# Hydrodynamic Fluctuations

**Mauricio Martinez Guerrero**

NCSU

The theory of hydrodynamic fluctuations was motivated originally to solve the tension between the fluctuation-dissipation theorem of statistical mechanics and the lack of noise stochastic terms in the Navier-Stokes equations. The origin of these type of 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 non-linear 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", which are non-analytic contributions to the time (or frequency) dependence of correlation functions. In this talk I review the recent developments on the propagation of hydrodynamic fluctuations. We discuss the emergence of long time tails in relativistic and non-relativistic systems and how the presence of these non-analytic terms leads to physical lower bounds of the transport coefficients. I will also present a novel method to study stochastic hydrodynamics for rapidly expanding systems by developing a new set of kinetic equations of the hydrodynamic correlations. For a charged fluid undergoing Bjorken flow expansion it is shown that the hydrodynamic fluctuations contributions to the macroscopic variables of the fluid are larger than the typical second order hydrodynamic corrections. If time allows it I shall discuss the propagation of fluctuations in QCD near a possible critical endpoint in the phase diagram.