"Quantum many-body physics" sounds terribly difficult, but in practice the basic approaches are well known and well investigated. This means that once you decided to do this "business" you can find many textbooks and papers for the approaches. This is particularly true to me, who was trained as a nuclear many-body theorist.
Ironically, "classical many-body physics" is really hard to do for me. As a matter of fact, I have never thought about this branch of physics before. (More precisely speaking, I have thought about it briefly only when I wrote my CV to send around to universities and research institutions....)
Today, when I read a textbook on the nuclear pairing, which was written by Brink and Broglia, there was a sentence mentioning about the mean field, in a confusing way (at least to me). So, I started to review what I know about the mean-field approximation.
Basically, the mean-field approximation is a way to sum all the contributions coming from the inter-particle interactions, V(i,j). The exact total potential is expressed as, V(1,2,....,N) = Σ i< jV(i,j). But in the mean-field approximation, V(1,2,....,N) = Σ iVMF(i), which implies VMF(i) = Σ j != i V(i,j), roughly speaking. So, away from an averaging factor 1/N, this last expression literally describes the "mean" field. (i != j, which means i is not equal to j, needs to be taken into account from the reason that the self-interaction is removed from the consideration. This is a long-standing pathetic problem inherited already from a classical mechanics, particularly of gravity.)
A problem of the mean-field approximation in classical many-body systems is that each particle is moving around following the equation of motion (or Newton's equation). In other words, due to a time-dependence of the position of particles, the way to sum the two-body interactions between an arbitrary pair of the particles will change in time. Namely, the mean field changes in time. When the number of particles becomes larger, the summation at a given instant of time becomes harder. And also, keep tracking this time evolution of the mean field requires tremendous numerical effort.
In quantum many-body systems, stationary states are of main interest of our research, so that time-dependence does not usually appear. So, the determination of the mean field is actually one single numerical process in principle, which simplifies numerical investigations extremely.
Today, I suddenly realised this difference in treating the mean field between classical and quantum many-body physics, and started to be interested in a numerical challenge to evaluate classical mean fields. I guess this is a very good exercise even for some undergraduate students with good skills in computation. I hope I can write some code to deal with this problem sometime in next year.... I will definite come back to this topic soon.
