Non-dissipative second-order transport, spin, and pseudo-gauge transformations in hydrodynamics
aa r X i v : . [ h e p - t h ] D ec Non-dissipative second-order transport, spin, and pseudo-gauge transformations inhydrodynamics
Shiyong Li, ∗ Mikhail A. Stephanov, † and Ho-Ung Yee ‡ Physics Department, University of Illinois at Chicago, Chicago, Illinois 60607, USA (Dated: November 2020)We derive a set of nontrivial relations between second-order transport coefficients which followfrom the second law of thermodynamics upon considering a regime close to uniform rotation of thefluid. We demonstrate that extension of hydrodynamics by spin variable is equivalent to modifyingconventional hydrodynamics by a set of second-order terms satisfying the relations we derived. Wepoint out that a novel contribution to the heat current orthogonal to vorticity and temperaturegradient reminiscent of the thermal Hall effect is constrained by the second law.
Introduction. — Relativistic hydrodynamics [1] is aneffective description, at large distance and time scales,of systems in local thermodynamic equilibrium param-eterized by slowly varying profiles of 4-velocity u µ ( x )( u µ u µ = − T ( x ) and chemical po-tential µ ( x ) for a conserved charge. The system of equa-tions based on the conservation laws is closed, and alldynamic information about the system in the hydrody-namic regime is contained in the hydrodynamic variables.Relativistic hydrodynamics has been successful in manybranches of physics, in particular, in describing dynami-cal evolution of the fireball created in relativistic heavy-ion collisions (RHIC) [2, 3].Recent developments include interesting attempts toincorporate spin polarization of microscopic constituentsas an additional hydrodynamic variable characterizingthe system, which led to consideration of “spin hydro-dynamics” [4–6]. This is motivated by importance ofspin observables in many applications of hydrodynamicsin condensed matter as well as nuclear physics. Specif-ically, each event in non-central RHIC carries a signifi-cant amount of initial orbital angular momentum ∼ ~ ,some of which is transferred to the spin polarization ofobserved hadrons [7–13]. However, in the strict sense ofhydrodynamics, spin polarization of plasma constituentsshould also be in local equilibrium, and must be deter-mined by conventional hydrodynamic variables. In this work we assume the standard local equilibrium,and show that the spin hydrodynamics and the conven-tional hydrodynamics are two equivalent descriptions ofthe same system. This not only reconciles the two formu-lations, but also leads us to find new constraints for cer-tain transport coefficients in conventional second-orderhydrodynamics.The central question we answer in this work is the Certain variants of spin hydrodynamics [5] could describe off-equilibrium dynamics of spin polarization in a system where re-laxation time of spin polarization is much slower than other mi-croscopic time scales. Similar extensions of hydrodynamics bynon-hydrodynamic, but nevertheless parametrically slow, vari-ables have been termed Hydro+ [14]. meaning of pseudo-gauge transformations [15–17] in spinhydrodynamics. Since hydrodynamics is based on lo-cal thermodynamics, this question can only be answeredafter properly addressing how thermodynamics trans-form under pseudo-gauge transformations. We show theequivalence of local thermodynamics between the spinand conventional hydrodynamics, which requires us togeneralize pseudo-gauge transformation to currents of en-tropy and conserved charge. We