Bulk viscosity from hydrodynamic fluctuations with relativistic hydro-kinetic theory

# Bulk viscosity from hydrodynamic fluctuations with relativistic hydro-kinetic theory

Yukinao Akamatsu Department of Physics, Osaka University, Toyonaka, Osaka 560-0043, Japan    Aleksas Mazeliauskas Institut für Theoretische Physik, Universität Heidelberg, 69120 Heidelberg, Germany    Derek Teaney Department of Physics and Astronomy, Stony Brook University, Stony Brook, New York 11794, USA
July 26, 2019
###### Abstract

Hydro-kinetic theory of thermal fluctuations is applied to a non-conformal relativistic fluid. Solving the hydro-kinetic equations for an isotropically expanding background we find that hydrodynamic fluctuations give ultraviolet divergent contributions to the energy-momentum tensor. After shifting the temperature to account for the energy of non-equilibrium modes, the remaining divergences are renormalized into local parameters, e.g. pressure and bulk viscosity. We also confirm that the renormalization of the pressure and bulk viscosity is universal by computing them for a Bjorken expansion. The fluctuation-induced bulk viscosity reflects the non-conformal nature of the equation of state and is modestly enhanced near the QCD deconfinement temperature.

## I Introduction

Ultra-relativistic heavy-ion collisions are a major experimental tool to study nuclear matter in an extremely hot environment. The energy density in heavy ion collisions at the Relativistic Heavy Ion Collider (RHIC) at BNL and the Large Hadron Collider (LHC) at CERN is so high that partonic degrees of freedom are liberated from nucleons and a deconfined quark-gluon plasma (QGP) is formed. The QGP then expands hydrodynamically as a fluid with very small shear viscosity over entropy ratio  Heinz and Snellings (2013); Luzum and Petersen (2014). The hydrodynamic paradigm for heavy-ion collisions has been very successful in explaining the various collective flow observables as dynamical response to event-by-event fluctuations of the initial fireball shape Heinz and Snellings (2013); Teaney (2010); Luzum and Petersen (2014); Gale et al. (2013); Romatschke (2010).

Recently, attention has been paid to another source of fluctuations in the hydrodynamic picture, namely, thermal fluctuations Gavin and Abdel-Aziz (2006); Kapusta et al. (2012); Yan and Grönqvist (2016); Young et al. (2015); Kapusta and Torres-Rincon (2012); Murase and Hirano (2016); Nagai et al. (2016). Thermal fluctuations are theoretically required by the fluctuation-dissipation theorem. Furthermore, thermal fluctuations play an important role in systems with a small number of particles and are essential near the critical point, which is the focus of the ongoing beam energy scan program at RHIC Kumar (2013).

A unique feature of hydrodynamic fluctuations in heavy-ion collisions is the rapidly expanding background flow along the beam direction, which at mid-rapidity is often modelled as one dimensional Bjorken flow Bjorken (1983). The distribution of fluctuations around such evolving background is characterized by a specific wave number scale , where the longitudinal expansion and (-dependent) relaxation rates balance, and the distribution function approaches a non-equilibrium steady state. In the previous publication, we developed an effective kinetic description for conformal hydrodynamic fluctuations around the characteristic scale and discussed how to deal with ultra-violet divergences associated with short wavelength fluctuations Akamatsu et al. (2017). Using the hydro-kinetic theory we obtained a universal renormalization of the pressure and shear viscosity in agreement with previous diagrammatic calculations around a non-expanding background Kovtun and Yaffe (2003); Kovtun et al. (2011). Furthermore, we applied the hydro-kinetic approach to the Bjorken expansion, and found the precise coefficient of the fractional-power-law tail arising due to the out-of-equilibrium distribution of hydrodynamic fluctuations.

