A couple of random questions about quantum stuff I learned a few years ago how fluids and solids behave in a classical way. Also how some thermodynamic properties of solids arise from electrons in condensed matter (Bloch thing and etc), and the same with gasses in quantum statistical mechanics.
Is there a quantum version of Navier-Stokes equation? What quantum theory says about approximate incomprehensibility of fluids? An explanation of viscosity?
Edit: I deleted the unrelated question. Also, I know we can get good predictions by knowing that classical mechanics is an approximation to QM, but I think that it's interesting enough this question to be asked anyway. I mean, this has to have been done somewhere, I want to know how it's called and how they have done that. Justifying material derivative and all that stuff in QM can be really weird (I think), but it's interesting for it's own sake (I think).
 A: Roughly speaking, the Navier-Stokes equations are a set of partial differential equations arising from certain conservation laws such as the conservation of momentum and mass for a portion of fluid and in general one also has to define a constitutive equation which is a (in general, a differential) equation between the Cauchy's stress tensor and the strain tensor. Finally, they are coupled in order to form a closed set of equations that can be solved.
The underlying assumption which allows to write these equations is that the matter is a continuum and therefore, in a representative point of a portion of fluid, there is a number of particles equivalent to the Avogadro number.
Hence, it is a big risk, in my opinion, to find a strict connection with the quantum formalism!
Nevertheless, there is a wonderful formalism introduced by Madelung which formally recasts the Schrödinger equation into a Navier-Stokes-like equation.
See for instance:
https://en.wikipedia.org/wiki/Madelung_equations and https://en.wikipedia.org/wiki/Quantum_hydrodynamics
Such a formalism is intriguing because allows to describe quantum system which manifests a fluid behavior as for instance superfluid helium.
