# Relativistic Hydrodynamic Fluctuations

## TL;DR

This paper develops a comprehensive formalism for analyzing fluctuations in relativistic hydrodynamic flows, incorporating feedback effects and providing a framework that resembles kinetic equations, with applications to numerical simulations.

## Contribution

It introduces a systematic approach to describe relativistic hydrodynamic fluctuations using equal-time correlation functions and confluent Wigner transforms, including renormalization techniques.

## Key findings

- Derived equations for confluent Wigner functions resembling kinetic equations.
- Established the equivalence between phonon propagation and Wigner function dynamics.
- Presented a renormalization procedure for short-distance singularities.

## Abstract

We present a general systematic formalism for describing dynamics of fluctuations in an arbitrary relativistic hydrodynamic flow, including their feedback (known as long-time hydrodynamic tails). The fluctuations are described by two-point equal-time correlation functions. We introduce a definition of equal time in a situation where the local rest frame is determined by the local flow velocity, and a method of taking derivatives and Wigner transforms of such equal-time correlation functions, which we call confluent. We find that the equations for confluent Wigner functions not only resemble kinetic equations, but that the kinetic equation for phonons propagating on an arbitrary background nontrivially matches the equations for Wigner functions, including relativistic inertial and Coriolis forces due to acceleration and vorticity of the flow. We also describe the procedure of renormalization of short-distance singularities which eliminates cutoff dependence, allowing efficient numerical implementation of these equations.