A kinetic regime of hydrodynamic fluctuations and long time tails for a Bjorken expansion

We develop a set of kinetic equations for hydrodynamic fluctuations which are equivalent to nonlinear hydrodynamics with noise. The hydro-kinetic equations can be coupled to existing second order hydrodynamic codes to incorporate the physics of these fluctuations, which become dominant near the critical point. We use the hydro-kinetic equations to calculate the modifications of energy momentum tensor by thermal fluctuations from the earliest moments and at late times in Bjorken expansion. The solution to the kinetic equations precisely determines the coefficient of the first fractional power ($\propto \tau^{-3/2}$) in the energy momentum tensor gradient expansion. Numerically, we find that the contribution to the longitudinal pressure from hydrodynamic fluctuations is comparable to the second order hydrodynamic terms for typical medium parameters used to simulate heavy ion collisions.


I. INTRODUCTION A. Overview
The purpose of the current paper is to develop a set of kinetic equations for hydrodynamic fluctuations, and to use these kinetic equations to study corrections to Bjorken flow arising from thermal fluctuations.The specific test case of Bjorken flow (which is a hydrodynamic model for the longitudinal expansion of a nucleus-nucleus collision [1]) is motivated by the experimental program of ultra-relativistic heavy-ion collisions at RHIC and the LHC.Detailed measurements of two particle correlation functions have provided overwhelming evidence that the evolution of the excited nuclear material is remarkably well described by the hydrodynamics of the Quark Gluon Plasma (QGP) with a small shear viscosity to entropy ratio of order η/s ∼ 2/4π [2,3].The typical relaxation times of the plasma, while short enough to support hydrodynamics, are not vastly smaller than the inverse expansion rates of the collision.For this reason the gradient expansion underlying the hydrodynamic formalism has been extended to include first and second order viscous corrections [4], and these corrections systematically improve the agreement between hydrodynamic simulations and measured two particle correlations [2].Additional corrections, which have not been systematically included, arise from thermal fluctuations of the local energy and momentum densities and could be significant in nucleus-nucleus collision where only ∼ 20000 particles are produced.This has prompted a keen practical interest in the heavy ion community in simulating relativistic hydrodynamics with stochastic noise [5][6][7][8][9][10].In a non-relativistic context such simulations have reached a fairly mature state [11][12][13].For a static fluid, thermal fluctuations give rise through the nonlinearities of the equations of motion to fractional powers in the fluid response function at small frequency, G R (ω) ∝ ω 3/2 .Indeed, the "long-time tails" first observed in moleculardynamics simulations [14][15][16] are a consequence of this non-analytic ω 3/2 behavior.For Bjorken flow, the same nonlinear stochastic physics leads to fractional powers in the gradient expansion for the longitudinal pressure of the fluid.One of the goals of this manuscript is to compute the coefficient of the first fractional power in this expansion.
The measured two particle correlations in heavy ion collisions reflect both the fluctuations in the initial conditions and thermal fluctuations.Thermal fluctuations are believed to be a small (but conceptually important) correction to non-fluctuating hydrodynamics [6][7][8].In addition, thermal fluctuations can become significant close to the QCD critical point [17] and in smaller colliding systems such as proton-nucleus and proton-proton collisions [7], which show remarkable signs of collectivity [18].
In the current manuscript, rather than simulating nonlinear fluctuating hydrodynamics directly, we will reformulate fluctuating hydrodynamics as non-fluctuating hydrodynamics (describing a long wavelength background) coupled to a set of kinetic equations describing the phase arXiv:1606.07742v1[nucl-th] 24 Jun 2016 space distribution of short wavelength hydrodynamic fluctuations.For Bjorken flow this set of equations can be solved to determine the first fractional powers in the gradient expansion.

B. Hydrodynamics with noise and fractional powers in the gradient expansion
Hydrodynamics is a long wavelength effective theory which describes the evolution of conserved quantities by organizing corrections in powers of gradients.For the hydrodynamic expansion to apply we require frequencies under consideration to be small compared to the the microscopic relaxation rates where we have estimated the microscopic relaxation time with the hydrodynamic parameters, τ R ≡ η/(e+p)c 2 s [19] and for later convenience defined ≡ ωτ R .
For definiteness, we follow precedent [4,20,21] and consider a conformal neutral fluid driven from equilibrium by a small metric perturbation h xy (ω) of frequency ω.Within the framework of linear response (see Sect.II and Ref. [22] for further details), the stress tensor at low frequency takes the form The first term is the prediction of ideal hydrodynamics δT xy = −ph xy ; the middle term is the prediction of first order viscous hydrodynamics [20], where η is the shear viscosity; finally, the last term is the prediction of second order hydrodynamics, where τ π and κ are the associated second order parameters [4].In writing Eq. ( 2) we have neglected additional contributions stemming from fluctuations which will be described below.Thermal fluctuations can be incorporated into the hydrodynamic description by including stochastic terms into the equations of motion [23,24] where variance of the noise, S µν S ρσ ∼2T ηδ(t − t ), is determined by the fluctuation dissipation theorem at temperature T and introduces no new parameters into the effective theory 1 .After including these stochastic terms, the correlators of momentum and energy evolve to their equilibrium values in the absence of the external force, h xy (ω).Specifically, the equilibrium two point functions of the energy and momentum densities, δe(t, x) ≡ T 00 (t, x) − T 00 and g i (t, x) ≡ T 0i respectively, approach the textbook result [23] where c s is the speed of sound, and δe(t, k) notates the spatial Fourier transform of δe(t, x).In the presence of an external force or a non-trivial expansion these correlations are driven away from equilibrium.The purpose of hydrodynamics with noise is to describe in detail these deviations from equilibrium.Due to the nonlinear character of hydrodynamics the thermal fluctuations change the evolution of the system.Indeed, a diagrammatic analysis of the hydrodynamic response at one-loop order shows that the stress in the presence of a weak external field (or the retarded Green function) is where p, e, and η are renormalized physical quantities (see Sect.II A and Sect.II B for further discussion of the 1 We follow a standard notation for hydrodynamics summarized in Ref. [19].dµ notates a covariant derivative using the "mostlyplus" metric convention.T µν ideal = (e+p)u µ u ν +pg µν and T µν visc = −ησ µν where σ µν = ∆ µρ ∆ νσ (dρuσ + dσuρ − 2 3 gρσdγ u γ ), with ∆ µν = g µν + u µ u ν .The noise correlator is fully specified in Eq. ( 15) of Sect.II.

