Modified Bogolyubov's derivation of the two-fluid hydrodynamics

A consistent microscopic derivation of the two-fluid hydrodynamics for superfluid helium-4 in the ideal approximation is represented. The starting point in our formalism is a system of Heisenberg's equation of motion for both normal and anomalous correlation functions. The use of a mixed Wigner representation allows us to perform the expansion of the equations of motion for correlation functions in gradients directly, very easily, and with a rigorous mathematics. To find the hydrodynamic flows, we have constructed a local equilibrium statistical operator for superfluid helium in the reference frame, where the condensate is at rest.