In this paper, we consider a relativistic non-conformal fluid, for which the speed of sound and the bulk viscosity is finite. The bulk viscosity determines the dissipative correction to the pressure in response to an isotropic expansion or compression and is a measure for scale symmetry breaking. For example, perturbative calculations in a high-temperature QGP show that it is proportional to the square of the scale symmetry breaking factors (the QCD running coupling and finite quark mass) Arnold et al. (2006). Also, lattice QCD simulations suggest a correlation between the bulk viscosity and the scale symmetry breaking realized in the equation of state Meyer (2008). Spectral sum rules in the bulk channel also indicate some correlation between the bulk viscosity and a non-conformal nature of the equation of state Kharzeev and Tuchin (2008); Karsch et al. (2008); Moore and Saremi (2008); Romatschke and Son (2009). Finally, near the critical point, the bulk viscosity diverges due to the critical slowing down Onuki (2002).

In the main part of the paper we apply our hydro-kinetic theory to a static system perturbed by an isotropic expansion and compute the response function of the energy-momentum tensor in the bulk channel. We discus the case of Bjorken expansion in the Appendix A. In a non-conformal fluid the two point correlation function of hydrodynamic fluctuations contributes to the trace of the energy momentum tensor, which gives rise to a renormalization of the bulk viscosity:

 ζ(T) =ζ0(T;Λ) (1) +TΛ18π2⎡⎢ ⎢ ⎢ ⎢ ⎢⎣(1+3T2dc2s0dT−3c2s0)2e0+p0ζ0+43η0+4(1−3c2s0)2e0+p02η0⎤⎥ ⎥ ⎥ ⎥ ⎥⎦.

Here, is a UV cut-off for the hydrodynamic fluctuations and is the bare bulk viscosity. The fluctuation contribution to the bulk viscosity is positive and proportional to the scale symmetry breaking factors in the equation of state. It is noteworthy that in order to arrive to eq. (1), the temperature of the background fluid must be shifted depending on the cut-off so as to include the energy of the non-equilibrium hydrodynamic modes (see Sec. III.2 for details).

The fluctuation induced renormalization in eq. (1) can be used to estimate a lower bound of the bulk viscosity of QCD – see ref. Kovtun et al. (2011) for a similar estimate of the shear viscosity. Very recently the approach was also used to estimate the bulk viscosity of a non-relativistic cold Fermi gas, where the renormalization was obtained with diagrammatic methods Martinez and Schaefer (2017). Using the lattice equation of state for entropy density and the speed of sound  Borsanyi et al. (2016) in eq. (1), we calculate the magnitude of bulk viscosity renormalization by setting , and choosing a representative values of the kinematic viscosity, , and the temperature dependent UV cut-off (see Fig. 1).

Due to small deviation from scale symmetry at high temperatures the bulk viscosity renormalization is vanishing small for . However, the degree of non-conformality peaks around the pseudo-critical temperature where the bulk viscosity reaches at .

The logic of the estimate in Fig. 1 is the following. The physical bulk viscosity (which is independent of ) arises from two contributions: the fluctuations above , which at weak coupling are dominated by single-particle excitations, and the fluctuations below , which are described by hydrodynamics. We have only included the hydrodynamic fluctuations here, and thus we expect the physical bulk viscosity to be larger than the estimate shown in Fig. 1.

The organization of this paper is as follows. In Sec. II, we derive the kinetic equations for hydrodynamic fluctuations for an isotropically expanding non-conformal fluid. Then in Sec. III, we compute the fluctuation contributions to the energy-momentum tensor, and discuss the subtle temperature shift. After the temperature shift, we renormalize the energy density, the pressure and the bulk viscosity, and find the finite long time tails for the weak isotropic expansion. The summary of the paper is given in Sec. IV. Finally, in Appendix A we repeat the computation of the temperature shift and the renormalization of hydrodynamic fields for Bjorken expansion.

## Ii Kinetic equations for hydrodynamic fluctuations

In this section we apply the formalism developed in ref. Akamatsu et al. (2017) to a non-conformal fluid under isotropic expansion (or compression). We will follow the same procedure to derive the relaxation type equations for the two point correlation functions under the presence of background perturbations.

