PeterDonis said:
Can you give a good general reference? I know this body of theory exists but I don't know of a good place to go to get a presentation of the essentials.
As a standard source, I'd recommend to start with Landau-Lifshitz vol. X.
Generally, you have a kind of "hierarchy of descriptions" going from the microscopic description via various steps of approximations "down" to more and more "coarse grained" effective theories describing macroscopic observables.
You start with the quantum many-body level. The most convenient way is to use quantum field theory, because this deals in the most convenient way with the (anti-)symmetrization for many-body boson (fermion) states. This can be done both in the non-relativistic realm (usually sufficient for most things concerning "condensed-matter physics") and the relativistic (needed mostly in my field of "relativistic heavy-ion collisions", where you deal with "fireballs" of rapidly expanding "blobs" of strongly interacting matter, which behaves astonishingly like a fluid close to local thermal equilibrium).
There are two main formalisms in the literature to get to classical equations. One of them is to use the socalled two-particle irreducible formalism in the Schwinger-Keldysh real-time method to get closed equations for the exact two-point Green's function, which in Wigner-transformed form, is close to a classical phase-space distribution function, though not positive definite. That's because these socalled Kadanoff-Baym equations still contain all quantum effects.
The next step is some "coarse-graining formalism", i.e., you "forget" some too-detailed information. The idea behind this is that there's a separation of space-time scales: On the one hand a relatively slow time scale of macroscopic properties and macroscopic distances along which the macroscopic quantities significantly change and on the other hand the rapid oscillations/fluctuations around the mean values of these quantities. That's why one formal mathematical way of coarse-graining is the gradient expansion, i.e., you expand in powers of the space-time gradients of the macroscopic quantities. From another point of view you can also understand this as an expansion in powers of ##\hbar##.
This then leads to Boltzmann-like transport equations, usually with non-Markovian (memory) effects for a true positive definite one-particle phase-space distribution function with a hierarchy of ##n##-body correlation functions in the collision term. The usual Boltzmann equation truncates this at the one-particle level employing what's known in the classical case the "molecular-chaos assumption", i.e., the two-body distribution function is approximated as the product of two one-body distribution functions. At this point you through away information and the H-theorem (increasing entropy) can be proven.
The next step then is to identify the local-thermal equilibrium states, leading to Maxwell-Boltzmann (or Fermi-Dirac/Bose-Einstein distributions if you take Pauli blocking or Bose enhancement in the collision term into account) distributions with time-dependent temperature and chemical potentials as well as a flow velocity. If you then assume that the evolution of the system is slow compared to the typical relaxation-time scales to equilbrium you are at the level of ideal hydro. Expanding the distribution function around the local-thermal-equilibrium solutions you get in a systematic way various versions of dissipative hydro dynamics, with the Navier-Stokes equations as the leading order correction. At this level the microscopic origin (cross sections in the collision term of the Boltzmann equation) is just lumped into the various transport coefficients like shear and bulk visicosity, heat conductivity, diffusion constants etc.
Of course there are similar approaches also not only for fluids but also solids, where however you can describe many things using a quasi-particle picture, where collective excitations like lattice vibrations of a crystal are described by annihilation and creation operators of the corresponding (quantized) modes, which leads to a formalism which looks like the description of particles. In the Green's-function formalism this is always possible, if you have sharp peaks in the spectral function (which are just the imaginary parts of the retarded one-body propagator/two-point function). Then you can describe these excitations as "quasi-particles"/long-lived resonances.