Application of functional integration to the derivation of the low-frequency asymptotic behaviour of Green's functions and kinetic equations for a nonideal Bose gas

A form of perturbation theory is developed which enables one to calculate the Green's functions of a nonideal Bose gas below the Bose condensation point for low energies and momenta (in the hydrodynamic region). The functional integral method is used to find the diagrams of perturbation theory that make the main contribution to the hydrodynamic asymptotic behavior of the Green's functions. It is shown that summation of these diagrams reduces to solution of kinetic-type equations. The solution of these equations by the Chapman–Enskog–Hilbert method in the first approximation gives the Green's functions with poles corresponding to first and second sound. The second approximation enables one to express the damping in terms of the transport coefficients of first and second viscosity and the thermal conductivity. These coefficients are determined by the collision integral, which is obtained automatically in the process of summation of the diagrams.