Hydrodynamic fluctuations and long-time tails in a fluid on an anisotropic background
aa r X i v : . [ h e p - t h ] J a n Hydrodynamic fluctuations and long-time tails in a fluid on an anisotropic background
Ashish Shukla ∗ Perimeter Institute for Theoretical Physics, 31 Caroline Street North, Waterloo, Ontario N2L 2Y5, Canada
The effective low-energy late-time description of many body systems near thermal equilibriumprovided by classical hydrodynamics in terms of dissipative transport phenomena receives importantcorrections once the effects of stochastic fluctuations are taken into account. One such physical effectis the occurrence of long-time power law tails in correlation functions of conserved currents. In thehydrodynamic regime k → ω . In this article, we consider a relativistic fluid with a conserved global U (1) chargein the presence of a strong background magnetic field, and compute the long-time tails in correlationfunctions of the stress tensor. The presence of the magnetic field renders the system anisotropic.In the absence of the magnetic field, there are three out-of-equilibrium transport parameters thatarise at the first order in the hydrodynamic derivative expansion, all of which are dissipative. Inthe presence of a background magnetic field, there are ten independent out-of-equilibrium transportparameters at the first order, three of which are non-dissipative and the rest are dissipative. Weprovide the most general linearized equations about a given state of thermal equilibrium involvingthe various transport parameters in the presence of a magnetic field, and use them to compute thelong-time tails for the fluid. I. INTRODUCTION
Hydrodynamics provides an effective description oflow-energy late-time dynamics of systems near thermalequilibrium. At long time scales, the evolution and re-laxation of the system is governed by the dynamics ofthe conserved currents associated to the system, suchas the energy-momentum tensor. In the hydrodynamicregime, the conserved currents can be expressed in anexpansion in number of derivatives acting on the hydro-dynamic variables, with transport parameters enteringas coefficients of various terms in this expansion (see [1–3] for pedagogical reviews). These transport parameterscapture important physical properties of the system un-der consideration, and can either be dissipative i.e. leadto entropy production, or non-dissipative.The equations of hydrodynamics are nothing but theconservation equations for the conserved currents of thesystem. A salient feature of the hydrodynamic equationsis the fact that they are highly nonlinear in nature. Thenonlinear terms give rise to interactions between the fluc-tuating hydrodynamic degrees of freedom. These interac-tions in turn have several important consequences for thedynamics of the system. One of these is the non-analyticdependence on frequency ω of the two-point correlationfunctions of the conserved currents of the system in thelong wavelength limit k →
0. In real time, this trans-lates to power-law dependence of correlation functionson time t at late times, aptly called long-time tails [4, 5],as opposed to an exponential fall-off one would expect ifthere were no interactions. In particular, the value of thepower-law exponent is independent of the microscopicdetails of the theory, and thus important macroscopic in-formation can be extracted from the long-time tails, evenfor strongly coupled systems. ∗ [email protected] One way to think about the occurrence of fluctua-tions is to realize that hydrodynamics is a coarse grainedmacroscopic description of the system. The fluctuationsthat occur in the hydrodynamic degrees of freedom canthus be thought of arising as a consequence of integrat-ing out the microscopic degrees of freedom, which maythemselves have an inherent thermal randomness. Thisapproach to hydrodynamic fluctuations, called stochas-tic hydrodynamics, models the fluctuations to be sourcedby random microscopic currents and stresses, which areGaussian correlated [6–8]. More recently, an effectivefield theory (EFT) approach based on the Schwinger-Keldysh closed time path for non-equilibrium systems hasbeen developed for studying fluctuating hydrodynamics,which takes into account the effects of fluctuations be-yond the assumption of Gaussianity [9–11]. See [12–14]for recent studies of analyticity properties of correlationfunctions and long-time tails in the Schwinger-KeldyshEFT approach.In this paper, we initiate the study of long-time tailsin a relativistic fluid with a global U (1) symmetry, inthe presence of a strong background magnetic field. Onemotivation for computing these tails could be the dynam-ics of the quark-gluon plasma (QGP) produced in heavyion collisions, where the magnetic fields produced in non-central collisions can reach magnitudes of the order 10 Tesla [15–17]. Our analysis will be similar in spirit tothe ones in [5, 18], which in turn assume the fluctuationsto be Gaussian in nature, just as in stochastic hydrody-namics. As pointed out in [18], in the small k regime,the distribution of fluctuations is arbitrarily close to be-ing Gaussian, with non-Gaussian contributions provid-ing only sub-leading corrections to the leading long-timetails. With this in mind, we compute the linearized hy-drodynamic equations about a given state of global ther-mal equilibrium for a relativistic fluid in the presence of abackground magnetic field, and use them to compute thelong-time tails in stress tensor correlation functions andtheir dependence on the magnetic field as well as someof the first-order transport parameters.This article is organized as follows. In section II,we start by reviewing the properties of a charged fluidin thermal equilibrium in the presence of a strongbackground magnetic field. For a parity preservingfluid in four spacetime dimensions, which we shall befocusing on, there is one transport parameter in thermalequilibrium at one-derivative order, the magneto-vorticalsusceptibility M Ω . Out of equilibrium, there are tenmore [19]. These are the three non-equilibrium non-dissipative parameters: the two Hall viscosities ˜ η ⊥ , ˜ η k ,and one Hall conductivity ˜ σ , and seven dissipativeparameters: the two electrical conductivities σ ⊥ , σ k , thetwo shear viscosities η ⊥ , η k , and four bulk viscosities ζ , ζ , η , η , of which only three are independent due tothe Onsager relation 3 ζ − η − η = 0. We review theconstruction of the equilibrium generating functional upto first order in derivatives, and vary it with respect tothe sources to obtain the associated energy-momentumtensor and the conserved current. In section III, wego beyond the assumption of thermal equilibrium, andinclude fluctuations about an equilibrium configuration.We write down the constitutive relations includingnon-equilibrium corrections, highlighting the role ofvarious transport parameters that appear at this order.Subsequently, in section IV, we construct the linearizedhydrodynamic equations, and compute the retardedtwo-point correlation functions of the hydrodynamicdegrees of freedom about a state of equilibrium, makinguse of linear response theory. Finally, in section V, weinclude nonlinear terms in the hydrodynamic constitu-tive relations, and compute the long-time tails in thecorrelation functions of the energy-momentum tensor.This requires us to make use of the symmetric correlationfunctions that follow from linear response theory. Thesecan be obtained from the retarded correlation functionsusing the fluctuation-dissipation relation. We illustratethe dependence of the long-time tails on two of themany transport parameters that occur for the fluid onthe anisotropic background: the shear viscosity η k andthe bulk viscosity ζ . The paper ends with a discussionin section VI. Notation and conventions. —The metric has signature( − +++). Spacetime indices are denoted by Greek letters µ, ν, . . . = (0 , , , i, j, . . . = (1 , , k , x etc. Scalar product of spatial vectorsis denoted in boldface, e.g. k · x , whereas the Lorentzinvariant inner product of four-vectors is denoted withoutboldface, e.g. A · B ≡ η µν A µ B ν . The Levi-Civita tensordensity is given by ǫ µνρσ = ε µνρσ √− g , ǫ µνρσ = √− g ε µνρσ , with ε µνρσ being the ordinary Levi-Civita symbol, with ε = 1. An important identity we make use of is the contraction identity, ǫ µ ...µ p ν ...ν m ǫ µ ...µ p λ ...λ m = ( − n p ! δ ν ...ν m λ ...λ m , where n is the number of minus signs in the metric sig-nature ( n = 1 for us), and p is the number of contractedindices. δ ν ...ν m λ ...λ m is the generalized Kronecker delta func-tion, which can be expressed as a determinant via δ ν ...ν m λ ...λ m = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) δ ν λ δ ν λ · · · δ ν λ m δ ν λ δ ν λ · · · δ ν λ m ... ... . . . ... δ ν m λ δ ν m λ · · · δ ν m λ m (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . II. THERMAL EQUILIBRIUM IN A STRONGMAGNETIC FIELD