use these results to provethe equivalence between the spin hydrodynamics and theconventional hydrodynamics. In particular, we find thatthe ideal limit of spin hydrodynamics is equivalent to theconventional hydrodynamics with certain non-dissipativesecond-order transport coefficients. Moreover, five ofthese second-order transport coefficients are uniquely de-termined by two thermodynamic functions, one of whichappears as the spin susceptibility in the spin hydrody-namics description.The existence of such constraints on certain second-order transport coefficients is an interesting fact by itself,independent of its physics connection to the spin hydro-dynamics. Within the conventional hydrodynamics, weshow that the same constraints can be derived directlyusing the second law of thermodynamics, and are there-fore universal. Our derivation is based on a new powercounting scheme for gradients of hydrodynamic variables,motivated by considering small deviations from one of theequilibrium states of uniformly rotating fluid, which existdue to conservation of total angular momentum.We consider dissipative gradients of fluid velocity andof α = µ/T , as being much smaller than the vorticity andthe temperature gradients neither of which appear in theentropy production rate at leading order in gradients.This allows us to reorganize the naive gradient expan-sion in the entropy production rate and to derive a set ofnontrivial constraints on certain second-order transportcoefficients by applying the second law of thermodynam-ics. Our method should be more generally applicableto some higher-order transport coefficients, as well as totransport coefficients involving external electromagneticfields, but we leave such generalizations to future work.Although similar constraints have been found for char-gless fluid [18–20] and charged fluid in Ref.[21] using dif-ferent approach, the constraints in Ref. [21] appear to beless stringent, leaving four unconstrained parameters inconstrast to the two coefficients we find. It would be in-teresting to establish relationship between the constraintswe derive and the ones in Ref. [21], which appears to bea nontrival task due to difference in choices of variablesand frames (we use conventional Landau frame). Non-dissipative second-order hydrodynamics. —Guided by the observation that vorticity in a uniformlyrotating fluid can take arbitrary values without entropyproduction, we consider fluid states where vorticity andtemperature gradients, ω µν = ( ∂ ⊥ µ u ν − ∂ ⊥ ν u µ ), ∂ ⊥ µ β ,while still being small, are larger than other, dissipa-tive gradients, θ µν = ( ∂ ⊥ µ u ν + ∂ ⊥ ν u µ ) and ∂ ⊥ µ α , where ∂ ⊥ µ ≡ ∆ µν ∂ ν with ∆ µν = u µ u ν + g µν . To this end, we in-troduce the following power counting scheme: ω µν ∼ ǫ ω , ∂ ⊥ µ β ∼ ǫ ′ , θ µν ∼ ∂ ⊥ µ α ∼ ǫ , while any further spa-tial derivative on ( ω µν , β ) and ( θ µν , α ) brings an ex-tra ǫ ′ and ǫ , respectively. For example, ∂ ⊥ ρ ω µν ∼ ǫ ω ǫ ′ , ∂ ⊥ µ ∂ ⊥ ν β ∼ ǫ ′ , and ∂ ⊥ ρ θ µν ∼ ∂ ⊥ µ ∂ ⊥ ν α ∼ ǫ . In addi-tion, we consider spatial gradients of thermal vorticityto be of the same order as the dissipative gradients, i.e. ∂ ⊥ ν ( βω µ ) ∼ ǫ ω ǫ rather than ǫ ω ǫ ′ , which means ∂ ⊥ ν ω µ = − ( ∂ ⊥ ν β ) ω µ /β + O ( ǫ ω ǫ ). From this and the ideal equationof motion, one can show that ∂ µ ω µ ∼ ω µ ∂ µ β ∼ ǫ ω ǫ .We then invoke the hierarchy, ǫ ′ ≪ ǫ ≪ ǫ ω ǫ ′ ≪ ǫ ω ≪ ǫ ′ ≪ ǫ ω ≪