renormalization), and
is the momentum diffusion coefficient [21,25].As emphasized and estimated previously, the fractional order ω 3/2 is parametrically larger than second order hydrodynamics [21].However, the coefficient of the ω 3/2 terms is vanishingly small in weakly coupled theories and in strongly coupled theories at large N c , and therefore second order hydrodynamics may be an effective approximation scheme except at very small frequencies.In the con- text of holography, the ω 3/2 term can only be determined by performing a one loop calculation in the bulk [26].
In the current paper we will rederive Eq. ( 5) using a kinetic description of short wavelength hydrodynamic fluctuations.For an external driving frequency of order ω, we identify an important length scale set by equating the damping rate and the external frequency We will refer to the k * as the dissipative scale below (see also Ref. [26]).Modes with wavenumbers significantly larger than the dissipative scale, k k * , are damped and reexcited by the noise on a time scale which is short compared to period 2π/ω, and this rapid competition leads to the equilibration of these shorter wavelengths, i.e. their equal time correlation functions are given by Eq. (4).By contrast, modes with wavenumbers of order k ∼ k * have equal time correlation functions which deviate from the equilibrium expectation values.
It is notable that the wavenumbers of interest k * are large compared to ω/c s , but still small compared to microscopic wavenumbers of order the inverse mean free path.Estimating the mean free path as mfp = c s τ R , we see that the strong inequalities can be written as and thus holds whenever hydrodynamics is applicable, 1.The scale separation illustrated in Fig. 1 can be used to set up an approximation scheme where modes of order k * are treated with a kinetic or WKB type approximation scheme.We will develop the appropriate kinetic equations in Sect.II.These kinetic equations can be solved and used to determine how the two point functions of energy and momentum with wavenumbers of order k * deviate from equilibrium when driven by an external perturbation.The ω 3/2 term in Eq. ( 5) roughly represents the contribution of k 2 dk∼k 3 * slightly out of equilibrium hydro-kinetic modes per volume, with each mode contributing 1  2 T of energy to the stress tensor.Note that the contribution to the stress tensor of modes outside of the kinetic regime k k * is suppressed by phase space.Similar kinetic equations can be derived for much more general flows.We will establish the appropriate kinetic equations for a Bjorken expansion [1], which is a useful model for the early stages of a heavy ion collision.The ideal, first, and second order terms in the gradient expansions have been given in Refs.[1], [27], and [4,28] respectively.For a conformal (non-fluctuating) fluid the longitudinal pressure during a Bjorken expansion takes the form The expansion rate is ∂ µ u µ = 1/τ , and each higher term in the gradient expansion is suppressed by an integer power of 1/τ T .For Bjorken flow the expansion rate plays the role of frequency, and the distribution of sound modes are characterized by a dissipative scale analogous to Eq. ( 7) of order At this scale the viscous damping rate balances the expansion rate.These hydrodynamic modes satisfy the inequality and this strong set of inequalities can be used to determine a kinetic equation for hydrodynamic modes of order k * .The equal time correlation functions for wavenumbers of this order deviate from their equilibrium form in Eq. ( 4), and the kinetic equations precisely determine the functional dependence of this deviation.Finally, these modes contribute to the longitudinal pressure and determine first fractional power in the longitudinal pressure of a conformal fluid (analogous to Eq. ( 5)).In Sect.III we 2 The quantities k * (τ ), γη(τ ), s(τ ), . . .are all functions of time for a Bjorken expansion, e.g. for a conformal equation of state and an ideal expansion, k * (τ Throughout the paper k * ,γη, s, . . .(without a time argument) will denote the physical quantity at the final time of consideration.The explicit time argument will be used when needed, e.g.
will establish that this nonlinear correction to the longitudinal component of the stress tensor is Noise also contributes to transverse momentum fluctuations, and this contributes at quadratic order to T τ τ as we discuss in Sect.III.Thus, a complete description of a Bjorken expansion with noise must also reexamine the relationship between the background energy density e, and the one point function T τ τ .An outline of the paper is as follows.In Sect.II we consider a static fluid perturbed by an external gravitational perturbation.The purpose of this section is to introduce the kinetic equations, and to reproduce the results of the diagrammatic analysis of Refs.[21,25] using the hydro-kinetic theory adopted here.In Sect.III B we linearize the hydrodynamic equations of motion to determine the appropriate kinetic equations for a Bjorken expansion.In Sect.III C and III D we determine the solutions to the kinetic theory and use these solutions to evaluate the contribution of hydrodynamic modes to the stress tensor.We give an intuitive physical interpretation of the main results of the paper in Sect.III E. Finally we conclude with results and discussion in Sect.IV.

