A kinetic regime of hydrodynamic fluctuations and long time tails for a Bjorken expansion

We develop a set of kinetic equations for hydrodynamic fluctuations which are equivalent to nonlinear hydrodynamics with noise. The hydro-kinetic equations can be coupled to existing second order hydrodynamic codes to incorporate the physics of these fluctuations, which become dominant near the critical point. We use the hydro-kinetic equations to calculate the modifications of energy momentum tensor by thermal fluctuations from the earliest moments and at late times in Bjorken expansion. The solution to the kinetic equations precisely determines the coefficient of the first fractional power (∝τ^−3/2) in the energy momentum tensor gradient expansion. Numerically, we find that the contribution to the longitudinal pressure from hydrodynamic fluctuations is comparable to the second order hydrodynamic terms for typical medium parameters used to simulate heavy ion collisions.