1. As we will see, this allows us to focus on thevorticity related terms arising from certain second-ordertransport coefficients as the leading contributions to theentropy production rate up to order ǫ ω ǫ ′ ǫ , while the dis-sipative terms from first-order transport coefficients areof order ǫ ≪ ǫ ω ǫ ′ ǫ , and are thus sub-leading. Note that ǫ ω ǫ ′ ǫ would naively be of higher order than ǫ in the con-ventional gradient expansion. By careful inspection of allpossible terms in the entropy production rate, potentiallylarger terms of ǫ ω , ǫ ω ǫ ′ and ǫ ω ǫ ′ can be shown to be ab-sent in parity even plasma that we focus on in this work.Then the second law of thermodynamics, i.e. the non- negativity of entropy production, should be applied tothese leading contributions involving second-order trans-port coefficients.We write the general parity even constitutive relationsfor symmetric energy-momentum tensor, as well as forcharge and entropy currents: T µν = ( ε + p ) u µ u ν + pg µν + ∆ T µν , (1) j µ = nu µ + ∆ j µ , (2) s µ = su µ + ∆ s µ , (3)where ∆ T µν , ∆ j µ and ∆ s µ contain all relevant secondorder terms in our hierarchy,∆ T µν = a ∆ µν ω λρ ω λρ + a ω µλ ω λν , (4)∆ j µ = c ∆ µρ ∂ ν ω νρ + c ω µν ∂ ν β, (5)∆ s µ + α ∆ j µ = b ∆ µρ ∂ ν ω νρ + b ω µν ∂ ν β + b ω µν ∂ ν α, (6)with seven second-order transport coefficients { a i , b i , c i } .We do not need to include the first-order transport termsas explained above, and we omit other possible second-order terms, such as ∂ ⊥ µ β∂ ⊥ ν β in ∆ T µν and ω µν ∂ ν α in∆ j µ , that do not contribute to the entropy productionrate to order ǫ ω ǫ ′ ǫ , and whose coefficients are thus notconstrained by our method. We also remark that onecould put a purely spatial gradient ∆ µρ ∆ γν ∂ γ ω νρ in placeof ∆ µρ ∂ ν ω νρ , but this would be equivalent up to a redef-inition of { b , c } due to the ideal equations of motionand the thermodynamic relation βdp = − wdβ + ndα .Introducing ω µ ≡ ǫ µναβ u ν ω αβ and using the identity ω µ Dω µ = ω µν ( ∂ µ ε )( ∂ ν p )2 w − ω µ ω µ Dpw − ω µα ω αν θ µν (7)which follows from the ideal equations of motion, where w = ε + p and D ≡ u · ∂ , one finds the entropy productionrate up to O ( ǫ ω ǫ ′ ǫ ) given by ∂ µ s µ = C (1) ω ν ω ν θ + C (2) ( ∂ ⊥ ν ω νµ ) ∂ ⊥ µ α + C (3) ( ∂ ⊥ ν ω νµ ) ∂ ⊥ µ β + C (4) ( ∂ µ β ) ω µν ( ∂ ν α ) + C (5) θ µν ω µα ω αν , (8)where θ ≡ θ µµ = ∂ · u and C ( i ) are given by C (1) = − a β + b + 2 b c s + b wβ ε + b α p wc s ) , (9a) C (2) = (cid:18) ∂b ∂α (cid:19) β + b − c , C (3) = (cid:18) ∂b ∂β (cid:19) α + b , (9b) C (4) = b β + (cid:18) ∂b ∂β (cid:19) α + nβw (cid:18) ∂b ∂β (cid:19) α + 1 β (cid:18) ∂b ∂α (cid:19) β + 2 b ∂∂β (cid:18) nβw (cid:19) α + b nβw − (cid:18) ∂b ∂α (cid:19) β − c β + c , (9c) C (5) = a β + 4 b , (9d) where c s = ( ∂p/∂ε ) s/n , α p = ( ∂α/∂p ) s/n and β ε =( ∂β/∂ε ) s/n are thermodynamic derivatives taken with s/n fixed, which appear naturally due to the ideal equa-tions of motion, ( u · ∂ )( s/n ) = 0.All five terms in Eq. (8) are independent and can haveeither sign for generic initial conditions. The second lawof thermodynamics thus requires that all C ( i ) vanish.This gives five constraints for seven unknowns { a i , b i , c i } ,which determines them up to two free functions. Choos-ing a and a as two given functions, one can solve for theother five transport coefficients without any integration,proceeding in the following order: b = − βa , b = − (cid:18) ∂b ∂β (cid:19) α , (10a) b = 1 α p wc s (cid:20) wβ ǫ (cid:18) ∂b ∂β (cid:19) α − b − b c s − βa (cid:21) , (10b) c = b + (cid:18) ∂b ∂α (cid:19) β , c = − (cid:18) ∂c ∂β (cid:19) α − b ∂∂β (cid:18) nβw (cid:19) α . (10c)As a nontrivial check of these relations we can con-sider conformal