II. HYDRODYNAMIC FLUCTUATIONS IN A STATIC FLUID
We will first derive the kinetic equations for hydrodynamic fluctuations in homogeneous flat space in Sect.II A. The purpose here is to introduce notation, and to discuss the kinetic approximations in the simplest context.Then in Sect.II B we will perturb the system with a gravitational field and derive the appropriate kinetic theory in this case.We then use this hydro-kinetic theory to reproduce the results of loop calculations [21,25] for the renormalization of the shear viscosity and the long-time tails which characterize the hydrodynamic response due to nonlinear noise effects.

A. Relaxation equations for hydrodynamic fluctuations
To illustrate the approximations that follow and to introduce notation, we first will derive kinetic equations for the two point functions for energy and momentum density perturbations around a static homogeneous background.The (bare) background quantities of the hydrodynamic effective theory, such as the energy density, pressure, and shear viscosity (e 0 (Λ), p 0 (Λ), and η 0 (Λ) respectively) are calculated by integrating out fluctuations above a scale Λ, i.e. by excluding the contributions of hydrodynamic fluctuations with wavenumber k < Λ to the stress tensor.This is important because modes with k < Λ will not be in equilibrium when the system is perturbed by a driving force.The relation between the bare parameters and the physical quantities (which may be computed in infinite volume with lattice QCD for instance) is discussed in Sect.II B and in Ref. [21], where η 0 (Λ) is referred to as η cl (p max ).
To derive a relaxation equation for the two point functions we linearize the equations of stochastic hydrodynamics and study the eigenmodes of the system.The correlations between eigenmodes with vastly different frequencies are neglected in a kinetic (or coarse graining) approximation.For the constant background e 0 = const, and to linear order in field perturbations and stochastic fluctuations, the equations of motion (Eq.( 3)) become where γ η ≡ η 0 /(e 0 + p 0 ) is computed with bare quantities, and −ξ i is the stochastic force, −ik j S j i (t, k).Here S j i (t, k) are spatial components of the noise tensor with equilibrium correlation given by [24] It is convenient to combine Eq. ( 14) into a single matrix equation for an amalgamated field φ a = (c s δe, g j ) where ideal and dissipative terms are and the stochastic noise ξ a satisfies correlation equation At the dissipative scale the acoustic matrix L ∼ c s k * originating from ideal equations of motion dominates over the competing dissipation D and fluctuation ξ a terms.L ab has four eigenmodes: two longitudinal sound modes with λ ± = ±c s |k| and two transverse zero modes (λ T1 = λ T2 = 0).Since L drives evolution of φ a , it will be convenient to analyze the dynamics in terms of eigenmodes of L ab : where k = k/|k|, and T 1 and T 2 are two orthonormal spatial vectors perpendicular to k k = (sin θ cos ϕ, sin θ sin ϕ, cos θ), Now we will derive a relaxation equation for the two point correlation function of hydrodynamic fluctuations by defining a density matrix N ab (t, k) and analyzing the time evolution of N ab (t, k).
The analysis is most transparent in the eigenbasis, φ A ≡ φ a (e A ) a with A = +, −, T 1 , T 2 , and below we will determine the equation of motion for N AB ≡ φ A φ B where A, B = +, −, T 1 , T 2 .We note that the positive and negative sound modes φ + and φ − are related since the hydrodynamic fields are real, φ * − (k, t) = φ + (−k, t).Using the equations of motion for φ A we calculate the infinitesimal change of N AB (t + ∆t) − N AB (t), and use the equal time correlator for the noise (Eq.( 18)) to find a differential equation for where We are interested in the evolution of two point correlation functions over time scales much larger than acoustic oscillations, ∆t 1/(c s k * ).On these timescales the off-diagonal matrix elements of the density matrix, N +T1 for example, rapidly oscillate reflecting the large difference in eigenvalues, λ + − λ T1 ∼ c s k * .In a coarse graining approximation the contributions of these off-diagonal matrix elements to physical quantities can be neglected when averaged over times long compared to 1/(c s k * ).This reasoning does not apply to the diffusive modes A, B = T 1 , T 2 where both eigenvalues are zero, but rotational symmetry in the transverse xy-plane requires N T1T2 to vanish 3 .
With these approximations, the non-trivial relaxation equations of two point correlation functions in Eq. ( 22) are where is the equilibrium value for N AA (c.f.Eq. ( 4)).In the absence of external perturbations, two point correlation functions relaxes to their equilibrium values.The next step towards general kinetic equations is to study how equal time correlations are driven out of equilibrium by the presence of external fields.

