^{3}.

We investigate the impact of hydrodynamic fluctuations on correlation functions in a scale invariant fluid with a conserved

The hydrodynamic description of relativistic heavy ion collisions at the Relativistic Heavy Ion Collider (RHIC) and the Large Hadron Collider (LHC) has been extremely successful

The role of thermodynamic fluctuations during the evolution of the system is much less explored. These fluctuations arise from the fact that fluid dynamics is a coarse-grained description, so that at any coarse graining scale unresolved microscopic degrees of freedom lead to fluctuations of the macroscopic variables. These effects are interesting, because hydrodynamics is a nonlinear theory, so that couplings between hydrodynamic modes lead to novel phenomena beyond just Gaussian fluctuations in the hydrodynamic variables. For example, it is known that fluctuations lead to hydrodynamic “tails”

More importantly, hydrodynamic fluctuations dominate the dynamics near a critical point, and any fully dynamical description of fluctuations of conserved charges in a heavy ion collision must include fluctuations in the hydrodynamic evolution equations. This implies that the study of hydrodynamic fluctuations is crucial for interpreting the results from the RHIC beam energy scan program.

There are a variety of techniques for taking int account hydrodynamic fluctuations. In homogeneous systems fluctuations have been studied using diagrammatic or effective action methods

In this work we adopt the latter approach, extending a formalism developed by Akamatsu

We use the following conventions: We use a bar to denote background values of the hydrodynamic variables. Four dimensional indices are denoted by greek letters

In this section we describe the theoretical framework for studying the evolution of hydrodynamic fluctuations on top of an evolving background for a charged fluid. We consider a relativistic fluid with an unbroken global

The local conservation laws for this system are

In the Landau frame where

In the previous equations the macroscopic quantities of the fluid are the isotropic pressure

In order to solve the conservation laws of stochastic hydrodynamics

The set of equations

In the leading order of the perturbation each hydrodynamical field entering in the right-hand side (RHS) of the previous equations is replaced by its mean field value. These approximations will allow us to recast the problem of solving the equations of stochastic hydrodynamics onto a set of coupled Langevin equations.

In this work we consider a conformally invariant fluid whose equation of state (EOS) is

Hydrodynamic tails appear in the long time behavior of response functions. We compute these response functions by coupling the hydrodynamic evolution in the presence of noise terms to external fields, and then determine the variational derivative of currents with respect to the sources. Specifically, the metric can be used to study response functions of the stress tensor, and the external gauge field determines the current response.

In order to study diffusive terms in the shear response in a static background it is sufficient to consider a metric of the type ^{1}

Equation

In this section we will derive the evolution equation for hydrodynamic fluctuations in the presence of noise, the linearized Navier-Stokes Langevin equation. We begin by linearizing the equations of motion of relativistic fluid dynamics around a static background. We write the hydrodynamic variables as
^{2}

The spatial Fourier transform is defined as

Here, we have defined the electric field by

The natural degrees of freedom of Eqs.

In this basis the momentum flux density

where we have defined the diffusion coefficient

At leading order (LO) the acoustic matrix

At next-to-leading order (NLO) nondiagonal terms in the matrix

The source matrix

The solutions of the NSL equations

The general solution of the NSL equations

If one performs a stochastic average over the solution

Information on hydrodynamic fluctuations is contained in two-point and higher n-point correlation functions. We define the symmetric two point correlation function at equal times as^{3}

In an effective hydrodynamic theory where the fields are purely classical there is no distinction between the symmetrized correlator

The equation of motion for the correlation functions

For a scale-invariant theory and in the limit when

In this limit the response functions

The operator

In hydrodynamics we can read off the variation of the spatial components of the energy-momentum tensor

Here, we have expressed the symmetric correlation matrix

As a first application of this formalism we compute the low frequency behavior of the response in the static case. For harmonic perturbations

In Eq.

Here, we have neglected terms that correspond to rapidly oscillating modes

We observe that the leading term in the stress tensor response, Eqs.

We have all the ingredients to calculate the response functions

Using the same methods we can determine the current response function ^{4}

It is important to mention that in Sec. III of Ref.

The imaginary part of the response function determines the spectral function, and it can be used to define frequency dependent diffusion constants. We find

where the renormalized momentum diffusion constant

We note that the spectral function has a

The renormalized transport coefficients, Eqs.