theory, such as the strongly coupled con-formal plasma described by AdS/CFT correspondencefor which some of the coefficients have been calculatedin Ref. [22]. Conformal invariance imposes certain con-straints on some of the thermodynamic quantities, suchas w = 4 ε/ c s = 1 / β ε = − β/ (4 ε ), α p = 0, as well ontransport coefficients: a = 3 a and ( ∂b /∂β ) α = − b /β .Substituting into Eq. (10b) we find that it is satisfied forany b because, while α p = 0, also the expression in thesquare brackets nontrivially vanishes, provided b is givenby Eq. (10a). Furthermore, conformal invariance requires( ∂ ( n/βw ) /∂β ) α = 0. Substituting into Eq. (10c), wefind a relationship between c and c which coincideswith a nontrivial constraint imposed by conformal Weylsymmetry [22, 23]. Finally, solving Eqs. (10a) and (10c)we can now predict the values of b , b and b whichhave not been calculated in Ref. [22], in terms of a and c which have been calculated. Spin hydrodynamics. — Spin hydrodynamics is based on the canonical energy-momentum tensor T µνc and therank-3 tensor S µαβ = − S µβα of spin current, which havemicroscopic field theory definitions. The total angularmomentum tensor consists of the orbital and the spinparts, J µαβ = ( x α T µβc − x β T µαc ) + S µαβ , and the formal-ism needs the additional conservation law, ∂ µ J µαβ = 0,corresponding to the introduction of additional spin de-grees of freedom. This relates the anti-symmetric partof T µνc to non-conservation of spin due to spin-orbit ex-change of angular momentum: T µνc − T νµc = − ∂ α S αµν .The constitutive relations are given by T µνc = εu µ u ν + p ∆ µν + ( u µ q ν + u ν q µ ) + τ µν − ∂ α S αµν , (11) j µ = nu µ + τ µ , S µαβ = u µ S αβ + σ µαβ , (12)where we do not assume that u µ is the Landau frame, q µ ( u · q = 0) is a contribution to energy current, S µν isthe spin density in local rest frame satisfying the Frenkelcondition u µ S µν = 0, and ( τ µν , τ µ , σ µαβ ) are dissipativegradient corrections. We will not be concerned with thesedissipative terms in our subsequent discussion of an ideallimit, because their inclusion will not affect our mainconclusion.Writing the entropy current as s µ = su µ + ∆ s µ ,( u µ ∆ s µ = 0) and adding 0 = β ν ∂ µ T µνc + α∂ µ j µ to ∂ µ s µ we obtain the following expression for the entropy pro-duction rate: ∂ µ s µ = [ Ds − βDε + αDn + 12 βω µν DS µν ] + θ [ s − β ( ε + p ) + αn + 12 βω µν S µν ] − βτ µν θ µν − τ µ ∂ µ α + ∂ µ [∆ s µ − β ν ∂ α S αµν − βq µ + ατ µ + 12 ( ∂ ρ β δ ) σ µρδ ] + [( − βDu ν + ∂ ν β )( q ν − β S νρ ∂ ρ β )] −
12 ( ∂ α ∂ µ β ν ) σ αµν , (13)where β ν ≡ βu ν .There exists an ideal limit of spin hydrodynamicswhere the right hand side of Eq.(13) vanishes. The van-ishing of the first two square brackets leads to the follow-ing thermodynamics relations [24], ds = βdε − αdn − β γ µν dS µν , s = β ( ε + p ) − αn − β γ µν S µν , (14)where the entropy density is a function of ε , n and S µν ,with the spin potential being equal to the fluid vorticityin local equilibrium: γ µν = ω µν . We emphasize that thespin density should be fixed by the spin potential as athermodynamic relation in equilibrium, i.e. S µν = χγ µν with the spin susceptibility χ [25]. This determines thespin density in terms of hydrodynamic variables, S µν = χω µν .Vanishing of other terms requires∆ s µ = 12 β ν ∂ α ( u α S µν ) + βq µ − ατ µ −
