Out-of-equilibrium hydrodynamic fluctuations in the expanding QGP

We develop a set of kinetic equations for a correlator of thermal fluctuations which are equivalent to nonlinear hydro-dynamics with noise. We first show that the kinetic response precisely reproduces the one-loop renormalization of the shear viscosity for a static fluid previously discussed by Kovtun, Moore and Romatschke. We then use the hydro-kinetic equations to analyze thermal fluctuations for a Bjorken expansion. The rapid Bjorken expansion of a medium drives the hydrodynamic fluctuations out of equilibrium prescribed by the fluctuation-dissipation theorem. The steady state solution to the kinetic equations determine the coefficient of the first fractional power of the gradient expansion ($\propto 1/(\tau T )^{3/2}$ ), which was computed for the first time for Bjorken expansion. Away from the conformal limit, such non-linear noise corrections also induce a non-vanishing bulk viscosity. The formalism of hydro-kinetic equations can be applied to more general background flows, non-conformal systems and coupled to existing viscous hydrodynamic codes to incorporate the physics of hydrodynamic fluctuations, which become dominant near the critical point.