How to obtain hydrodynamics from many-body quantum mechanics?

It is known that the Schrodinger equation is equivalent to the Euler equation (with a "quantum potential" term) and to the probability conservation equation (which is formally identical to the mass conservation equation of the usual hydrodynamic theories). The calculation is quite simple and it’s based on the so-called Madelung transformation, i.e. on writing the wave function as
$$
psi(mathbf{x},t) = sqrt{n(mathbf{x},t)} e^{phi(mathbf{x},t)}
$$
and inserting it into the Schrodinger equation.

Now, this works for a single-particle system. If there is a Bose condensate, then we can interpret $psi(mathbf{x},t)$ as the "collective wave function" (in the Hartree–Fock approximation, the total wave-function of the system of $N$ bosons is taken as a product of $N$ identical single-particle functions $psi(mathbf{x},t)$). Hence, the same stuff of the single-particle case applies and we can end up with the hydrodynamic theory of a condensate.

All this is extremely cool and kind of textbook-physics to date… but how to derive the hydro equations for a system that is comprised of many particles and it’s not a condensate? References/hints?

What concerns me: where does the temperature or the entropy (that are typically important quantities in hydrodynamics) come from? I am aware of the possibility of doing thermal-many body quantum mechanics, but there is no "real time" there (it’s a theoretical framework for the statistical averages at equilibrium, or close to equilibrium, as far as I know).