When a system in thermal equilibrium is perturbed, itrelaxes back to equilibrium via dynamical processes thatare generically governed by two types of variables: slowand fast. The slow variables are typically associated withconserved quantities of the system, that can relax backonly by physical transport, and thus govern the dynam-ics of the perturbed system over long time scales. Thefast variables, on the other hand, are associated withnon-conserved quantities, and can relax back to the newequilibrium state very quickly. Hydrodynamics capturesthe long-time low-energy behaviour of conserved quanti-ties of the system, as the system relaxes back to thermalequilibrium after being perturbed.Before jumping into hydrodynamics, we would like tobetter understand the state of thermal equilibrium itself.Mathematically, thermal equilibrium is quantified via thepresence of a timelike vector V µ , which in the fluid restframe takes the form V µ rest = (1 , ). The intuitive factthat things in equilibrium do not change with time is im-posed by the condition that the Lie derivative of the hy-drodynamic variables as well as the sources with respectto V µ vanishes. In fact, in thermal equilibrium, the hy-drodynamic variables, which are the fluid four-velocity u µ , temperature T , and chemical potential µ , can be de-fined in terms of the vector V µ and the sources as [20, 21] T = T √− V , u µ = V µ √− V , µ = V µ A µ + Λ V √− V , (1)where V ≡ g µν V µ V ν is the squared norm of V µ . Themetric g µν and the vector field A µ are external sources,which respectively source the energy-momentum tensorand the conserved U (1) current for the system. The nor-malization of the temperature is set by T , and Λ V en-sures that the chemical potential is gauge invariant [22].The fluid velocity is normalized such that u µ u µ = − thermodynamic frame .We will be interested in a charged fluid with a global U (1) symmetry kept in the presence of a strong back-ground magnetic field. By the background magnetic fieldbeing strong, we imply that various susceptibilities andtransport parameters of the fluid will depend not justupon the temperature and the chemical potential, butalso on the strength of the magnetic field, as discussedfurther below. Given the fluid four-velocity u µ , one canexpress the antisymmetric field strength tensor F µν as F µν = u µ E ν − u ν E µ − ǫ µνρσ u ρ B σ , (2)where the electric and magnetic fields are defined via E µ ≡ F µν u ν , B µ = ǫ µνρσ u ν F ρσ . Note that both theelectric and magnetic fields are orthogonal to the fluidvelocity, u · E = u · B = 0. Electric and magnetic fields arenot independent, but linked to each other via the Bianchiidentity ǫ µνρσ ∇ ν F ρσ = 0, which in thermal equilibriumbecomes [19] ∇ · B = B · a − E · Ω , (3a) u µ ǫ µνρσ ∇ ρ E σ = u µ ǫ µνρσ E ρ a σ , (3b)where a µ = u ν ∇ ν u µ is the acceleration and Ω µ = ǫ µνρσ u ν ∇ ρ u σ is the vorticity of the fluid. Note that eqs.(3) are the curved space analogues of the flat space equi-librium identities ∇ · B = 0 , ∇ × E = 0.In thermal equilibrium, the correlation functions ofconserved currents can be obtained from an equilibriumgenerating functional W [ g µν , A µ ], which is a functional ofthe external sources [20, 21]. The variation of W withrespect to the sources gives δ W = 12 Z d x √− g T µν δg µν + Z d x √− g J µ δA µ , (4)which implies that the one-point functions of the con-served currents are T µν = 2 √− g δ W δg µν , J µ = 1 √− g δ W δA µ . (5)Higher point correlation functions in equilibrium can beobtained by taking further derivatives with respect to thesources. The conservation equations ∇ µ T µν = F νρ J ρ , (6a) ∇ µ J µ = 0 , (6b)follow from the diffeomorphism and gauge invariance ofthe generating functional. The generating functional canbe expressed as an integral over a local free energy density F via W [ g µν , A µ ] = Z d x √− g F ( g µν , A µ ) . (7) As an aside, the equilibrium generating functional can be pro-moted to be the equilibrium effective action, and can be usedto derive effective Einstein’s equations incorporating the effectsof derivative corrections beyond the perfect fluid approximation[23]. See [24] for a similar analysis for electromagnetism.
When the sources vary on length scales much longer thanthe inherent microscopic scales in the system, such as thecorrelation length, the density F admits an expansion interms of the derivatives of the hydrodynamic variablesand the sources [20, 21]. At zeroth order in the derivativeexpansion, for instance, we have F (0) = p ( T, µ, B ) , (8)where p is the equilibrium pressure.An important point here is the choice of the deriva-tive counting scheme. The metric is naturally countedas O (1). For conducting fluids in the presence ofstrong magnetic fields, the appropriate derivative count-ing scheme is B µ ∼ O (1). Electric field in equilibriumsatisfies the condition ∂ λ µ = E λ − µa λ , and hence it isappropriate to consider E µ ∼ O ( ∂ ), which is another wayto say that the electric field is screened in charged flu-ids. Since the magnetic field is O (1), the thermodynamicsusceptibilities and transport parameters of the fluid willdepend upon its strength as well. For instance, the equa-tion of state in eq. (8) has an explicit dependence on B . A. The equilibrium constitutive relations
The energy-momentum tensor for the fluid can be de-composed into components along and orthogonal to thefluid four-velocity u µ via T µν = E u µ u ν + P ∆ µν + Q µ u ν + Q ν u µ + T µν , (9)where ∆ µν = g µν + u µ u ν is the projector orthogonal tothe fluid velocity, the heat current Q µ is orthogonal tothe fluid velocity, and T µν is symmetric, traceless andtransverse to the fluid velocity. Given T µν one can ex-tract the coefficients via E = u µ u ν T µν , P = 13 ∆ µν T µν , Q µ = − ∆ µν u ρ T νρ T µν = 12 (cid:18) ∆ µα ∆ νβ + ∆ µβ ∆ να −
23 ∆ µν ∆ αβ (cid:19) T αβ . (10)In the hydrodynamic regime, the quantities E , P , Q µ , T µν can be expanded in a series in the number of derivativesacting on the hydrodynamic variables and the sources,which are called the constitutive relations for the fluid.Just like the energy-momentum tensor, the conserved U (1) current can also be decomposed as J µ = N u µ + J µ , (11)where J µ is transverse to the fluid velocity. Given thecurrent J µ , one can extract the coefficients in eq. (11)via N = − u µ J µ , J µ = ∆ µν J ν . (12)
1. The zeroth order
As an illustration, let us compute the constitutive re-lations for the fluid at the zeroth order in the derivativeexpansion [19]. The equilibrium generating functional atthe zeroth order in derivatives is given by W [ g µν , A µ ] = Z d x √− g p ( T, µ, B ) . (13)Varying it with respect to the sources g µν and A µ givesthe zeroth order constitutive relations, E = − p + T ∂p∂T + µ ∂p∂µ ≡ ǫ ( T, µ, B ) , (14a) P = p − α BB B , (14b) N = ∂p∂µ ≡ n ( T, µ, B ) , (14c) T µν = α BB (cid:18) B µ B ν −
13 ∆ µν B (cid:19) , (14d)where ǫ ( T, µ, B ) is the equilibrium energy density and n ( T, µ, B ) is the equilibrium charge density. Also, α BB ( T, µ, B ) = 2 ∂p/∂B denotes the magnetic suscep-tibility. At the zeroth order, one has Q µ = J µ = 0.