The governing equations for non-conformal hydrodynamics with noise are given by Landau and Lifshitz (1980); Lifshitz and Pitaevskii (1980); [forarecentreview:]Kovtun:2012rj

 dμTμν =0,Tμν=Tμνideal+Tμν% visc.+Sμν, (2a) Tμνideal =(e+p)uμuν+pgμν, (2b) Tμνvisc. =−ησμν−ζΔμνΔαβdαuβ, (2c) σμν =ΔμρΔνσ(dρuσ+dσuρ−23gρσdγuγ), (2d) Δμν =gμν+uμuν, (2e)

where denotes a covariant derivative using the “mostly-plus” metric convention. Below we notate the divergence of the flow velocity as . The variance of the stochastic noise is determined by the fluctuation-dissipation theorem:

 ⟨Sμν(x1)Sαβ(x2)⟩ =2T⎡⎢ ⎢⎣η(ΔμαΔνβ+ΔμβΔνα)+(ζ−23η)ΔμνΔαβ⎤⎥ ⎥⎦δ(x1−x2)√−detgμν. (3)

Differently from the conformal case, both shear and bulk viscosities are now present in the equation of motion and noise correlator.

### ii.1 Background fluid

Dynamics of hydrodynamic fluctuations on a background fluid in a weak isotropic expansion (or compression) is conveniently studied in the reference frame of the fluid. In the comoving frame for the isotropic expansion, the metric is time dependent

 ds2=−dt2+(1+h(t))d→x2,   (|h(t)|≪1) (4)

and the background fluid satisfies

 0=˙e0(t)+3˙h2[e0(t)+p0(t)]+O(h2). (5)

The second term on the right hand side represents the change of energy density due to the expansion and the associated work done by the pressure. Throughout this paper, denotes a quantity of the background fluid in a perturbed metric (). As discussed previously Akamatsu et al. (2017), and denote the background energy density and pressure from modes with wavenumbers greater than a cut-off . In Sec. III.2 we detail how and are related to the lattice equation of state.

Solving perturbatively in , the energy density for the background fluid evolves as

 e0(t)=¯e0−3h(t)2(¯e0+¯p0)+O(h2), (6)

where denotes the energy density of the background fluid in an unperturbed state (). Again, throughout this paper denotes a quantity of the background fluid in an unperturbed state ().

### ii.2 Evolution of hydrodynamic fluctuations

For the expanding background described by eq. (6), the hydrodynamic fluctuations excited by thermal noise and evolve according to the following equations in -space:

 0 =∂tδe+ikigi+3˙h2(1+c2s0)δe, (7a) 0 =∂tgi+ic2s0kiδe+3˙h2gi +γη0(klklδji−kikj)gj+γζ0kikjgj+ξi, (7b)

with noise correlation given by

 ⟨ξi(t,k)ξj(t′,−k′)⟩ =2T0(e0+p0)√−detgμν(2π)3δ(k−k′)δ(t−t′) ×[γη0(klklgij−kikj)+γζ0kikj]. (8)

Here and are kinematic viscosities. Analysis becomes simpler by utilizing a vielbein formalism. We introduce new variables

 G^i ≡(1+12h(t))gi, (9a) K^i ≡(1−12h(t))ki, (9b) Ξ^i ≡(1+12h(t))ξi, (9c)

which give , , and . We define a four component vector of hydrodynamic fluctuations. The equation of motion for is

 −˙ϕa(t,k) =iLabϕb+Dabϕb+Ξa+Pabϕb, (10a) L=(0cs0→Kcs0→K0), (10b) D=(000γη0(K2δ^i^j−K^iK^j)+γζ0K^iK^j), (10c) P=˙h⎛⎜ ⎜ ⎜ ⎜ ⎜ ⎜⎝32(1+¯c2s0+¯T02d¯c2s0d¯T0)222⎞⎟ ⎟ ⎟ ⎟ ⎟ ⎟⎠, (10d)

with noise correlation given by

 ⟨Ξa(t,k)Ξb(t′,−k′)⟩ =2T0(e0+p0)√−detgμνDab(2π)3δ(k−k′)δ(t−t′). (11)

The matrices and originate from ideal and viscous parts of the hydrodynamic equations respectively, while arises from remaining interactions between the fluctuations and the background fluid. Note that the term in derives from the time dependence of in . In the kinetic regime, drives the evolution of so that it will be more convenient to analyze eq. (10a) in terms of eigenmodes of :

 (e±)a=1√2(1±^K),(eT1)a=(0→T1),(eT2)a=(0→T2).