B. Linear response to gravitational perturbations
In this section we will study the evolution of two point energy and momentum correlators in the presence of time varying gravitational field.We determine the kinetic equations in the time dependent background, and use these equations to reproduce the modifications of the retarded Green function (Eq.( 5)) due to thermal fluctuations, which were previously found by a one-loop calculation [21,25].
A straightforward way of introducing an external source to equations of motion is to study fluctuating hydrodynamics in the presence of a small metric perturbation, g µν = η µν + h µν .The Green function records the response of T µν to the metric perturbation For a constant homogeneous background with time dependent metric perturbation h ij (t), symmetry constrains the form of the retarded Green function and therefore we can obtain the Green function in Eq. ( 5), i.e.GR (ω), by considering a diagonal traceless metric perturbation, In the presence of metric perturbations and thermal fluctuations, the energy momentum tensor is where the nonlinear term stems from the constitutive relation of ideal hydrodynamics, T ij = p 0 δ ij +(e 0 +p 0 )u i u j .The averaged squared momentum, g i (t, x)g j (t, x) , is related to the two-point functions of g i in k space as In this integral, the equilibrium value of N ij and its first viscous correction will renormalize p 0 and η 0 (see below), while the finite remainder will determine the first fractional power in the stress tensor correlator ∝ ω 3/2 .Studying the hydrodynamic equations in Eq. ( 3), and neglecting metric perturbations of the dissipative terms, we find that the linearized equations of motion are identical to flat background Eq. ( 14), but now there is a difference between covariant and contravariant indices To avoid this complication, we use a vielbein formalism and scale the spatial components of momentum and wavenumber by √ g ij , i.e. g i and k j are replaced by where now the position of hatted indices is unimportant.Analogously to Eq. ( 16), we obtain a matrix equation for with an additional metric dependent source term which drives the hydrodynamic fluctuations away from equilibrium.The eigenbasis of L (see Eq. ( 19)) is now defined with respect to the time dependent vector K(t), but remains orthonormal at all times.Furthermore, the metric perturbation preserves rotational symmetry in the transverse xy-plane, and this guarantees that the T 1 and T 2 modes are not mixed by the time-dependent perturbation.Thus, the only non-trivial diagonal components of the symmetrized energy and momentum two point functions are We can find a perturbative solution to these equations for a small periodic metric perturbation, e.g.
(34) To find the correction to the energy momentum tensor due to the nonlinear momentum fluctuations in Eq. ( 27), we need to perform the k space integral in Eq. ( 28) Note, care should be taken when transforming the zeroth order value N AA = N 0 to original unhatted basis as it produces terms linear in metric perturbation.The modification of the response function GR (ω) due to the momentum fluctuations (i.e. the last term in Eq. ( 27)) is Performing K-space integral with UV cutoff, K max = Λ, and adding the remaining terms in Eq. ( 27), we find in agreement with previous work [21,25].The first two terms in Eq. ( 37) are the renormalized pressure (p ≡ p 0 (Λ) + O(T Λ 3 )) and shear viscosity (η ≡ η 0 (Λ) + O(ΛT 2 )) as discussed previously [21].Further discussion of the renormalization of these quantities is given in the next section when the expanding case is presented.The last term is the finite nonlinear modification of the medium response, and agrees with loop calculations in equilibrium.The kinetic approach outlined in this section has the advantage that it can be readily applied to more general backgrounds, and we will exploit this advantage to calculate the analogous correction for a Bjorken expansion in the next section.In contrast to the linear response described here, the deviation from equilibrium in the expanding case is of order unity.Consequently, computing the first fractional power in an expanding system with the diagrammatic formalism would require an extensive resummation, which would invariably reproduce kinetic calculation described in the next section [29].

III. HYDRODYNAMIC FLUCTUATIONS FOR A BJORKEN EXPANSION
In this section we will derive the kinetic evolution equations for hydrodynamic fluctuations during a Bjorken expansion.We consider a neutral conformal fluid, for which c 2 s = 1/3, ζ = 0, and µ B = 0.In Bjorken coordinates the energy and momentum conservation laws are with Γ τ ηη = τ and Γ η τ η = Γ η ητ = 1/τ [19].For hydrodynamics without noise the background flow fields are independent of transverse coordinates and rapidity and satisfy where roman indices, i, j . .., run over transverse coordinates x, y.The transverse momentum T τ i is constant, and can be chosen to be zero.In hydrodynamics T τ τ and τ 2 T ηη are related by constitutive equations Note that in τ 2 T ηη the viscous correction is of order = η/(e + p)τ 1 smaller than the ideal part, and the solution is approximately e(τ ) = e(τ 0 ) We will consider the evolution of linearized fluctuations on top of this background.The effect of these fluctuations on the background evolution can then be included as a correction after the two point functions are known, i.e.
where the constitutive relations take the form Here and below e(τ ) is the average rest frame energy density 4 ; G is the momentum density G = (T τ x , T τ y , τ T τ η ), and all quantities are renormalized as explained more completely below.
There are two sorts of fluctuations to consider: fluctuations in the initial conditions (which are long range in rapidity), and hydrodynamic fluctuations stemming from thermal noise (which are short range in rapidity).The average over the initial conditions and noise are denoted with . . .τ0 and . . .respectively, while the average over both fluctuations is denoted with the double brackets . . . .Since the transverse momentum per rapidity is conserved for boost invariant fields, approximately boost invariant initial fluctuations in τ T τ i remain important at late times.In Sect.III A we study initial transverse momentum fluctuations, while in remainder of the paper we complete our study of thermal fluctuations during a Bjorken expansion.