2. The first order
For a parity violating fluid, at the first order in deriva-tives in thermal equilibrium, there are five independentscalars that can contribute to the generating functional W [ g µν , A µ ], which consequently takes the form [19] W = Z d x √− g p ( T, µ, B ) + X i =1 M i ( T, µ, B ) S i ! , (15)where the one-derivative scalars S i are S = B µ ∂ µ (cid:18) B T (cid:19) , S = ǫ µνρσ u µ B ν ∇ ρ B σ ,S = B · a, S = B · Ω , S = B · E. (16)Out of these, only S is parity preserving, and the restare parity violating. Also, M i ( T, µ, B ) are thermody-namic susceptibilities and need to be determined fromthe microscopic theory, just like the pressure p .Our focus in the present work is on parity preservingfluids in 3+1D. Thus, for our purposes, the equilibrium Note that when the magnetic field is weak, B µ ∼ O ( ∂ ), deriva-tive contributions to the equilibrium generating functional beginat the second order [25]. See also [26] for the behaviour of secondorder susceptibilities for free quantum field theories as a functionof µ, T . See [27] for computations in 3+1D where parity violation is con-sidered. Additionally, see [28–30] for the effects of strong back-ground magnetic fields in 2+1D. generating functional up to first order in derivatives isgiven by (with the susceptibility M denoted more sug-gestively by M Ω ) W = Z d x √− g (cid:2) p ( T, µ, B ) + M Ω B · Ω (cid:3) . (17)Varying this with respect to the metric g µν gives us thefollowing first order constitutive relations for the energy-momentum tensor eq. (9), E = ǫ + ( T M Ω ,T + µM Ω ,µ − M Ω ) B · Ω , (18a) P = p − α BB B − (cid:0) M Ω + 4 B M Ω ,B (cid:1) B · Ω , (18b) Q µ = M Ω ǫ µνρσ u ν ∂ ρ B σ − M Ω ,B ǫ µνρσ u ν B ρ ∂ σ B + (2 M Ω − T M Ω ,T − µM Ω ,µ ) ǫ µνρσ B ν u ρ a σ + (cid:0) M Ω ,µ − M Ω ,B B · Ω − α BB (cid:1) ǫ µνρσ u ν E ρ B σ + M Ω ǫ µνρσ Ω ν E ρ u σ , (18c) T µν = α BB (cid:18) B µ B ν −
13 ∆ µν B (cid:19) + 2 M Ω ,B B · Ω B h µ B ν i + 2 M Ω B h µ Ω ν i , (18d)with α BB = 2 ∂p/∂B as defined earlier. A comma inthe subscript denotes a partial derivative with respect tothe argument that follows. In deriving the constitutiverelations in eqs. (14) and (18), we made use of the fol-lowing intermediate results for the variation of variousquantities that enter the generating functional under avariation of the metric. These are δT = T u µ u ν δg µν , (19a) δµ = µ u µ u ν δg µν , (19b) δB µ = − B µ ∆ αβ δg αβ + u µ u α B β δg αβ + ǫ µαρσ u β u ρ E σ δg αβ , (19c) δB = B µ B ν δg µν − B ∆ µν δg µν − ǫ µνρσ u ν E ρ B σ u α δg µα , (19d) δ Ω µ = Ω µ (cid:18) u α u β − g αβ (cid:19) δg αβ + ǫ µνρσ u α ∇ ρ u σ δg να + ǫ µνρσ u ν ∂ ρ u α δg ασ + ǫ µνρσ u ν u α ∂ ρ δg ασ (19e) δ ( B · Ω) = B µ Ω ν δg µν + (cid:18) u µ u ν − g µν (cid:19) B · Ω δg µν − ǫ µαβρ B α ∇ β u ρ u ν δg µν − ǫ µαβρ B α u β ∂ ρ u ν δg µν − ǫ µαβρ B α u β u ν ∂ ρ δg µν . (19f) The terms in the last line of eq. (18d) have extra factors of 2 whencompared with the results in [19]. This is so because the authorsof [19] use two different definitions for the projection A h µ B ν i ,whereas we uniformly use the definition given in eq. (20). Also, the angular bracket notation used above helps indefining the symmetric transverse traceless part of a ten-sor, given by A h µ B ν i ≡ (cid:18) ∆ µα ∆ νβ + ∆ µβ ∆ να −
23 ∆ µν ∆ αβ (cid:19) A α B β . (20)For computing the constitutive relations for the U (1)current, it is convenient to regard the generating func-tional as a functional of A µ and its derivative F µν . Thevariation of the generating functional is then defined as[24, 25] δ A,F W = Z d x √− g (cid:18) J µf δA µ + 12 M µν δF µν (cid:19) , (21)where J µf is the current of free charges and M µν is theantisymmetric polarization tensor. In terms of these thetotal current can be written as J µ = J µf − ∇ λ M λµ . (22)The ambiguity in defining the free charge current is fixedby choosing J µf = ρu µ , with ρ = ∂ F /∂µ being the freecharge density. Comparing eqs. (11), (12) with eq. (22)above, we find the constitutive relations for the U (1) cur-rent to be N = ρ − ∇ · p + p · a − m · Ω , (23a) J µ = ǫ µνρσ u ν ( ∇ ρ + a ρ ) m σ , (23b)where we have made use of the electric and magneticpolarization vectors, denoted respectively by p µ and m µ ,in writing the constitutive relations eqs. (23), with p µ = u ν M νµ , m µ = 12 ǫ µνρσ u ν M ρσ . (24)For the first order generating functional eq. (17) wefind the polarization vectors p µ = 0 , (25a) m µ = (cid:0) α BB + 2 M Ω ,B B · Ω (cid:1) B µ + M Ω Ω µ , (25b)which lead to the constitutive relations N = n + ∂M Ω ∂µ B · Ω − m · Ω , (26a) J µ = ǫ µνρσ u ν ( ∇ ρ + a ρ ) m σ . (26b) III. BEYOND THERMAL EQUILIBRIUM
Let us now move on and introduce deviations from thestate of thermal equilibrium introduced in section II. Acomprehensive analysis of the out-of-equilibrium consti-tutive relations in the presence of a strong magnetic fieldappears in [19], which we shall make use of. It is im-portant to notice that hydrodynamic variables such as the temperature, chemical potential or the fluid velocityhave no a priori definition once one considers a state outof equilibrium. In other words, hydrodynamic variablesare well defined only for a system in equilibrium. Thefreedom to choose a definition for hydrodynamic vari-ables out of equilibrium, called a choice of hydrodynamicframe, can be used to fix some of the constitutive rela-tions to a simple form. Following [19], we choose to workin the thermodynamic frame. In this frame, the freedomto choose the temperature T and the chemical potential µ out of equilibrium can be used to fix E and N to theirequilibrium values, given by eqs. (18a) and (26a). Also,the freedom to redefine the fluid velocity can be used tofix Q µ to its equilibrium value (18c). The other threequantities P , J µ and T µν in the constitutive relationsreceive derivative corrections beyond their equilibriumvalues. The thermodynamic frame constitutive relationsfor P , J µ and T µν with first-order derivative correctionsout of equilibrium are given by P = ¯ P − ζ ∇ · u − ζ b µ b ν ∇ µ u ν , (27a) T µν = ¯ T µν − η ⊥ σ µν ⊥ − η k ( b µ Σ ν + b ν Σ µ ) − b h µ b ν i ( η ∇ · u + η b α b β ∇ α u β ) − ˜ η ⊥ ˜ σ µν ⊥ − ˜ η k ( b µ ˜Σ ν + b ν ˜Σ µ ) , (27b) J µ = ¯ J µ + (cid:18) σ ⊥ B µν + σ k B µ B ν B (cid:19) V ν + ˜ σ ˜ V µ . (27c)Here ¯ P , ¯ J µ , ¯ T µν are the equilibrium values, given in eqs.