PeterDonis said:
Because the distribution ##\rho## might not be uniform over all of the microstates (or more precisely over all of the points in phase space, which is what "microstates" actually means in this context). And if it's not uniform, then the subsystem (not system--see below) will have different probabilities of being at different points of phase space.The distribution ##\rho## is for a subsystem, not for the whole system.
Not necessarily. The microcanonical ensemble describes a closed system.
The argument that in classical mechanics the occupation density of phase-space volume elements are the measure for the probabilities of a closed system is that the phase-space volume elements' volume is unchanged under time evolution (Liouville's theorem). Of course, knowing the exact ##N##-body phase-space distribution function at initial time implies knowing the complete state of the system, which in practice is impossible. That's why you have to "coarse grain" the description to describe the "relevant macroscopic degrees" of freedom for the situation at hand.
The most simple case is equilibrium, which for the microcanonical case, I've described in #22. That's in fact the description of minimal knowledge you can have, i.e., the system has "forgotten" anything about the initial state.
The next simple case used in practice is to truncate the practically infinitely many equations of motion for the ##n##-particle phase-space distribution functions (##n \leq N## with ##N## the total number of particles in your closed system) at the one-body level. This is Boltzmann's idea, known as the "molecular-chaos assumption" (or in German the "Stoßzahlansatz"). The equation of motion for the one-particle distribution, depends on the two-particle distribution, and the molecular-chaos assumption simply assumes that two particles are uncorrelated, i.e., the two-particle distribution function can be approximated by the product of the two one-particle distribution functions. At this point, in fact, you "coarse grain" in the sense that you neglect some information about the system, namely two-particle correlations. This then leads to the Boltzmann equation and to the H-theorem, i.e., that the macroscopic entropy is increasing with time, defining the "thermodynamical arrow of time", which however in fact is by construction identical with the fundamental "causal arrow of time".
The quantum description is similar and in some sense less complicated, because there the single-particle phase-space volume has a natural measure, i.e., ##h^3=(2 \pi \hbar)^3##. Then you can describe the system in the 2nd-quantization (QFT) formalism and Green's functions. For the general off-equilibrium case this leads to the so-called Schwinger-Keldysh (SK) real-time contour formalism. Again the exact equations of motion for the many-body SK Green's functions build a coupled set of (practically) infinitely many Green's functions, which you never can solve. Also here the idea is to truncate this "BBGKY hierarchy" (names after Born, Boguliuobov, Green, Kirkwood, and Yvon). On the one-particle level you only consider the one-body Green's function, i.e., the two-point function (and maybe also some one-point function, i.e., some mean field, as, e.g., for a Bose Einstein condensate or the magnetization of a ferromagnet,...).
You can derive such approximation from a variational principle, known as the ##\Phi##-functional formalism (going back to Luttinger, Ward, Baym, and Kadanoff in condensed-matter physics and to Cornwall, Jackiw, and Tomboulis). In diagrammatic Language the ##Phi## functional is the sum over all closed two-particle irreducible diagrams, i.e., diagrams with lines symbolizing exact, interacting two-point propagators, which don't get disconnected when cutting any pair of lines. Then the exact self-energy is given by opening each diagram by omitting one line, and the exact Green's function then is given by the Dyson equation with these self-energies.
Of course also this cannot be solved for non-trivial interacting theories, and you have to truncate the ##\Phi## functional in some way to a finite number of diagrams or an infinite subset of diagrams, which can be "resummed" somehow. Usually this is achieved by expanding formally in powers of some parameter (coupling constants or ##\hbar##, the latter based on the assumption that a macroscopic system is well described by classical physics with "small" quantum corrections).
This then leads to the socalled Kadanoff-Baym equations for the two-point (one-particle) Green's functions. The next step towards a classical transport equation is then to do a so-called "Wigner transformation", which is a Fourier transformation wrt. ##t_1-t_2## and ##\vec{x}_1-\vec{x}_2##, where ##t_1,t_2## and ##\vec{x}_1,\vec{x}_2## are the arguments of the two-point Green's function. Then you get the Green's function in terms of the corresponding Fourier-transformed function, which now is considered as a function of the "momentum" and ##\tau=(t_1+t_2)/2## and ##\vec{X}=(\vec{x}_1+\vec{x}_2)/2##. One of the so Wigner-transformed functions, ##G^{<}##, then is something very similar to a phase-space distribution function, but it's not yet really one, because it's real but not positive definite, but integrating over ##\vec{p}## gives the one-particle density and integrating over ##\vec{X}## gives the momentum distribution.
To get to a phase-space distribution function and a semiclassical quantum-transport equation you do a gradient expansion, which formally is also an ##\hbar##-expansion. This indeed leads to a Boltzmann-like quantum transport equation.
You find all this in Landau-Lifshitz vol. X, which is a very good book on kinetic theory (both classical and quantum).
