Eigenfunction method of collision Boltzmann integral in kinetic theory of long wave fluctuations

Linearized general equations of long wave fluctuation kinetics are solved (by utilizing eigenfunctions and eigenvalues of the linearized collision Boltzmann integral) in the asymptotic field $t \gg \tau_{r}$ ($\tau_{r}$ is a relaxation time). A general form of linearized equations of fluctuation hydrodynamics is obtained. Effective initial conditions for the fluctuation hydrodynamics equations are derived for the case when arbitrary order fluctuations are absent at the initial instant. A time asymptotics of the one – particle distribution function is found in the fluctuation hydrodynamic evolution regime when the fluctations of hydrodynamic quantities play an essential role. It is compared with the early obtained results of the “long hydrodynamic tails” theory.