(18b), (26b) and (18d), respectively. Also, we have usedthe notation b µ ≡ B µ / √ B , and B µν ≡ ∆ µν − B µ B ν B is the projector orthogonal to both u µ and B µ . Also,we use the shorthand notation V µ ≡ E µ − T ∆ µν ∂ ν (cid:0) µT (cid:1) ,and ˜ V µ = ǫ µναβ u ν b α V β . A decomposition similar to eq.(27) for the first-order out-of-equilibrium constitutive re-lations in the Landau frame appears in [31].The shear tensor σ µν , given by σ µν = ∆ µα ∆ νβ (cid:18) ∇ α u β + ∇ β u α − g αβ ∇ · u (cid:19) , has been decomposed as σ µν = σ µν ⊥ +( b µ Σ ν + b ν Σ µ )+ 12 b h µ b ν i (3 b α b β ∇ α u β −∇· u ) . (28)Here σ µν ⊥ is transverse to both u µ and B µ , σ µν ⊥ = 12 (cid:0) B µα B νβ + B µβ B να − B µν B αβ (cid:1) σ αβ . Similarly, Σ µ ≡ B µα σ αβ b β is also transverse to both u µ and B µ .The tensor ˜ σ µν is another rank-two one-derivative ten-sor that enters the constitutive relations above. It isgiven by˜ σ µν = 12 √ B (cid:16) ǫ µραβ u ρ B α σ νβ + ǫ νραβ u ρ B α σ µβ (cid:17) . (29)Inserting the decomposition eq. (28) for σ µν into eq. (29),one gets ˜ σ µν = ˜ σ µν ⊥ + (cid:16) b µ ˜Σ ν + b ν ˜Σ µ (cid:17) , (30)where˜ σ µν ⊥ = 12 (cid:0) B µα B νβ + B µβ B να − B µν B αβ (cid:1) ˜ σ αβ , and ˜Σ µ = B µα ˜ σ αβ b β .The first-order out-of-equilibrium transport parame-ters appearing in the constitutive relations eq. (27) arenot all dissipative. In general, every dissipative trans-port parameter is necessarily an out-of-equilibrium con-tribution, but every out-of-equilibrium parameter is notnecessarily dissipative. The eleven transport parametersappearing in eq. (27) can be classified as follows. Parameter Name Nature η ⊥ Transverse shear viscosity Dissipative η k Longitudinal shear viscosity Dissipative σ ⊥ Transverse conductivity Dissipative σ k Longitudinal conductivity Dissipative ζ , ζ , η , η Bulk viscosities Dissipative˜ η ⊥ Transverse Hall viscosity Non-diss.˜ η k Longitudinal Hall viscosity Non-diss.˜ σ Hall conductivity Non-diss.An important point to note is that not all of the fourbulk viscosities are independent. The time reversal in-variance of the underlying microscopic theory imposesan Onsager relation on the bulk viscosities above, andgives the constraint3 ζ − η − η = 0 . (31)Thus, the number of independent out-of-equilibriumtransport parameters at one-derivative order is ten. Also,if one assumes the fluid to be conformal, then the bulkviscosities ζ = ζ = 0. IV. LINEARIZED HYDRODYNAMICS
We now move on to study the behaviour of small fluc-tuations and their correlation functions about a state ofthermal equilibrium, which is the main thrust of this pa-per. The equilibrium state we consider coincides withthe fluid at rest u µ = (1 , ), at a constant global tem-perature T and chemical potential µ . The equilibriumenergy density is ǫ , pressure p and charge density n .The background magnetic field can be taken to pointalong the + Z direction without any loss of generality,so B µ = (0 , , , B ). All other background sources areturned off. We work with fluctuations in the energy den-sity δǫ , charge density δn , and momentum density π i . These are defined via T = ǫ + δǫ + . . . ,T i = π i + . . . ,J = n + δn + . . . , (32)where the . . . in the expressions above denote terms whichdepend upon the background magnetic field B . Notethat the momentum density fluctuations are related tothe velocity fluctuations via π i = ( ǫ + p ) δu i = w δu i ,where w = ǫ + p is the equilibrium enthalpy density.Using the thermodynamic frame constitutive relationsfor the energy-momentum tensor and the conserved cur-rent discussed in sections II and III, one can write downthe hydrodynamic equations (6) to linear order in thefluctuations δn, δǫ, π i about the chosen state of equilib-rium. The full set of linearized equations about the equi-librium state of interest are presented in appendix A.The primary reason to work out the linearized fluctua-tion equations is to be able to make use of linear responsetheory, which would allow us to obtain retarded correla-tion functions of the fluctuations about the given state ofequilibrium. As per the linear response approach, for aset of hydrodynamic variables Φ A ( t, x ) satisfying coupledlinear equations in momentum space of the form ∂ t Φ A ( t, k ) + M AB ( k ) Φ B ( t, k ) = 0 , (33)the retarded two-point correlation functions are given by G Ret ( ω, k ) = − (cid:0) + iω K − (cid:1) χ , (34)where the matrix K = − iω + M ( k ). Also, if the sourcesfor the hydrodynamic fluctuations Φ A are φ A , then thesusceptibility matrix χ is given by χ AB = ∂ Φ A ∂φ B . (35)The matrix χ has been computed in appendix B. Linearresponse theory is thus good enough if one is interestedonly in the causal linear response of the system to exter-nal sources. However, the richness of hydrodynamics iscontained in the nonlinear nature of its equations, and weexplore some of their consequences in section V below.Making use of the linear response approach outlinedabove, in principle one can work with the entire set oftransport parameters that appear at the first-order andcompute the linear response of the system following fromthe hydrodynamic equations in appendix A. However, The retarded and symmetric correlation functions are defined via G Ret O O ( t − t , x − x ) ≡ − i Θ( t − t ) h [ O ( t , x ) , O ( t , x )] i , G Sym O O ( t − t , x − x ) ≡ h{O ( t , x ) , O ( t , x ) }i , where Θ( t ) is the step function, and square and curly bracesrespectively denote a commutator and an anticommutator. this is quite cumbersome and not analytically tractable.A simpler strategy is to compute the linear response whenone of the