Abstract
We investigate the impact of hydrodynamic fluctuations on correlation functions in a scale invariant fluid with a conserved $\text{U}(1)$ charge. The kinetic equations for the two-point functions of pressure, momentum, and heat energy densities are derived within the framework of stochastic hydrodynamics. The leading nonanalytic contributions to the energy-momentum tensor as well as the $\text{U}(1)$ current are determined from the solutions to these kinetic equations. In the case of a static homogeneous background we show that the long time tails obtained from hydrokinetic equations reproduce the one-loop results derived from statistical field theory. We use these results to establish bounds on transport coefficients. We generalize the stochastic equation to a background flow undergoing Bjorken expansion. We compute the leading fractional power $\mathcal{O}({(\ensuremath{\tau}T)}^{\ensuremath{-}3/2})$ correction to the $\text{U}(1)$ current and compare with the first-order gradient term.
Highlights
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 [1,2,3]
Several authors have derived deterministic equations for the time evolution of hydrodynamic n-point functions [14,15,16]. These methods rely on linearizing the fluid dynamic equations, and on truncations in the number of moments, but the resulting equations are easier to evolve in time, and the comparison to analytical results for homogeneous systems is more straightforward
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
Summary
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 [1,2,3]. 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. Several authors have derived deterministic equations for the time evolution of hydrodynamic n-point functions [14,15,16] These methods rely on linearizing the fluid dynamic equations, and on truncations in the number of moments, but the resulting equations are easier to evolve in time, and the comparison to analytical results for homogeneous systems is more straightforward. We denote the spatial measure of the wd3hkil/e[(th2eπt)h3r√ee−-dgi]m≡enskio. nal measure
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