More formally, we can view this bound as arising from a matching procedure. Consider a UV scale

The largest possible cutoff scale is determined by the regime of validity of fluid dynamics. Consider a diffusive shear mode. The decay rate is proportional to

For

By extremizing the first of these expressions with respect to

We now come to the central problem studied in this work: extending the calculation of diffusive tails to an expanding background. We will consider a conformal U(1) charged fluid undergoing longitudinal boost invariant expansion, known as Bjorken flow. The symmetries of the Bjorken flow are manifest in Milne coordinates

In the absence of external sources and noise the conservation laws provide the evolution equations for the background hydrodynamic fields. Bjorken flow corresponds to the well known equations

where we have neglected dissipative terms. For the conformal EOS

The evolution equations for the hydrodynamic variables in the presence of external fields and noise can be obtained using the procedure discussed in Sec. ^{5}

Both

The acoustic matrix has the same structure as its static counterpart

The evolution equations for the symmetric correlation functions are determined by following the procedure, discussed in Sec.

Here we assume that the background is slowly evolving, so that

The equations of motion for the Bjorken flow have a mathematical structure similar to their static counterparts

For the Bjorken flow the particle current in stochastic fluid dynamics is

In Eq.

where the previous expressions are determined by comparing orders with the same power law

In this section we determine finite cutoff-independent contributions to the particle current. We follow closely the procedure outlined in

From the formalism discussed in Sec.

Using the exact solution

The coefficients appearing in the RHS of the previous expressions were calculated numerically, as indicated in Appendix

In this work we have studied the role of hydrodynamic fluctuations in a conformal fluid with a conserved

Hydrodynamic fluctuations imply the existence of lower bounds on the transport coefficients. These bounds are purely classical and they are not universal. In fact, the lower bounds depend on the thermodynamic properties and the typical momentum scale at which hydrodynamics breaks down. The lower bound on the diffusion coefficient is enhanced when the value of the ratio of shear viscosity to entropy density is small, and thus the experimental determination of this type of transport coefficient has the potential to determine the impact of hydrodynamic fluctuations in relativistic heavy ion collisions.

In the case of a background undergoing boost invariant Bjorken flow, the constitutive relations for the particle current in the presence of hydrodynamic fluctuations are

Renormalized thermodynamic variables and transport coefficients due to hydrodynamic fluctuations in different approaches.

We are now in a position to estimate the relative size of fluctuation corrections to the

There are a number of issues that can be studied in the future. On the technical side it would be interesting to study convergence properties and higher order asymptotics of the long time expansion in the presence of fluctuations. This is related to resurgent asymptotics, which has received a considerable amount of attention in the context of both kinetic theory and holographic dualities. From a phenomenological point of view it is crucial to extend this formalism to include critical behavior

We would like to thank D. Teaney, M. Stephanov, J. Kapusta, A. Cherman, G. Basar, and Ho-Ung Yee for enlightening discussions, and the members of the Nuclear Theory Group at NCSU for advice on the regularization of divergent integrals. This work was supported in part by the U.S. Department of Energy Grant No. DE-FG02-03ER41260 and by the BEST (Beam Energy Scan Theory) DOE Topical Collaboration.

In this Appendix we describe the thermodynamic relations of a relativistic fluids with a conserved

The thermal expansion coefficient is

Thus one can express the thermodynamic coefficients

Now for the conformal EOS given by

It is straightforward to show that for the EOS

In this section we present the calculation of the time symmetrized (unordered) correlator

Now, in the absence of external and noise sources the linearized hydrodynamical equations for

For

The explicit functional form of each of the components of

In order to make connection with correlators one follows the well known procedure of linear response theory

Thus the

On the other hand we can reconstruct the two-point

where again one considers that close to equilibrium the fluctuations of the pressure and heat energy density as well as the pairs of momentum flux components are independent

In order to calculate the independent cutoff contributions to the charge current

where

with

In the derivation of Eqs.

By following a similar procedure we find that the differential equations for the response ratio functions

Now we define the residual function

In the following we present the generic method used in this work to obtain the residual contributions to the particle current. We calculate explicitly the mixed diffusive-sound contribution appearing in Eqs.

By using the same regularization procedure for each integral in Eq.

The sum of the different contributions gives rise to the numerical values of the particle current quoted in Eq.