A. A Bjorken expansion with initial transverse momentum fluctuations
After the initial passage of two large nuclei in a specific event, each rapidity interval contains a finite amount of transverse momentum, although the event-averaged transverse momentum per rapidity is zero.This initial transverse momentum is spread over a large rapidity range by the subsequent re-scatterings in the initial state.Ultimately, this dynamical process can be described by (transverse) momentum diffusion in rapidity, and can be modeled with hydrodynamics and noise -see Sect.III D.Here we will determine how long-range transverse momentum fluctuations in the initial state influence the evolution of the background energy density at late times.
As a model for the initial conditions in the x, y plane, we take Gaussian statistics for the initial transverse momentum fluctuations where g i ⊥ (τ, x ⊥ ) ≡ T τ i is approximately independent of rapidity, so that each (large) rapidity interval is approximately boost invariant.Integrating over the transverse area A, the total transverse momentum per rapidity, 4 e(τ ) notates the average rest frame energy density and does not fluctuate; T τ τ is the average energy density.In general, the rest frame energy density e + δe in a finite volume would be estimated from sample estimate of T τ τ and G through the (ideal) constitutive equations, e+δe T τ τ − G 2 (1+c 2 s )T τ τ .Thus e is given by Eq. ( 44), and δe δT τ τ − δ( G 2 /T τ τ )/(1 + c 2 s ) δT τ τ .
fluctuates from event to event with a scaled variance of To find out how this fluctuating initial condition changes the evolution of the system, we linearize the equations of motion of viscous hydrodynamics and Fourier transform with respect to the transverse coordinates The full equations of motion are given in the next section, see Eq. ( 56).Decomposing the transverse momentum fluctuation into longitudinal and transverse pieces with ki ⊥ g i T = 0 and g i L = ki ⊥ kj ⊥ g j ⊥ , we find that the transverse piece obeys a two dimensional diffusion equation with initial conditions specified by Eq. ( 46) Solving the diffusion equation with a time dependent diffusion constant γ η ∝ τ c 2 s , we see that the variance at a specified space time point due to the fluctuating initial conditions is5 Thus, we see that a fluctuating initial conditions contributes quadratically to the average stress tensor where p = c 2 s e.

B. Kinetic equations of hydrodynamic fluctuations
To derive the kinetic equations we will follow the strategy of Sect.II A, and expand all fluctuations in Fourier modes conjugate to transverse coordinates and rapidity, e.g.
Following the procedure outlined in Sect.II we rewrite Eqs.(56) in a compact matrix notation.We define with noise correlator Here and the factor of 1/τ stems from the Jacobian of the coordinate system δ 4 (x − x )/ g(x).
The kinetic equation for the two-point functions is obtained similarly to Sect.II (62) The eigenvectors of L are of the same form as before, Eq. ( 19), However, now the wavenumber vector K is time dependent The azimuthal angle ϕ K is independent of time due to the residual rotational symmetry of the background in xy-plane.Following the same arguments as in Sect.II, we arrive at the kinetic equations for diagonal components The first terms on the right hand side describe relaxation of N AA toward local equilibrium T (e 0 + p 0 )/τ , and the second terms drive N AA out of equilibrium through the interaction with the background flow.
We derived these equations relying on the scale separation given in Eq. ( 12).The off-diagonal components between gapped modes (such as between the ± and T 1 and T 2 modes) are ignored because they rapidly rotate as discussed in Sect.II A. Note that the transverse mode φ T1 is so chosen that it does not mix with the other modes.This is possible because of the residual rotational symmetry in the xy-plane in the Bjorken expansion.Therefore the kinetic equation for N T1T1 , Eq. (65b), holds without the scale separation in Eq. ( 12) and is applicable for all wavenumbers k from to zero to 1/ mfp .
hydro-kinetics visc.approx FIG. 2. Steady state solutions of Eq. ( 65) for the two point energy-momentum correlation functions during a Bjorken expansion at late times, τ τ0.The correlations are plotted as a function of K[γητ ] 1/2 for final time angle cos θK = 0.1.For comparison leading order viscous solutions in 1/(γηK 2 τ ) are also shown, Eq. ( 66).The differences of the steady state solutions from their asymptotic forms induces finite corrections to energy-momentum tensor, Eq. (74).

