# Stochastic hydrodynamics and long time tails of an expanding conformal charged fluid

## TL;DR

This paper studies how hydrodynamic fluctuations affect correlation functions in a conformal charged fluid, deriving kinetic equations and analyzing long-time tails, with applications to static and expanding backgrounds.

## Contribution

It derives kinetic equations for correlation functions in stochastic hydrodynamics of a conformal charged fluid and analyzes long-time tails and transport bounds.

## Key findings

- Long time tails match one-loop field theory results.
- Bounds on transport coefficients are established.
- Fractional power corrections in expanding backgrounds are computed.

## Abstract

We investigate the impact of hydrodynamic fluctuations on correlation functions in a scale invariant fluid with a conserved $U(1)$ 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 non-analytic contributions to the energy-momentum tensor as well as the $U(1)$ 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 hydro-kinetic 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 $\mathcal{O}((\tau T)^{-3/2})$ correction to the $U(1)$ current and compare with the first order gradient term.