first-order transport parameters dominatesover the rest. In such a scenario, one can assume that itis only this transport parameter that is non-zero, whilethe rest are set to vanish. This approximation of deal-ing with a single transport parameter and its effects onthe system at a given time renders the setup analyticallytractable. Needless to say, one can work with all or anyother suitable subset of non-zero transport parametersdepending upon the situation of interest. However, thiswill inevitably require the use of numerical approachesto the problem, which we leave out for a future study.In section V below, we will illustrate the effect of non-linearities and the emergence of long-time tails for thetwo cases when the dissipative viscosities η k and ζ arethe ones that dominate the transport and relaxation pro-cess. We make this choice for the relative simplicity of theequations and loop-computations that follow. Further, itis useful to assume that we are working with a fluid whichis perturbed around a neutral state i.e. n = 0. With thisin mind, we now move on to compute the linear responseand the associated retarded correlation functions for thetwo cases of interest. A. Linear response when only η k = 0 We first consider the case when the dominant trans-port parameter is the shear viscosity η k , and the othertransport parameters are negligibly small. Since we areinterested in the shear modes, an additional simplifica-tion one can impose is the assumption of incompressibil-ity, ∂ i π i ≈
0, which decouples the sound modes of thesystem and further simplifies the coupled equations inappendix A. In fact, with these simplifications put in,the energy density and charge density fluctuations de-couple from the velocity fluctuations, leaving behind amuch simpler system of equations to be analyzed for theshear modes. Linear response theory then leads to thefollowing set of retarded two-point correlators. G Ret π x π x ( ω, k ) = η k k k h iω − η k w ( k − k x ) i F ( ω, k k ) F ( ω, k ) , (36a) G Ret π y π y ( ω, k ) = η k k k h iω − η k w ( k − k y ) i F ( ω, k k ) F ( ω, k ) , (36b) G Ret π z π z ( ω, k ) = η k k ⊥ F ( ω, k ) , (36c) G Ret π x π y ( ω, k ) = η k k k k x k y w F ( ω, k k ) F ( ω, k ) , (36d) G Ret π x π z ( ω, k ) = η k k x k z F ( ω, k ) , (36e) G Ret π y π z ( ω, k ) = η k k y k z F ( ω, k ) , (36f) where F ( ω, k k ) = iω − η k w k k ,F ( ω, k ) = iω − η k w k . Note that since the magnetic field points along the + Z direction, in the above we have made use of the intuitivenotation k k ≡ k z , k ⊥ = q k x + k y , k ≡ | k | . B. Linear response when only ζ = 0 This is the second of the cases we will consider, whenthe bulk viscosity ζ is the dominant transport param-eter, while the rest are negligibly smaller. The set ofnon-vanishing retarded two-point functions for this casethat arise from linear response theory are as follows. G ret ǫǫ ( ω, k ) = w k ( ω − ω + )( ω − ω − ) , (38a) G ret ǫπ i ( ω, k ) = w ω k i ( ω − ω + )( ω − ω − ) , (38b) G ret π x π x ( ω, k ) = k x (cid:0) w v s − iωζ (cid:1) ( ω − ω + )( ω − ω − ) , (38c) G ret π y π y ( ω, k ) = k y (cid:0) w v s − iωζ (cid:1) ( ω − ω + )( ω − ω − ) , (38d) G ret π z π z ( ω, k ) = k z (cid:0) w v s − iωζ (cid:1) ( ω − ω + )( ω − ω − ) , (38e) G ret π x π y ( ω, k ) = k x k y (cid:0) w v s − iωζ (cid:1) ( ω − ω + )( ω − ω − ) , (38f) G ret π x π z ( ω, k ) = k x k z (cid:0) w v s − iωζ (cid:1) ( ω − ω + )( ω − ω − ) , (38g) G ret π y π z ( ω, k ) = k y k z (cid:0) w v s − iωζ (cid:1) ( ω − ω + )( ω − ω − ) , (38h)where ω ± are the two sound modes, given by ω ± = ± v s k − i ζ w k . Note that v s is the speed of sound, which in the neutralequilibrium state of interest is given by v s = ( ∂p/∂ǫ ). V. INTERACTIONS AND LONG-TIME TAILS
With the retarded correlation functions in hand, wenow move on to compute the long-time tails, and their de-pendence on the viscosities η k , ζ . As we have mentionedearlier, the long-time tails are a consequence of the non-linear terms in the hydrodynamic constitutive relations,which correspond to interactions between the fluctuat-ing degrees of freedom. The leading long-time tails willcome from the leading nonlinear terms in the energy-momentum tensor eq. (18), which have the form T ij = δ ij (cid:18) Υ δǫ + Υ δn − v s w π + 12 α BB B w π z (cid:19) + (cid:0) w − α BB B (cid:1) π i π j w + ( α BB − M Ω ,µ ) B w × (39) (cid:2) π x ( δ ix π j + δ jx π i ) + π y ( δ iy π j + δ jy π i ) (cid:3) + · · · where Υ = (cid:16) ∂ p∂ǫ (cid:17) n , Υ = (cid:16) ∂ p∂n (cid:17) ǫ . We have droppednonlinear terms involving any derivatives, denoted by the · · · , as their contribution in the hydrodynamic limit ω → , k → J . For concreteness, let us as-sume that the constitutive relation has the generic form J quad ∼ Φ Φ , where Φ and Φ are two fluctuatinghydrodynamic degrees of freedom. Then the symmetrictwo-point correlation function of J receives a nontrivialcontribution from the quadratic fluctuation terms, givenby G int J J ( t, x ) ∼ h Φ ( t, x )Φ ( t, x )Φ (0)Φ (0) i . (40)Now assuming that the small fluctuations about the stateof equilibrium are Gaussian in nature, the four-point cor-relation function above factorizes into a product of twotwo-point correlation functions, and is given byG int J J ( ω, k ) ∼ Z dω ′ π d k ′ (2 π ) G symΦ Φ ( ω ′ , k ′ ) G symΦ Φ ( ω − ω ′ , k − k ′ ) , (41)where G sym O O is the symmetric two-point correlationfunction that follows from linear response theory. Thesecan be computed from the retarded two-point correlatorsof section IV by using the fluctuation-dissipation relation, G Sym O O ( ω, k ) = − T ω Im G Ret O O ( ω, k ) . (42)Now the integral in eq. (41) is essentially a one-loop in-tegral, as depicted in the diagram below. J J Φ Φ Φ Φ ( ω ′ , k ′ )( ω − ω ′ , k − k ′ ) It is the evaluation of this one-loop integral that resultsinto the non-analytic long-time tails in the
J J two-pointfunction, as we will explore now for the stress tensor cor-relation functions in our system.
