  Hydrodynamic theories offer successful approaches that are capable of
simulating the otherwise difficult-to-compute dynamics of quantum many-body
systems. In this work we derive, within the positive-P phase-space formalism, a
new stochastic hydrodynamic method for the description of interacting Bose
gases. It goes beyond existing hydrodynamic approaches, such as superfluid
hydrodynamics or generalized hydrodynamics, in its capacity to simulate the
full quantum dynamics of these systems: it possesses the ability to compute
non-equilibrium quantum correlations, even for short-wavelength phenomena.
Using this description, we derive a linearized stochastic hydrodynamic scheme
which is able to simulate such non-equilibrium situations for longer times than
the full positive-P approach, at the expense of approximating the treatment of
quantum fluctuations, and show that this linearized scheme can be directly
connected with existing Bogoliubov approaches. Furthermore, we go on to
demonstrate the usefulness and advantages of this formalism by exploring the
correlations that arise in a quantum shock wave scenario and comparing its
predictions to other established quantum many-body approaches.