C. Nonlinear fluctuations in the energy momentum tensor
Now let us investigate the solution of the kinetic equations close to the cutoff and isolate the UV divergent contribution.Solving Eq. (65) in series of 1/(γ η K 2 τ ) we obtain an asymptotic solution for large where we used ∂ τ [T (e 0 + p 0 )] −(1 + 2c 2 s )[T (e 0 + p 0 )]/τ which is adequate for the desired accuracy of the present analysis.For a given K 2 γ η τ = (K/k * ) 2 and θ K at final time τ , we can solve Eq. ( 65) numerically and find a steady state solution at late time τ τ 0 .We compare this steady state solution to the asymptotic form Eq. (66) in Fig. 2.
Eq. ( 66) is analogous to the ideal and first viscous correction to the thermal distribution function, f 0 + δf , which are used in heavy ion phenomenology and in determining the shear viscosity [19].At large K/k * the distribution N AA attains its equilibrium value, T (e 0 + p 0 )/τ , up to viscous corrections of order τ R /τ , where τ R is a typical relaxation time for a mode of momentum The energy-momentum tensor averaged over fluctuations is given by Calculating Note that the momentum integral is done in final time variables, K(τ ).As shown below, the integration in Fourier space is divergent in the ultraviolet.Therefore, we regulate the integral by introducing a cutoff at | K| ∼ Λ.In turn, the background quantities such as e 0 and η 0 must be renormalized and depend on Λ so that the total result is independent of Λ.The choice of Λ is arbitrary as long as k * Λ 1/ mfp so that the nonlinear contribution with | K| ∼ Λ is independent of the background flow.
The integration in Eq. ( 68) includes the soft fluctuations for which the kinetic equation may not be applicable.However, this contribution is suppressed by phase space and the kinetic result can be extrapolated into this regime with negligible errors.
Combining Eqs.(66), ( 67) and (68), the energy momentum tensor is obtained as where the finite contributions ∆T τ τ , ∆T xx , ∆T yy , and τ 2 ∆T ηη are discussed in the next section.By comparing terms with the same explicit τ dependence, the ultraviolet divergences are absorbed into the renormalized hydrodynamic variables T (e 0 (Λ) + p 0 (Λ)) Note that we do not assign a cut-off dependence to the temperature.The coefficients of the cubic and linear renormalizations of the pressure and shear viscosity are independent of the background expansion, and match the static fluid results of Sect.II B.Here e, p, and η are physical quantities at a given temperature T in an infinite volume.Using the physical quantities, the energymomentum tensor is given as If the two-point functions of the fluctuations were completely determined by the first two terms in Eq. ( 66), their contributions would be completely absorbed by the renormalization of the background flow parameters such as p 0 (Λ) and η 0 (Λ).However, the kinetic equations yield residual contributions, since the full solution deviates from its asymptotic form for K ∼ k * as seen from Fig. 2.
The purpose of hydrodynamics with noise is to capture this contribution.Physically, the parameters e 0 (Λ), p 0 (Λ), and η 0 (Λ) in fluctuating hydrodynamics reflect the equilibrium properties of modes above a cutoff Λ, which have been already integrated out.Equivalently, these parameters are determined by modes contained in a cell of size a ∼ 2π/Λ.For example, p 0 (Λ) is the partial pressure from equilibrated modes above the cutoff (inside a cell), while the partial pressure from the modes below the cutoff (larger than a cell size) is determined dynamically with fluctuating hydrodynamics.The second terms on the right hand sides of Eq. (70) are the contributions to each quantity from the modes below the cutoff, when all of these long wavelength modes are in perfect equilibrium in infinite volume.

D. Out of equilibrium noise contributions to energy momentum tensor
In this section we determine the residual contributions to the energy momentum tensor, ∆T µν , after the hydro-dynamic parameters have been renormalized.We evaluate the precise numerical factors of the long-time tail terms for a Bjorken expansion (which is the main result of this paper), and identify additional contributions from the noise at early times.The mathematical procedure is somewhat involved, so here we outline the calculation and present results, delegating the technical details to the Appendix A.
To find the full out of equilibrium correlators we need to solve Eq. ( 65), which can be written in the following general form where f (τ, k) has contributions from both the dissipative and external forcing terms, and g(τ, k) is the inhomogeneous term coming from the equilibrium correlation functions.A formal solution of Eq. ( 72) is given by The first term describes the evolution of the initial correlation density matrix N AA (τ 0 , k) to final time τ .The second term in Eq. ( 73) is the contribution from thermal fluctuations.As we will see, only the N T1T1 contribution is sensitive to the initial conditions and the thermal fluctuations at early times.For the T 1 T 1 correlator we will take the initial conditions described by Eq. (52) in Sect.III A, τ 2 0 N T1,T1 (τ 0 , k) = χ gg τ0 2πδ(κ).
Substituting the formal solution for N AA in Eq. (67) and Eq. ( 68) we can determine the stress tensor at time τ τ 0 .The integral d 3 k in Eq. ( 68) diverges, but after subtracting Λ 3 and Λ divergences discussed in the previous section, we obtain the finite correction to energy momentum tensor with numerically determined coefficients of the long-time tails, 1/(γ η τ ) 3/2 .Here χ gg τ0 + δχ gg τ0 records the initial variance in transverse momentum in a given rapidity slice (see Eq. (46) and Eq. ( 48)) together with the thermal contribution where the brackets (. ..) τ0 indicate that all contained quantities are to be evaluated at the initial time, τ 0 .We will discuss the result in the next section.
Squared fluctuations in equilibrium are of order δe( x)δe( y) eq /e 2 ∼ v i ( x)v j ( y) eq ∼ s −1 δ( x− y).Thus a fluctuation with wavenumber k is suppressed by k 3 /s.The suppression factor k 3 /s is roughly the inverse of the degrees of freedom inside a box of volume ∆V ∼ (1/k) 3 , which must be a huge number for local thermodynamics to apply.This is why the linear analysis of the hydrodynamic fluctuations is justified.
The energy momentum tensor in viscous hydrodynamics is expanded in powers of gradients, leading to corrections in powers of ≡ η/(e + p)τ 1.In addition, the fluctuations with wavenumber of order | K| ∼ k * provide a nonlinear correction to the stress tensor, which is suppressed by s∆V ≡ s/k 3 * 1.This correction to the longitudinal pressure reflects the equipartition of energy, with 1  2 T of energy per mode, and the number of non-equilibrium modes per volume ∼ k 3 * .To summarize, the reasoning in this paragraph leads to the following parametric estimate for the longitudinal stress which is reflected by Eq. (74).