12 ( ∂ ρ β δ ) σ µρδ , (15)and the following relation q µ − wn τ µ = 12 β S µν ∂ ν β = χ β ω µν ∂ ν β. (16)Eq.(16) is independent of the choice of the hydrody-namic frame u µ . However, one can show, by introducingan impurity as in Ref.[26], that τ µ vanishes in the “no-drag frame”. This is a non-trivial example, similar toChiral Vortical Effect [26], where the entropy flows pasta static impurity without generating a drag. One couldrefer to this non-dissipative heat current we find as thevorticity driven thermal Hall effect.As a nontrivial check of Eq. (16) we can calculate theheat current in the no-drag frame for the microscopic chi-ral kinetic theory of massless Dirac fermion. As detailedin Ref. [27], we choose the fluid rest frame as the spinframe n µ = u µ so that the Frenkel condition is satisfied.With n µ = (1 , , , s is proportionalto the axial current, s i = ~ j i = ~ ¯ ψγ i γ ψ . Therefore, s = R p ,λ ~ λ j p , where j p is the phase space (Liouville) currentand R p ,λ ≡ P λ = ± / R d p / (2 π ~ ) includes the sum overhelicities λ . According to Ref. [27], to order O ( ~ ), j p =( ˆ p − ( ~ λ/p ) ˆ p × ∇ ) f eq , where p = | p | . The secondterm in j p not only accounts for 2 / f eq = 1 / (exp { β ( − p · u +(1 / S µνn ω µν ) } +1),where S µνn = λǫ µναβ p α n β / ( p · n ) and µ = 0 for simplicity.The spin density S ij can then be computed as S ij = ǫ ijk s k = ω ij ~ β + O ( ~ ) . (17)On the other hand, the canonical energy-momentumtensor is given by T µνc = R p ,λ j µp p ν . Using the knownresult for j µp , now up to O ( ~ ) from Ref. [29], j p = ˆ p − ~ λp ˆ p × ∇ + ( ~ λ ) p ( ˆ p × ∇ ) × ∇ ! f eq + ( ~ λ ) p { p · [( ˆ p × ∇ ) × ∇ ] } ∂∂p (cid:18) f eq p (cid:19) , (18)and j p = ˆ p · j p , we find that the symmetric part of T ic contains the vorticity driven thermal Hall effect q i = 12 Z p ,λ ( j p p i + j ip p ) = ω ij ∂ j β ~ β + O ( ~ ) . (19)Combined with Eq. (17), this agrees with Eq. (16). It canalso be checked that a similar term in the charge current τ = R p ,λ j p vanishes, in accordance with our expectationin the no-drag frame. Equivalence between spin hydrodynamics and non-dissipative second-order hydrodynamics. — It is wellknown that the canonical energy-momentum tensor canbe transformed into the symmetric Belinfante-Rosenfeldenergy-momentum tensor by a specific pseudo-gauge transformation with Σ αµν = S αµν [15–17],˜ T µν = T µνc + 12 ∂ α (Σ αµν − Σ µαν − Σ ναµ )= 12 ( T µνc + T νµc ) − ∂ α ( S µαν + S ναµ ) . (20)As a result, the spin tensor no longer appears in the totalangular momentum tensor, i.e., ˜ S αµν = S αµν − Σ αµν = 0.This leaves the conservation of energy and momentumunchanged, ∂ µ ˜ T µν = ∂ µ T µνc = 0, and the two descrip-tions of the system based on each energy-momentum ten-sor should be equivalent. This suggests that the corre-sponding hydrodynamic descriptions based on the samepremise of local equilibrium, i.e. the spin hydrodynam-ics and the conventional hydrodynamics, should also beequivalent to each other. We will establish this equiva-lence and show that the hydrodynamic variables betweenthe two descriptions are related quite non-trivially. In thefollowing, quantities in the spin hydrodynamics will bedenoted without tilde symbol, while those in the conven-tional hydrodynamics will be written with tilde symbol.A central question in showing the equivalence is howthe first law of thermodynamics used in hydrodynamicstransforms under the pseudo-gauge transformation. Theobservation crucial for answering this question is that wecan generalize the pseudo-gauge transformation to thecurrents of charge and entropy, without affecting theirconservation˜ j µ = j µ − ∂ ν (cid:18) a χ S µν (cid:19) = j µ − ∂ ν ( aω µν ) , (21a)˜ s µ = s µ − ∂ ν (cid:18) b χ S µν (cid:19) = s µ − ∂ ν ( bω µν ) , (21b)with thermodynamic functions a ( ε, n ), b ( ε, n ). An intu-itive understanding of physics of these transformations isobtained by noting that the spatial part of − ∂ ν ( aω µν ) / ∇ × M with vorticity induced magnetization M = − a ω /