Here , , and form an orthonormal basis. The subscripts stand for the two sound modes and for the two transverse diffusive modes. The corresponding eigenvalues are and .

### ii.3 Kinetic equations for hydrodynamic fluctuations

The two-point correlation functions of with are defined as

 (13)

We will determine the equations of motion for using the formalism of ref. Akamatsu et al. (2017). In the rotating wave approximation, the off-diagonal part of the density matrix can be neglected because of its rapid phase rotation111 has a stationary phase but vanishes due to the rotational symmetry. , while the diagonal part evolves according to

 ˙NAA=−2DAA[NAA−T0(e0+p0)√−detgμν]+2PAANAA,

where we have defined and similarly . The isotropic system does not distinguish the two transverse modes and , and thus we only have two independent kinetic equations: one for the sound modes (), and one for the transverse modes (). Using the matrices and eigenvectors of the previous section, eq. (II.3) evaluates to

 ˙NL =−γζ0K2[NL−T0(e0+p0)√−detgμν] −˙h2(3¯c2s0+3¯T02d¯c2s0d¯T0+7)NL, (15a) ˙NT =−2γη0K2[NT−T0(e0+p0)√−detgμν]−4˙hNT. (15b)

The kinetic equations (15) and (15b) describe how the distribution of fluctuations evolves on the isotropically expanding background. Perturbative solutions of the kinetic equations for take the form

 NL/T(t,k) =Neq(t)+δNL/T(t,k)+O(h2), (16)

where the equilibrium contribution is

 Neq(t) =T0(e0+p0)√−detgμν ≃[1−(3+3¯c2s0)h(t)]¯T0(¯e0+¯p0), (17)

and the non-equilibrium correction is

 δNL(ω,k) =12iωh(ω)−iω+¯γζ0K2¯Cζ0¯T0(¯e0+¯p0), (18a) δNT(ω,k) =iωh(ω)−iω+2¯γη0K2¯Cη0¯T0(¯e0+¯p0). (18b)

Here and below we have defined

 Cζ(T) ≡1+3T2dc2sdT−3c2s, (19a) Cη(T) ≡1−3c2s. (19b)

Note that when the background fluid is scale invariant , the corrections vanish. Therefore in conformal case, the isotropic expansion or compression does not drive the hydrodynamic fluctuations from the equilibrium distribution given by eq. (II.3).

## Iii Energy-momentum tensor with nonlinear fluctuations

In this section we compute the nonlinear contributions of hydrodynamic fluctuations to the statistically averaged energy momentum tensor . The main difference from the conformal case Akamatsu et al. (2017) is additional contributions to the averaged energy density , which are absorbed by a shift in the background temperature .

### iii.1 Averaged energy-momentum tensor

The averaged stress tensor consists of contributions from the background fluid and from the two-point functions of the hydrodynamic fluctuations:

 ⟨Tij⟩ =[1−h(t)]p0δij−32˙h(t)ζ0δij+Tijfluct, (20a) Tijfluct ≃1−h(t)e0+p0⎡⎢ ⎢⎣⟨G^i(t,x)G^j(t,x)⟩+δijT02dc2s0dT0⟨(cs0δe(t,x))2⟩⎤⎥ ⎥⎦. (20b)

The energy density fluctuations originate from the second order derivative , which is finite for a non-conformal equation of state. The trace of the stress tensor from the fluctuations is determined by the two-point functions :

 Tiifluct =1−h(t)e0+p0∫d3k(2π)3 (21) ×[(1+3T02dc2s0dT0)NL(t,k)+2NT(t,k)].

