationIt does not seem inconceivable that these new terms couldshow up in a laboratory experiment some day.

Shiraz Minwalla




We have derived what we believe to be the most generalset of equations for superfluid hydrodynamics at first orderin the derivative expansionOur results are based on the requirement of the existenceof a positive divergence entropy current in an arbitrarycurved background spacetime and the Onsager principleThe superfluid entropy current is forced to agree with the‘Landau Lifshitz’ canonical form assuming parityinvariance, but in general deviates from this form (with a 2parameter ambiguity) in the parity odd caseOur new framework agrees with the results of explicitgravitational computations, to the extent that we havetested it. It is conceivable that our new constitutiveparmeters may have observable consequencessomewhere

Shiraz Minwalla




A crazy conjecture

Recall that the possible forms of the equations of motion offluid and superfluid hydrodynamics are significantlyconstrained by the requirement that the entropy neverdecreases.Speculation: Could the same be true of gravity? Could thespace of allowed higher derivative corrections to Einstein’sequations be significantly constrained (i.e. cut down innumber of parameters) by the same requirement?Of course when entropy production is positive in Einsteingravity then small higher derivative results cannot changethis. However when entropy production vanishes inEinstein gravity, it seems like a possibility that some higherderivative corrections will push the balance the other way.

Shiraz Minwalla




Crazy Conjecture

However we have already seen that there exist nontrivialgravitational configurations in which the entropy productionvanishes in Einstein gravity: e.g. gravity duals to fluid flowsin which V2 and V3 are turned on but all other firstderivative data is zero.Conjecture: The requirement that entropy remain eitherconstant, or increase in such configurations will yieldnontrivial constraints on possible higher derivativecorrections to Einstein’s equation.This suggests the exciting possibility of an infinite numberof completely new purely low energy constraints onstructure of higher derivative corrections to Einstein gravitywhose origin lies in the thermodyanical nature of gravity inconfigurations with a horizon.

Shiraz Minwalla

Documents

The Entropy current in Hydrodynamics, Superfluid Hydrodymics … · 2011-07-06 · The Theory of charged dissipative hydrodynamics Superﬂuidity from AdS/CFT A Theory of Superﬂuid