Wavefunction branches are probably approximate eigenstates of hydrodynamic variables (and maybe more generally of slow local variables). Hydro variables are local averages of approximately locally conserved quantities, not restricted to fluids. They are most useful when the remaining degrees of freedom are all able to locally thermalize quickly conditional on the hydro variables, so that hydro variables follow closed equations of motion (with noise, generally described by a Lindbladian) that are derived by treating the remaining degrees of freedom as a heat bath.
Although the most famous hydro variables corresponds to exactly conserveda quantities like energy and momentum, there are many systems where approximately locally conserved quantities are the natural way to describe macroscopic behavior. This post just collects a few examples:
- Center-of-mass position for rigid object: I think there may be a few ways to characterize this, but for a system composed of atoms bound into rigid objects, the hydro variables are the densities of momentum (which is exactly conserved) and particle number (which is approximately conserved, and which maps to the location and orientation of objects), along with additional constraints imposed by the atomic bonds in the object. Particle number and the atomic bonds are only approximately conserved because bonds can break, chemical reactions can take place, nuclei can decay, etc.
- Chemical species concentration during slow reactions: Consider a reactive flow, i.e., a fluid composed of molecules that are chemically reacting. If all reaction timescales for a certain chemical species are much longer than the timescale of the microscopic dynamics, then the concentration of that species will be approximately conserved.