2.
Transverse momentum diffusion in rapidity: 1/γητ Additional corrections to the stress in Eq. (74) decrease as 1/γ η τ , in contrast to the long time tails.As described in Sect.III A, long range (in rapidity) initial transverse momentum fluctuations correct the mean transverse pressures, T xx and T yy , by a term proportional to χ gg τ0 /γ η τ (see Eq. ( 74b)).Hydrodynamic noise in the initial state adjusts this correction by adding to the long range fluctuations of transverse momentum (see Eqs. (74b) and ( 75)).The goal of this section is to explain this process qualitatively, and to quantitatively explain the adjustment, χ gg τ0 → χ gg τ0 + δχ gg τ0 .Formally, the N T1T1 correlation function is sensitive to the noise at the initial time τ 0 , which arises from a restricted region of K-space integration, k ⊥ ∼ k * and κ/τ ∼ k * (τ 0 )(τ 0 /τ ) ∼ k * (τ 0 /τ ) 1/3  k * .In this region the longitudinal momentum κ/τ 0 reflects the dissipative scale k * (τ 0 ) at the initial time τ 0 , while the transverse momenta reflect the dissipative scale at final time τ .
The dynamics in this phase space region is the following.During the initial moments, thermal fluctuations lead to a local fluctuation of transverse momentum in a given rapidity slice for each cell in the transverse plane Here (as before) the brackets (. ..) τ0 indicate that all contained quantities should be evaluated at τ 0 .During an initial time of order τ 0 , the momentum per rapidity diffuses to a finite longitudinal width [5] (see below) The process is diffusive because the transverse momentum per rapidity is conserved.The rapidity width is finite because the longitudinal expansion shuts off the diffusion process.σ η (τ 0 ) is broader than the rapidity width of subsequent interest, which is of order σ η (τ ).Thus, after an initial transient, the transverse momentum per rapidity may be considered approximately constant in time and rapidity, though localized in this transverse plane At much later times these transverse momentum fluctuations diffuse transversely (as described in Sect.III A) leading to a correction of order which qualitatively reproduces the correction in Eq. (74b).Now we will briefly sketch this reasoning with equations.At the early time moments τ ∼ τ 0 , the wave vector is predominantly longitudinal K ( 0 ⊥ , κ/τ ) and the transverse momentum correlator can be reconstructed from N T1T1 and N T2T2 since T 1 and T 2 form a basis for the transverse plane.In this limit, the equations of motion for N T1T1 and N T2T2 (see Eq. (65b) and Eq. ( 65c)) are the same, and N ij satisfies a one dimensional diffusion equation with a source at early times (83) The lhs of Eq. ( 83) represents the diffusion of transverse momentum in rapidity, while the rhs represents the thermal transverse momentum fluctuations at the earliest moments, which act as a source.The source for the fluctuations, 2T η (κ/τ ) 2 , is a rapidly decreasing function of time, and is dominant for times of order τ 0 .
The Green function propagating data from τ to τ for the lhs of Eq. ( 83) is for τ τ .Thus, a fluctuation localized in rapidity at time τ 0 will diffuse to a finite rapidity width of σ η (τ 0 ) = 6γ η (τ 0 )/τ 0 at late times 6 [5,30].This is a small rapidity width in absolute units (since γ η (τ 0 )/τ 0 1 when hydrodynamics is a good approximation), but much broader than the rapidity width of interest at the final time, γ η (τ 0 )/τ 0 γ η (τ )/τ . 6In Refs.[5,30] the authors consider an initial distribution which is Gaussian in rapidity of width σ 0 .During the expansion the width is broadened by the diffusion process These authors considered constant η/(e + p) and found a factor of 4 rather than 6 in Eq. (85).
Returning to Eq. ( 83), we solve the equation, and determine the transverse momentum correlation function (in the same rapidity slice) at an intermediate time τ which is large compared to τ 0 but much much less than the final time τ , τ 0 τ τ Implementing these steps we find This has the same form as the initial conditions described in Sect.III A, and fluctuations at the earliest moments simply increase the variance of long range transverse momentum fluctuations by a constant amount reproducing Eq. ( 75).In a sense, this constant shift simply finalizes the thermalization process described at the start of Sect.III A. The correction δχ gg τ0 scales as τ −1/3 0 and is therefore small compared to the first term in Eq. (75) if τ 0 is large compared to a typical thermalization time.