This integral is divergent and is regularized by introducing a cut-off for (not ). Substituting the solution (16), we write the fluctuating contribution as a sum of two terms

 Tiifluct(t,Λ)=TiiNeq(t;Λ)+TiiδN(t;Λ). (22)

The first term arises from equilibrium fluctuations (eq. (II.3))

 TiiNeq(t;Λ)≡[1−h(t)](1+T02dc2s0dT0)T0Λ32π2, (23)

while the second term arises from the non-equilibrium distribution functions, in eq. (18). In frequency space this non-equilibrium contribution reads

 TiiδN(ω;Λ)≡ h(ω)¯T04π2(1+3¯T02d¯c2s0d¯T0)¯Cζ0f(ω,¯γζ0,Λ) +h(ω)¯T0π2¯Cη0f(ω,2¯γη0,Λ). (24)

Here we have defined a function

 f(ω,γ,Λ) ≡∫Λ→∞0p2dpiω−iω+γp2 (25) =iωγΛ−(|ω|γ)3/2π2√2(1+isgn(ω)).

Next, we calculate the averaged energy density in a similar manner. It also consists of contributions from the background fluid and from the two-point functions of the fluctuations:

 ⟨Ttt⟩ =e0+Tttfluct, (26a) Tttfluct =⟨→G2⟩e0+p0 (26b) =1e0+p0∫d3k(2π)3[NL(t,k)+2NT(t,k)],

The contribution from the fluctuations is again divergent and we regularize with the same cut-off on . Substituting the perturbative solutions (16), we find

 Tttfluct(t;Λ)= TttNeq(t;Λ)+TttδN(t;Λ), (27)

where the first term arises from the equilibrium distribution (eq. (II.3))

 TttNeq(t;Λ)≡T0Λ32π2, (28)

while the second term (in frequency space) arises from

 TttδN(ω;Λ)≡ h(ω)¯T04π2¯Cζ0f(ω,¯γζ0,Λ) +h(ω)¯T0π2¯Cη0f(ω,2¯γη0,Λ). (29)

As will be described in the next section, the divergences in and are absorbed by renormalizing the background fields, e.g.  and . This renormalization procedure requires a clearer understanding how these bare parameters are defined, and how they depend on the cut-off .

### iii.2 Temperature shift

The bare parameters are determined by modes (such as particle-like excitations) with wavenumbers above the cut-off, , which are not explicitly propagated by the statistical hydrodynamic system. The goal of this section is to carefully explain how these parameters are defined and related to the physical equation of state (from lattice QCD) and the cut-off .

First consider the density matrix for non-hydrodynamic modes with wavenumbers above the cut-off . When the system is driven slightly out of equilibrium by the periodic compression and expansion the density matrix for these modes can be decomposed as an equilibrium density matrix and a viscous correction

 ρ(Λ)=ρeq(T0;Λ)+δρneq(Λ). (30)

The temperature parameter (which will depend on time and ) is chosen so that the average energy density above the cut-off equals the energy from the equilibrium density matrix alone222Because of this constraint we can and will drop the “eq” label below, i.e. .

 ⟨Ttt(t)⟩k>Λ=eeq,0(T0(t;Λ);Λ), (31)

i.e. is adjusted so that the energy moment associated with is zero (otherwise the rhs of eq. (31) would have a correction proportional to ). In kinetic theory this constraint is imposed by requiring that the viscous correction to the distribution function does not change the energy in the system Arnold et al. (2006); Moore and Saremi (2008). Once this prescription for is adopted the stress computed with the density matrix is given by333In the current setup .

 ⟨Tij⟩k>Λ=(1−h)p0(T0;Λ)δij−ζ0(T0;Λ)∇⋅uδij, (32)

where the partial pressure from modes above is determined by the equilibrium density matrix, , while the bulk term comes from the viscous correction, . Implicit in the definition of (which measures the deviation of the stress relative to a specific time-dependent equilibrium expectation) is that the parameter is defined according to the Landau constraint given by eq. (31Arnold et al. (2006); Moore and Saremi (2008). Later in this section we will define a temperature by imposing the Landau constraint on the whole system (including the energy of hydrodynamic fluctuations below the cut-off), and this will lead to a difference between and physical temperature .

