Hydrodynamical variables associated with approximately conserved quantities

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 conservedI will say “conserved” rather than “locally conserved” for brevity, but local conservation — that they obey a continuity equation — is crucial for ensuring their slow behavior relative to the microscopic other degrees of freedom.^a 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. Like exactly conserved quantities such as momentum, the local species concentration at a point can still change through flow and diffusion, but this will happen very slowly compared to the time on which the atoms in the fluid locally thermalize conditional on the local concentration and other hydrovariables like local velocity.
• Local average orientation of a liquid crystal in a nematic phase: The molecules in such liquid crystals are energetically constrained to approximately align with their neighbors, like a ferromagnet. The overall dynamics are not just symmetric under rotation of all atoms together (the exact conservation of angular momentum), but also approximately symmetric under the rotation of the molecules with their center of masses fixed. In normal fluids, there is enough energy for the orientation of each molecule to vary independently, so they can thermalize, but in a nematic liquid crystal all the molecules in a local region are constrained to vary together.
• Local average magnetization of a ferromagnet: see preceding, with the difference that anti-aligned neighboring spins are high energy while aligned and anti-aligned molecules in a nematic liquid crystal are both low energy, so the symmetry group is different.
• Defects in crystalline solids: Crystal defects (e.g., vacancies and dislocations) move through a lattice and may be created/annihilated much more slowly than phonons propagate, creating a separation of timescales. Even in cases where individual defects move quickly and thermalize, approximate number conservation can mean that the local defect density acts as a hydro variable. Relevant to plastic deformation and material aging.