General Idea

We know how to solve single particle problem or the problem of many non-interacting particles--even in external field (ideal gas!). We want to deal with the problem of many interacting particles.

Problem:

Interactions between particles makes the calculation of partition function (nearly) insolvable

Idea:

Substitute the forces acting on a given particle by an effective external field

This is mean-field theory. This is a very popular approach in Statistical Physics, Quantum Mechanics (adiabatic approximation), Mechanics, Fluid Mechanics...