Note that the partial energy density and pressure, and , are not quite the equilibrium energy density and pressure, and , as measured on a lattice. Indeed, they are cut-off dependent quantities and are determined by an equilibrium density matrix which excludes equilibrium hydrodynamic fluctuations below the scale . The contribution of such equilibrium hydrodynamic fluctuations to the energy density and pressure are given by eq. (28) and eq. (23) respectively, and thus the physical energy density and pressure are:

 e(T0)= e0(T0;Λ)+T0Λ32π2, (33a) p(T0)= p0(T0;Λ)+(1+T02dc2sdT0)T0Λ36π2. (33b)

At a practical level these equations serve to define the and parameters that should be used in a stochastic hydro-code with a given cut-off and physical equation of state .

As discussed above, the temperature for the complete system (background+fluctuations) is adjusted so that the energy density calculated from the lattice equation of state matches the energy of the partially equilibrated system

 ⟨Ttt(t)⟩=e(T(t)). (34)

After imposing this constraint, the time dependent stress of the driven system will deviate from its equilibrium expectation, , and these deviations are described (up to long-time tails) by the bulk viscosity. Combining eqs. (26a),(27),(28), and (33a), we see that

 e(T(t)) =e(T0(t;Λ))+TttδN(t;Λ). (35)

where was defined in eq. (III.1). Thus, the temperature for the whole system (which is independent of the cut-off) is related to the temperature parameter of the subsystem by a small shift

 T0(t;Λ)=T(t)+ΔT(t;Λ), (36)

so that eq. (35) is satisfied. The temperature shift is given in frequency space by

 −dedTΔT(ω;Λ) =h(ω)¯T4π2¯Cζ0f(ω,¯γζ0,Λ) (37) +h(ω)¯Tπ2¯Cη0f(ω,2¯γη0,Λ).

and clearly depends on the cut-off since the was defined with respect to a specific subsystem labeled by . The temperature shift in the time domain takes the form

 −dedTΔT(t;Λ)=−¯TΛ6π2[¯Cζ0¯γζ0+4¯Cη02¯γη0]∇⋅u+finite, (38)

where for this example. The divergent piece of the temperature shift is universal, but the finite corrections are not. This is verified by explicit calculation of the temperature shift for the Bjorken background in Appendix A. From practical perspective, eqs. (36) and (38) define how must be chosen for a stochastic hydro code (with a specified cut-off ) to reproduce the correct physical bulk viscosity for long wavelength hydrodynamic modes and a physical equation of state. This is detailed in the next section444In defining from , , and , the finite remainder in eq. (38) can be chosen in any convenient way. .

### iii.3 Renormalized background and long time tails

Once the temperature shift is obtained, the remaining divergences in can be absorbed by pressure and bulk viscosity renormalization. Using eqs. (20a),(22), (23), and (33b), the statistically averaged spatial stress tensor trace reads

 ⟨Tii⟩(t)=3[1−h(t)]p(T0(t;Λ))−92˙h(t)ζ0+TiiδN(t;Λ). (39)

where is given in eq. (III.1).

Now we will shift the temperature parameter in the pressure to the physical temperature determined by Landau matching (35), . The fluctuation contribution and the temperature parameter both diverge as . These two terms gracefully combine to produce a positive definite renormalization of bulk viscosity in the term

 ζ(T)=ζ0(T;Λ)+TΛ18π2[C2ζ0γζ0+4C2η02γη0]. (40)

In this step the coefficients in front of the linear divergences in and have neatly come together to complete the squares of and defined by eq. (19). Thus the renormalization of the bulk viscosity is positive and only necessary in a system with broken scale symmetry. We have confirmed that the bulk viscosity renormalization is universal by computing it for a Bjorken expanding background (see Appendix A).

Once all divergences are absorbed by renormalization, the stress tensor becomes finite and cut-off independent. In the presence of background expansion, there are remaining finite corrections from the fluctuations in . The total stress tensor is

 ⟨Tii⟩(t) =3[1−h(t)]p(T(t))−92˙h(t)ζ(T(t)) −∫dω2πe−iωth(ω)|ω|3/2π2√2(1+isgn(ω)) ×¯T4π2⎡⎢⎣¯C2ζ0(1¯γζ0)3/2+4¯C2η0(12¯γη0)3/2⎤⎥⎦,

and has a term with , which cannot be expressed by local time derivatives. This term is not analytic at and derives from the out-of-equilibrium fluctuations in the kinetic regime .