A. Long-time tails in stress tensor correlators
With the retarded correlation functions known for thetwo cases η k = 0 and ζ = 0, it is straight forward tocompute the long-time tails in stress tensor two-pointfunctions using the general methodology outlined above.Consider first the case of η k = 0. As a consequence of thenonlinear interaction terms in eq. (39), we get the follow-ing contribution to the symmetric two-point function ofthe T xz component of the stress tensor, eq. (40),G int T xz T xz ( t, x ) = κ h π x ( t, x ) π z ( t, x ) π x (0) π z (0) i , (43)where κ = 1 w (cid:0) w − M Ω ,µ B (cid:1) . (44)The assumption of Gaussianity for the fluctuations thenleads toG int T xz T xz ( ω, k )= κ Z dω ′ π d k ′ (2 π ) G Sym π x π x ( ω ′ , k ′ ) G Sym π z π z ( ω − ω ′ , k − k ′ ) . (45)Putting in the symmetric correlators in eq. (45) that fol-low from the retarded two-point functions eqs. (36a) and(36c) by the use of the fluctuation-dissipation relation eq.(42), and performing the integrals leads toG int T xz T xz ( ω, k ) = κ T w / πη / k Coth − √ √ − ! p | ω | + · · · (46)The appearance of the non-analytic term proportional to p | ω | is the long-time tail we were interested in findingout. If we Fourier transform the above expression toreal time, then we indeed obtain a long-time power-lawtail in the two-point function, proportional to | t | − / .This indicates that after sufficiently long time the two-point correlation function does not die off exponentiallyrapidly, as would be the case if there were no interactionspresent, but rather falls off as a power-law.Similarly, for the case when only ζ = 0, the samestress tensor two-point function receives the non-analyticcontributionG int T xz T xz ( ω, k ) = κ T w / πζ / p | ω | + · · · (47)which is the long-time tail we were after. One can sim-ilarly compute the tails in correlation functions of othercomponents of the stress tensor, and verify that the non-analytic structure present in eqs. (46) and (47) persists.Note that the leading corrections in the presence of themagnetic field are proportional to M Ω ,µ B for the stresstensor correlators considered above. For other correla-tors, the leading dependence on the magnetic field willsimilarly follow from eq. (39), as well as from the termsin the linearized equations in appendix A.Another important point to stress is that even thoughwe set up our calculations in the thermodynamic frame,the final results for the long-time tails are independent ofthis choice. It is so because physically measurable quan-tities such as the stress tensor are frame invariant ob-jects, and therefore their correlation functions are frameinvariant too.The non-analytic dependence on ω computed in eqs.(46) and (47) above is for the symmetric correlationfunctions of stress tensor components. Following thefluctuation-dissipation relation eq. (42), this implies thatthe retarded correlation functions of the stress tensor willhave a non-analytic dependence which behaves as | ω | / .It is the retarded correlation functions which capture thecausal response of a system to perturbations, and canbe computed by varying the hydrodynamic constitutiverelations for the conserved currents with respect to theexternal sources. Now, in a linearized analysis of fluctu-ations about a given equilibrium state, one would con-clude that the retarded correlators of conserved currentsin the limit k → will have a leading behaviour pro-portional to ω , coming from one-derivative terms. Thiswould be followed by a subleading term proportional to ω that comes from two-derivative terms in the hydrody-namic derivative expansion. However, as we saw above,the presence of nonlinear interaction terms actually con-tributes a term proportional to | ω | / to the retardedcorrelator, which is more significant than the terms thatcome from the second order in the hydrodynamic deriva-tive expansion. We are thus lead to include the effectsof nonlinear interaction terms before considering higherderivative terms in the derivative expansion.Note that the one-loop integral present in thefrequency-momentum integrals of eq. (41) is in generalUV divergent. We have regulated this divergence by in-troducing a UV cutoff Λ in our computations. Thesecutoff dependent terms have been suppressed in eqs. (46)and (47), to focus on the cutoff independent long-timetails in the correlation functions. The cutoff dependencecan be absorbed in the renormalization of the associatedtransport parameters. VI. DISCUSSION
In this paper, we studied fluctuations in a relativisticfluid with a conserved global U (1) current in the pres-ence of a background magnetic field, and wrote down themost general linearized hydrodynamic equations for thefluctuations in energy density, charge density and mo- mentum density about a state of global thermal equilib-rium (see appendix A). The background magnetic fieldrenders the setup anisotropic, which results into severalnew transport parameters. If there were no anisotropy,there would be three dissipative transport parameters atthe first-order in the derivative expansion: the shear vis-cosity η , the bulk viscosity ζ , and the conductivity σ .However, the anisotropic background leads to seven dis-sipative transport parameters in place of three. Theseare the two shear viscosities η k , η ⊥ , four bulk viscosities ζ , ζ , η , η , and two conductivities σ k , σ ⊥ . Only three ofthe four bulk viscosities are independent due to the On-sager relation that provides a linear condition betweenthree of them. In addition, there are three first-ordernon-dissipative transport parameters: the Hall conduc-tivity ˜ σ , and the Hall viscosities ˜ η k , ˜ η ⊥ .With the full set of linearized equations in hand, wewere able to focus on different subsectors of transport inthe system, in each of which one of the transport param-eters dominates over the rest. We in particular focusedon two cases, when the shear viscosity η k = 0 and whenthe bulk viscosity ζ = 0. In these two subsectors, wecomputed the retarded two-point correlation functionsof the hydrodynamic fluctuations under suitable simpli-fying assumptions. Further, with the retarded correlatorsin hand, we went on to compute the effects of nonlinearinteraction terms in the hydrodynamic constitutive re-lations on the correlation functions of the stress tensorfor the fluid. We found the appearance of non-analyticlong-time tails in these correlation functions, which leadto power-law fall off for the stress tensor correlation func-tions after long times, as opposed to an exponential de-cay.Several comments are in order. To maintain analyticalcontrol we focused on transport in sectors where one ofthe transport parameters is non-zero, and the rest canbe set to vanish. For the purpose of applications toQGP and other physical situations such as dense quarkmatter in magnetic fields, it would be useful to considertransport when several of the transport parameters arenon-zero. This would require a sophisticated numeri-cal approach, which we leave out for a future work. Inparticular, with applications to QGP in mind, a logi-cal next step is to develop the formalism and computethe long-time non-analytic behaviour when the magneticfield is dynamical i.e. to make use of the full formalism ofrelativistic magnetohydrodynamics [19], which can alsobe done in the dual formulation in terms of generalizedglobal symmetries [32]. Another possible future directionwould be to study the long-time tails in the presence ofa magnetic field about a state where the fluid is under-going longitudinal boost invariant expansion, known asBjorken flow [33]. Finally, it would also be interesting tounderstand the non-analytic long-time tails in the pres-ence of a magnetic field from a holographic perspective[34]. We hope to report on some of these directions inthe near future.0 Appendix A: The full set of linearized equations