2, i.e.the Barnett effect.Since the local charge and entropy densities, ( n, s ),are defined by n = − u µ j µ and s = − u µ s µ respectively,transformations in Eqs. (21) redefine them ˜ n = n − ∆ n ,˜ s = s − ∆ s , where∆ n = − u µ ∂ ν ( aω µν ) = − aω µ ω µ , ∆ s = − bω µ ω µ . (22)Taking ˜ T µν in Eq. (20) obtained from T µνc in the idealspin hydrodynamics in the previous section with S αµν = u α S µν = χu α ω µν , we work out the Landau’s condi-tion for the local energy density and the fluid velocity,˜ T µν ˜ u ν = − ˜ ε ˜ u µ , to obtain ˜ ε and ˜ u µ as ˜ ε = ε − ∆ ε ,˜ u µ = u µ − ∆ u µ with∆ ε = 2 χω µ ω µ , ∆ u µ = − βw ∆ µα ∂ λ ( βχω αλ ) . (23)In addition, we allow a redefinition of pressure ˜ p = p − ∆ p with ∆ p = 2 a ω µ ω µ , where a is a free thermody-namic function. In terms of these variables, the energy-momentum tensor in conventional hydrodynamics reads˜ T µν = ˜ ε ˜ u µ ˜ u ν + ˜ p ˜∆ µν + ˜ τ µν , (24)where ˜ τ µν denotes certain second-order transport terms˜ τ µν = 12 χ (( θ µα + ω µα ) ω αν + ( µ ↔ ν )) + 2 a ∆ µν ω λ ω λ . (25)Similarly, the charge and the entropy currents in the con-ventional hydrodynamics are given by˜ j µ = ˜ n ˜ u µ − n βw ∆ µλ ∂ ν ( βχω λν ) −
12 ∆ µλ ∂ ν ( aω λν ) , (26)˜ s µ = ˜ s ˜ u µ − s ∆ µλ ∂ ν ( βχω λν )2 βw + nχω µν ∂ ν α w − ∆ µλ ∂ ν ( bω λν )2 , (27)with other second-order transport terms. A similar ob-servation was made in Ref.[30]. It should be empha-sized that the ideal limit of spin hydrodynamics with ∂ µ s µ = 0 that we start with guarantees that the con-ventional hydrodynamics with the above second ordertransport terms is also ideal, i.e. ∂ µ ˜ s µ = 0.However, to make the conventional hydrodynamicstruly conventional, the thermodynamics relation of spinhydrodynamics in Eq. (14) should transform into conven-tional thermodynamic relations, d ˜ s = ˜ βd ˜ ε − ˜ αd ˜ n, ˜ s = ˜ β (˜ ε + ˜ p ) − ˜ α ˜ n. (28)We now show that there exists unique choice of ( a, b ) toachieve this equivalence, with ( a, b ) expressed in terms of( χ, a ) without any integrations.We start from the entropy density s in the spin hydro-dynamics as a function of density variables, s ( ε, n, σ ),where S µ ≡ ǫ µναβ u ν S αβ / σ ≡ S µ S µ /
2. Thefirst law of thermodynamics in Eq. (14), ds = βdε − αdn − βγ µν dS µν / βdε − αdn − βγ µ dS µ , then givesus β ≡ ( ∂s/∂ε ) n,σ , α ≡ − ( ∂s/∂n ) ε,σ and βγ µ ≡− ( ∂s/∂σ ) ε,n S µ . In local equilibrium, γ µ = ω µ , and thespin susceptibility is identified as χ ≡ − β ( ∂s/∂σ ) − ε,n from S µ = χω µ .To find the first law of thermodynamics in the con-ventional hydrodynamics, we express ˜ s in terms of thevariables in the conventional hydrodynamics as˜ s (˜ ε, ˜ n, ω µ ω µ ) = s (˜ ε + ∆ ε, ˜ n + ∆ n, σ ) − ∆ s (29)where σ = χ ω µ ω µ /
2, with the same function s and(∆ ε, ∆ n, ∆ s ) given by Eqs. (22) and (23).It is now straightforward to find the first law of ther-modynamics d ˜ s = ˜ βd ˜ ε − ˜ αd ˜ n + Aω µ dω µ (30) with ˜ β = β + ( βχ ε + b ε + αa ε ) ω µ ω µ , (31)˜ α = α − ( βχ n + b n + αa n ) ω µ ω µ , (32) A = 3 βχ + 2 αa + 2 b, (33)where f n ≡ ( ∂f /∂n ) ε and f ε ≡ ( ∂f /∂ε ) n . From s = β ( ε + p ) − αn − βχω µ ω µ in (14), we also find straightfor-wardly ˜ s = ˜ β (˜ ε + ˜ p ) − ˜ α ˜ n + Bω µ ω µ , (34)with B = βχ + αa + b − w ( βχ ε + b ε + αa ε ) − n ( βχ n + b n + αa n ) + 2 βa , (35)The conventional thermodynamics relations in Eq. (28)are obtained by imposing the conditions A = B = 0. Itis easy to see that these conditions determine ( a, b ) interms of ( χ, a ) without any integrations, and we skiptheir explicit expressions.With ( a, b ) given in terms of ( χ, a ), we see thatall second-order transport coefficients in the energy-momentum tensor, Eq. (25), in the charge current,Eq. (26) and in the entropy current, Eq. (27), can beexpressed in terms of two free thermodynamic functions( χ, a ). With the identification of a = χ , one can non-trivially check that these second-order transport coef-ficients agree precisely with those we find in the non-dissipative second-order hydrodynamics in the previoussection once they are also expressed in terms of ( a , a ).The conditions A = 0 and B = 0 correspond to the con-straint C (5) = 0 and a linear combination of C ( i ) = 0,respectively. In the special case of conformal system,condition B = 0 follows from A = 0 and conformality.This completes the proof that the ideal spin hydrody-namics is equivalent to the non-dissipative second-orderhydrodynamics by pseudo-gauge transformation. Conclusion and discussion. — In this Letter, we intro-duce a novel power counting scheme for gradients of hy-drodynamic variables and discover nontrivial constraintson certain non-dissipative second-order transport coef-ficients imposed by the second law of thermodynamics.We also show that the spin hydrodynamics and the con-ventional hydrodynamics with these second-order trans-port coefficients are two equivalent descriptions of thesame system related by pseudo-gauge transformation. Ina more concrete form, one can express the hydrodynamicvariables in one description in terms of those in the otherdescription.Furthermore, one can construct infinitely many equiv-alent spin hydrodynamics descriptions for the same sys-tem by performing pseudo-gauge transformations usingan arbitrary fraction of the spin tensor, i.e., with Σ αµν = tS αµν , where t = 1. This transformation changes the spinsusceptibility χ → (1 − t ) χ ≡ χ ( t ) in thermodynamic re-lations, while a ( t ) + χ ( t ) remains invariant. The othersecond-order transport coefficients are related to a ( t ) byEqs.(10). The conventional hydrodynamics is a specialchoice in this infinite family corresponding to t = 1. Ingeneral, the vorticity driven thermal Hall effect is givenby Eq.(16) with χ → a ( t ) + χ ( t ).What meaning should one then assign to the spin den-sity in a given spin hydrodynamics description? Our con-clusion naturally suggests that the answer to this ques-tion cannot be found within hydrodynamics itself. Forexample, a plasma may contain different microscopic con-stituents carrying their own spins, and it is a matter ofchoice what to include in the hydrodynamic description.All different choices are equivalent and describe the samesystem, while the non-dissipative second-order transportcoefficients corresponding to each choice are related inthe specific way we described.We thank Masaru Hongo, Xu-Guang Huang and En-rico Speranza for discussions and P. Kovtun for bringingRefs.[18–21] to our attention. This work is supported bythe U.S. Department of Energy, Office of Science, Officeof Nuclear Physics, Grant No. DE-FG0201ER41195, andwithin the framework of the Beam Energy Scan Theory(BEST) Topical Collaboration. ∗ Electronic address: [email protected] † Electronic address: [email protected] ‡ Electronic address: [email protected][1] L. Landau and E. Lifshitz,
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