With and known, we can write down the hydrodynamic equations for statistically averaged hydrodynamics with noise

 0=ddt⟨Ttt⟩+32˙h⟨Ttt⟩+12˙h⟨Tii⟩. (41)

Since the non-analytic term in is of , the rest frame energy density evolves according to

 0=˙e(t)+3˙h2[e(t)+p(t)], (42)

and we obtain the solution:

 e(t)=¯e−3h(t)2(¯e+¯p), (43)

which will be used to calculate the response function in the next section.

### iii.4 Response function in the bulk channel

The non-analytic behavior in is also present in the response function in the bulk channel. In the frequency space, the linear response of stress tensor to the external gravitational field is given by

 ⟨Tii⟩(ω)=Gii,jjR(ω,k=0)12h(ω). (44)

The response function is defined by

 Gii,jjR(t,x) ≡iθ(t)⟨[^Tii(t,x),^Tjj(0,0)]⟩, (45a) Gii,jjR(ω,k) =∫d4xGii,jjR(t,x)eiωt−ik⋅x. (45b)

Then from our results, the response function is obtained as

 Gii,jjR(ω)=δδh(ω)[2⟨Tii⟩(ω)]∣∣h=0 =−6(¯p+32¯c2s(¯e+¯p))+9iω¯ζ−1+isgn(ω)4√2π|ω|3/2¯T ×⎡⎢⎣¯C2ζ(1¯γζ)3/2+4¯C2η(12¯γη)3/2⎤⎥⎦, (46)

and the spectral function as

 ρii,jj(ω)=2ImGii,jjR(ω) (47) =18ω¯ζ−ω|ω|1/2¯T2√2π⎡⎢⎣¯C2ζ(1¯γζ)3/2+4¯C2η(12¯γη)3/2⎤⎥⎦.

This spectral function is consistent with a previous diagrammatic computation of the symmetrized correlation function (see the Appendix of ref. Kovtun and Yaffe (2003)) using the fluctuation-dissipation relation:555 The term in corresponds to a correlation of thermal noise in the stress tensor, which is not explicitly written in the calculation of Kovtun and Yaffe (2003).

 ρii,jj(ω) =ωTCii,jj(ω,k=0), (48a) Cii,jj(t,x) ≡12⟨{^Tii(t,x),^Tjj(0,0)}⟩conn. (48b)

## Iv Summary

In this paper we applied the kinetic theory of hydrodynamic fluctuations developed in ref. Akamatsu et al. (2017) to a relativistic non-conformal fluid. We calculated the contribution of out-of-equilibrium hydrodynamic fluctuations to the energy momentum tensor, which renormalize the background hydrodynamic fields and the bulk viscosity . The bulk viscosity renormalization is proportional to the scaling symmetry breaking in the equation of state and can be used to estimate the minimal bulk viscosity value in a hot QCD medium.

In the main body of the paper, we considered a non-conformal charge-neutral fluid, which is driven out of equilibrium by a weak isotropic expansion (or compression). Analogous calculations for a Bjorken expanding system is summarized in the appendix. The relaxation of hydrodynamic fluctuations to equilibrium is disturbed by the expansion and the deviation of two-point correlations from equilibrium becomes appreciable for wavelengths , where is the frequency of the background expansion and defines the hydro-kinetic regime.

We derive the hydro-kinetic equations for the two-point correlation functions , eq. (13), of energy and momentum density fluctuations in the presence of the expansion. The nonlinear fluctuations contribute to the statistically averaged energy-momentum tensor . The divergent part of the fluctuation contributions is regulated by an ultraviolet cut-off . The cut-off dependences of is (partially) absorbed by a universal renormalization of the background energy density , the pressure and the bulk viscosity (the same terms are found for the far-from-equilibrium Bjorken expansion666 Since Bjorken expansion is anisotropic, there is an additional linear divergence which renormalizes the background shear viscosity (63d):