In this appendix, we present the complete set of lin-earized hydrodynamic equations about the equilibriumstate with temperature T , chemical potential µ , energydensity ǫ , pressure p and fluid velocity u µ = (1 , ). Themagnetic field is assumed to be in the + Z -direction with magnitude B . The equations are written focusing onthe genuine transport parameters that appear at first or-der in the derivative expansion in an out-of-equilibriumstate, ignoring the effects from the static susceptibilitiessuch as α BB , M Ω etc. as their effects become sublead-ing when the system is dominated by fluctuations. Theequations take the following form, with the fluctuationsFourier transformed i.e. δǫ ( t, x ) = δǫ ( t ) e i k · x , and so on. ∂ t δǫ + i k · π = 0 , (A1) ∂ t δn + α (cid:0) σ k k z + σ ⊥ ( k x + k y ) (cid:1) δn + α (cid:0) σ k k z + σ ⊥ ( k x + k y ) (cid:1) δǫ + iw [( n + B ˜ σ ) k x − B σ ⊥ k y ] π x + iw [( n + B ˜ σ ) k y + B σ ⊥ k x ] π y + in w k z π z = 0 , (A2) ∂ t π x + 1 w (cid:2) B σ ⊥ + ( ζ + η ⊥ − η ) k x + η ⊥ k y + η k k z (cid:3) π x − w (cid:2) B ( n + B ˜ σ ) + ˜ η ⊥ ( k x + k y ) + ˜ η k k z − ( ζ − η ) k x k y (cid:3) π y + 1 w (cid:2) ζ + ζ + η k − ( η + η ) (cid:3) k x k z π z − ˜ η k w k y k z π z + i (cid:2)(cid:0)(cid:0) ∂P∂n (cid:1) ǫ + B α ˜ σ (cid:1) k x + B α σ ⊥ k y (cid:3) δn + i (cid:2)(cid:0) ∂P∂ǫ (cid:1) n k x + B α ( k x ˜ σ + k y σ ⊥ ) (cid:3) δǫ = 0 , (A3) ∂ t π y + 1 w (cid:2) B σ ⊥ + ( ζ + η ⊥ − η ) k y + η ⊥ k x + η k k z (cid:3) π y + 1 w (cid:2) B ( n + B ˜ σ ) + ˜ η ⊥ ( k x + k y ) + ˜ η k k z + ( ζ − η ) k x k y (cid:3) π x + 1 w (cid:2) ζ + ζ + η k − ( η + η ) (cid:3) k y k z π z + ˜ η k w k x k z π z + i (cid:2)(cid:0)(cid:0) ∂P∂n (cid:1) ǫ + B α ˜ σ (cid:1) k y − B α σ ⊥ k x (cid:3) δn + i (cid:2)(cid:0) ∂P∂ǫ (cid:1) n k y + B α ( k y ˜ σ − k x σ ⊥ ) (cid:3) δǫ = 0 , (A4) ∂ t π z + 1 w (cid:2) ( ζ + ζ + ( η + η )) k z + η k ( k x + k y ) (cid:3) π z + 1 w (cid:2) ( ζ + η k + η ) k x + ˜ η k k y (cid:3) k z π x + 1 w (cid:2) ( ζ + η k + η ) k y − ˜ η k k x (cid:3) k z π y + i (cid:0) ∂P∂n (cid:1) ǫ k z δn + i (cid:0) ∂P∂ǫ (cid:1) n k z δǫ = 0 , (A5)with α = (cid:18) ∂µ∂ǫ (cid:19) n − µT (cid:18) ∂T∂ǫ (cid:19) n , and α = (cid:18) ∂µ∂n (cid:19) ǫ − µT (cid:18) ∂T∂n (cid:19) ǫ . It is straightforward to compute the eigenmodes ofsmall oscillations about the chosen equilibrium state, us-ing the linearized equations above. One gets five eigen-modes, of which two are gapped, given by ω = ± B n w − iB w ( σ ⊥ ± i ˜ σ ) + O ( k ) , (A6)while the other three are gapless. For k k B , the threegapless modes are two sound waves and one diffusive mode, given by ω = ± kv s − i k k , (A7a) ω = − iD k k , (A7b)where the speed of sound is given by v s = (cid:18) ∂p∂ǫ (cid:19) n + n w (cid:18) ∂p∂n (cid:19) ǫ and Γ k = 1 w (cid:18) ζ + ζ + 23 ( η + η ) (cid:19) + 1 v s (cid:18) ∂p∂n (cid:19) ǫ (cid:18) α + n w α (cid:19) σ k ,D k = 1 v s (cid:18)(cid:18) ∂p∂ǫ (cid:19) n α − (cid:18) ∂p∂n (cid:19) ǫ α (cid:19) σ k . For k ⊥ B , the gapless modes consist of one diffusive,1one shear and one sub-shear mode, given by ω = − iD ⊥ k , (A8a) ω = − i η k w k , (A8b) ω = − i Γ ⊥ ( k ) , (A8c)where D ⊥ = w (cid:16)(cid:16) ∂p∂ǫ (cid:17) n − n α (cid:17) σ ⊥ B σ ⊥ + ( n + B ˜ σ ) , Γ ⊥ = (cid:16)(cid:16) ∂p∂ǫ (cid:17) n α − (cid:16) ∂p∂n (cid:17) ǫ α (cid:17) η ⊥ B (cid:16)(cid:16) ∂p∂ǫ (cid:17) n − n α (cid:17) . In [19], the authors study the linearized hydrodynamicequations in terms of the fluctuations δT, δµ in the tem-perature and chemical potential. On the other hand,our analysis above works with the fluctuations δǫ, δn inthe energy and charge density. The two approaches areequivalent, and in particular the modes we have foundabove are in agreement with the ones reported in [19], af-ter an appropriate transformation from derivatives withrespect to ǫ, n to derivatives with respect to µ, T , whichare the independent variables in the grand canonical en-semble.
Appendix B: The susceptibility matrix χ In the main text, we focus on two subsectors of thefull set of linearized equations in appendix A: first whenonly η k = 0, and second when only ζ = 0. For boththe cases we compute the retarded two-point correlationfunctions of hydrodynamic fluctuations using linear re-sponse theory, see IV A and IV B. For this we need tomake use of the susceptibility matrix χ defined in eq.(35). This can be computed as follows. The complete setof hydrodynamic fluctuations isΦ = ( δn, δǫ, π x , π y , π z ) . (B1)Now the partition function in the grand canonical ensem-ble is given by Z = Tr exp [ − β ( H − µN )] , (B2)where H is the Hamiltonian and N is the total charge.Under infinitesimal variations δT, δµ and δu i in temper-ature, chemical potential and velocity, we get the newpartition function to be Z ′ = Tr exp (cid:20) − β ( H − µN )+ δTT ( H − µN ) + δµT N + 1 T δ u · P (cid:21) (B3) For use in linear response theory, the quantity of interestis the perturbation to the Hamiltonian, which for slowlyvarying sources can be read off from eq. (B3) as δH ( t ) = − Z d x (cid:18) δT ( t, x ) T ( ǫ ( t, x ) − µ n ( t, x ))+ δµ ( t, x ) n ( t, x ) + δu i ( t, x ) π i ( t, x ) (cid:19) . (B4)Therefore the sources for the hydrodynamic variables eq.(B1) are given by φ = (cid:18) δµ − µT δT, δTT , δu x , δu y , δu z (cid:19) . (B5)One can now compute the susceptibility matrix eq. (35)by differentiating the hydrodynamic variables in eq. (B1)with respect to their sources in eq. (B5), keeping in mindthat δǫ = (cid:18) ∂ǫ∂µ (cid:19) T (cid:16) δµ − µT δT (cid:17) + " µ (cid:18) ∂ǫ∂µ (cid:19) T + T (cid:18) ∂ǫ∂T (cid:19) µ δTT ,δn = (cid:18) ∂n∂µ (cid:19) T (cid:16) δµ − µT δT (cid:17) + " µ (cid:18) ∂n∂µ (cid:19) T + T (cid:18) ∂n∂T (cid:19) µ δTT , and π i = w δu i . This gives the susceptibility matrix as χ = (cid:16) ∂n∂µ (cid:17) T µ (cid:16) ∂n∂µ (cid:17) T + T (cid:0) ∂n∂T (cid:1) µ (cid:16) ∂ǫ∂µ (cid:17) T µ (cid:16) ∂ǫ∂µ (cid:17) T + T (cid:0) ∂ǫ∂T (cid:1) µ w w
00 0 0 0 w . For computing the long-time tails, we further special-ize to an equilibrium state with µ = 0 i.e. the equilib-rium charge density vanishes n = 0. For systems thatrespect charge conjugation invariance, this further leadsto the vanishing of thermodynamic derivatives such as( ∂n/∂T ) µ = 0 , ( ∂ǫ/∂µ ) T = 0, since they are proportionalto h HN i conn , which vanishes by charge conjugation in-variance. Further ∂ǫ/∂T ≡ c v = w /T v s , further simpli-fying the susceptibility matrix above, which can then beused to compute the retarded correlation functions viaeq. (34) in linear response theory. ACKNOWLEDGMENTS
The author would like to thank Pavel Kovtun andJuan Hernandez for very helpful discussions, and for theircomments on a draft version of the paper. Research atPerimeter Institute is supported in part by the Govern-ment of Canada through the Department of Innovation,Science and Economic Development Canada and by theProvince of Ontario through the Ministry of Colleges andUniversities.2 [1] P. Kovtun, Lectures on hydrodynamic fluctuationsin relativistic theories,
INT Summer School onApplications of String Theory Seattle, Washington,USA, July 18-29, 2011 , J. Phys.