IV. RESULTS AND DISCUSSION
In this paper we determined a set of kinetic equations which describe the evolution of hydrodynamic fluctuations during a Bjorken expansion.We used these equations to find the first fractional power correction to the longitudinal pressure, ∝ 1/(τ T ) 3/2 , at late times.The evolution equations can be extended to much more general flows, and ultimately coupled to existing hydrodynamic codes.
The kinetic equations for hydrodynamic fluctuations are a WKB (or rotating wave) type approximation of the full stochastic hydrodynamic evolution equations.This approximation is justified because the relevant hydrodynamic modes have wavenumbers of order which is large compared to the inverse expansion rate, 1/τ .For example, the kinetic equation for the sound mode with wavenumber K = ( k ⊥ , κ/τ ) interacting with the Bjorken background takes the form of a relaxation type equation N ++ (τ, k) are short wavelength (symmetrized) two point functions of conserved stress tensor components, φ + ≡ (c s δe + K • G)/ √ 2 in an evolving Bjorken hydrodynamic background (see Sect.III and Eq. ( 65) for the remaining modes).At high wavenumbers K k * , the distribution function N ++ reaches its equilibrium form T (e 0 + p 0 )/τ , up to first viscous corrections which may be found by solving Eq. (90) order by order at large K/k * (see Eq. ( 66a)).This asymptotic form is responsible for the renormalization of the pressure and shear viscosity.For wavenumbers of order k * the hydrodynamic fluctuations are not in equilibrium at all, but reach a nonequilibrium steady state at late times.A graph of this non-equilibrium steady state is given in Fig. 2.
The deviation of hydrodynamic fluctuations from equilibrium has consequences for the evolution of the system.Indeed, the longitudinal pressure τ 2 T ηη receives a correction from the unequilibrated modes where we have repeated Eq. (74a) for convenience.The correction to the pressure ∼T /(γ η τ ) 3/2 is of order ∼T k 3 * , reflecting the number of modes of order k * and the energy per mode, 1  2 T .In contrast to all previous analyses of long-time tails [21,25], the hydrodynamic fluctuations in the expanding case are not close to equilibrium, and a loop expansion around equilibrium is not an appropriate approximation scheme.Formally, the noise correction is lower order than the correction due to second order hydrodynamics, which is proportional to a particular combination of second order parameters, λ 1 − ητ π .To quantify the importance of thermal fluctuations in practice, we take representative numbers for the entropy from the lattice [31,32], estimates for the second order hydrodynamic coefficients based on weakly and strongly coupled plasmas [4,28,33], and an estimate for τ T at τ ∼ 3.5 fm based on hydrody-namic simulations 7 For the smaller shear viscosity, Eq. ( 93), the nonlinear noise contribution completely dominates over the second order hydro contribution.For the larger shear viscosity, Eq. ( 94), the noise remains three times larger than second order hydro, but this contribution is only a ∼ 10% of the first order viscous term.The evolution of the average energy density of the system obeys where the double brackets notate an average over (long range in rapidity) initial conditions and thermal noise 8 .To close the system of equations, the relationship between average energy density T τ τ and the average rest frame energy density e(τ ) must be specified, this relation is given in Eq. (74).T τ τ , T xx , and T yy are sensitive to hydrodynamic noise at the earliest moments in addition to the long-time tails.In these cases thermal noise in the initial state adds to the long-range rapidity correlation functions of transverse momentum, which are already present without noise.This result is encapsulated by Eq. ( 75) and is discussed in Sect.III A and Sect.III E 2. 7 We take an estimate for the (approximately constant) average entropy in the transverse plane from a recent LHC simulation for PbPb collisions at √ s = 2.76 TeV/nucleon, τos(τo) 4.0 GeV 2 [34].We take a time of τ ∼ 3.5 fm (which is the time at which the elliptic flow develops [19]), where T 250 MeV. 8 The longitudinal pressure in Eq. ( 91) is independent of fluctuations in the initial conditions at late times.Thus, only the average over the noise is relevant in this case, τ 2 T ηη = τ 2 T ηη Although the analysis of hydrodynamic fluctuations in this paper was limited to conformal neutral fluids and a Bjorken expansion, the techniques developed here can be applied to much more general flows, such as the Hubble expansion.In addition, it will be phenomenologically important to extend this work to non-conformal systems with net baryon number.Near the QCD critical point the noise will continue to grow without bound, leading to a critical renormalization of the bulk viscosity.In an expanding system these fluctuations will not be fully equilibrated.We believe the formalism set up in this paper provides the first steps towards quantitatively analyzing this rich dynamical regime.AA integrals for energy momentum tensor corrections.For the special case of d 3 rR (r) T 1 T 1 the remaining one dimensional time integral can be done analytically.

FIG. 1 .
FIG. 1.The hydro-kinetic description of noise is based on the separation of scales between the long wavelength hydrodynamic background (with k ∼ 1/csτ ), and shorter wavelength hydrodynamic fluctuations (with k ∼ k * ≡ 1/ √ γητ ).The wavelengths of the hydrodynamic fluctuations are still much longer than microscopic mean free path.The hydrodynamic fluctuations are driven out of equilibrium by the expanding background, and this deviation is the origin of the long-time tail correction to the stress tensor.