A45 , 473001 (2012),arXiv:1205.5040 [hep-th].[2] S. Jeon and U. Heinz, Introduction to Hydrody-namics, Int. J. Mod. Phys. E , 1530010 (2015),arXiv:1503.03931 [hep-ph].[3] P. Romatschke and U. Romatschke, Relativistic Fluid Dynamics In and Out of Equilibrium ,Cambridge Monographs on MathematicalPhysics (Cambridge University Press, 2019)arXiv:1712.05815 [nucl-th].[4] Y. Pomeau and P. R´esibois, Time dependent cor-relation functions and mode-mode coupling theories,Physics Reports , 63 (1975).[5] P. B. Arnold and L. G. Yaffe, Effective the-ories for real time correlations in hot plasmas,Phys. Rev. D , 1178 (1998), arXiv:hep-ph/9709449.[6] L. D. Landau and E. M. Lifshitz, Hydrodynamic fluctu-ations, JETP , 512 (1957).[7] P. C. Martin, E. D. Siggia, and H. A. Rose, Statistical dy-namics of classical systems, Phys. Rev. A , 423 (1973).[8] P. C. Hohenberg and B. I. Halperin, Theory of dynamiccritical phenomena, Rev. Mod. Phys. , 435 (1977).[9] M. Crossley, P. Glorioso, and H. Liu, Effectivefield theory of dissipative fluids, JHEP , 095,arXiv:1511.03646 [hep-th].[10] F. M. Haehl, R. Loganayagam, and M. Rangamani,Topological sigma models and dissipative hydrodynam-ics, JHEP , 039, arXiv:1511.07809 [hep-th].[11] K. Jensen, N. Pinzani-Fokeeva, and A. Yarom, Dis-sipative hydrodynamics in superspace, JHEP , 127,arXiv:1701.07436 [hep-th].[12] X. Chen-Lin, L. V. Delacr´etaz, and S. A.Hartnoll, Theory of diffusive fluctua-tions, Phys. Rev. Lett. , 091602 (2019),arXiv:1811.12540 [hep-th].[13] A. Jain, P. Kovtun, A. Ritz, and A. Shukla, Hydrody-namic effective field theory and the analyticity of hydro-static correlators, (2020), arXiv:2011.03691 [hep-th].[14] A. Jain, K. Jensen, P. Kovtun, A. Ritz, and A. Shukla,Long-time tails in diffusive hydrodynamic effective fieldtheory, In preparation .[15] D. E. Kharzeev, L. D. McLerran, and H. J. War-ringa, The Effects of topological charge changein heavy ion collisions: ’Event by event P andCP violation’, Nucl. Phys. A , 227 (2008),arXiv:0711.0950 [hep-ph].[16] V. Skokov, A. Illarionov, and V. Toneev, Esti-mate of the magnetic field strength in heavy-ioncollisions, Int. J. Mod. Phys. A , 5925 (2009),arXiv:0907.1396 [nucl-th].[17] A. Bzdak and V. Skokov, Event-by-event fluctu-ations of magnetic and electric fields in heavyion collisions, Phys. Lett. B , 171 (2012),arXiv:1111.1949 [hep-ph]. [18] P. Kovtun and L. G. Yaffe, Hydrodynamic fluc-tuations, long time tails, and supersymmetry,Phys. Rev. D , 025007 (2003), arXiv:hep-th/0303010.[19] J. Hernandez and P. Kovtun, Relativistic magnetohydro-dynamics, JHEP , 001, arXiv:1703.08757 [hep-th].[20] N. Banerjee, J. Bhattacharya, S. Bhattacharyya,S. Jain, S. Minwalla, and T. Sharma, Constraints onFluid Dynamics from Equilibrium Partition Functions,JHEP , 046, arXiv:1203.3544 [hep-th].[21] K. Jensen, M. Kaminski, P. Kovtun, R. Meyer, A. Ritz,and A. Yarom, Towards hydrodynamics without anentropy current, Phys. Rev. Lett. , 101601 (2012),arXiv:1203.3556 [hep-th].[22] K. Jensen, R. Loganayagam, and A. Yarom, Anomalyinflow and thermal equilibrium, JHEP , 134,arXiv:1310.7024 [hep-th].[23] P. Kovtun and A. Shukla, Einstein’s equa-tions in matter, Phys. Rev. D , 104051 (2020),arXiv:1907.04976 [gr-qc].[24] P. Kovtun, Thermodynamics of polarized relativisticmatter, JHEP , 028, arXiv:1606.01226 [hep-th].[25] P. Kovtun and A. Shukla, Kubo formulas forthermodynamic transport coefficients, JHEP , 007,arXiv:1806.05774 [hep-th].[26] A. Shukla, Equilibrium thermodynamic sus-ceptibilities for a dense degenerate Diracfield, Phys. Rev. D , 096010 (2019),arXiv:1906.02334 [hep-th].[27] M. Ammon, S. Grieninger, J. Hernandez, M. Kamin-ski, R. Koirala, J. Leiber, and J. Wu, Chiralhydrodynamics in strong magnetic fields, (2020),arXiv:2012.09183 [hep-th].[28] S. A. Hartnoll, P. K. Kovtun, M. Muller, andS. Sachdev, Theory of the Nernst effect near quan-tum phase transitions in condensed matter, andin dyonic black holes, Phys. Rev. B , 144502 (2007),arXiv:0706.3215 [cond-mat.str-el].[29] A. Amoretti, D. K. Brattan, N. Magnoli, andM. Scanavino, Magneto-thermal transport impliesan incoherent Hall conductivity, JHEP , 097,arXiv:2005.09662 [hep-th].[30] A. Amoretti, D. Arean, D. K. Brattan, and N. Magnoli,Hydrodynamic magneto-transport in charge density wavestates, (2021), arXiv:2101.05343 [hep-th].[31] X.-G. Huang, A. Sedrakian, and D. H. Rischke,Kubo formulae for relativistic fluids in strongmagnetic fields, Annals Phys. , 3075 (2011),arXiv:1108.0602 [astro-ph.HE].[32] S. Grozdanov, D. M. Hofman, and N. Iqbal, Gen-eralized global symmetries and dissipative magne-tohydrodynamics, Phys. Rev. D95 , 096003 (2017),arXiv:1610.07392 [hep-th].[33] M. Martinez and T. Sch¨afer, Stochastic hydrody-namics and long time tails of an expanding con-formal charged fluid, Phys. Rev. C , 054902 (2019),arXiv:1812.05279 [hep-th].[34] S. Caron-Huot and O. Saremi, Hydrodynamic Long-Time tails From Anti de Sitter Space, JHEP11