The systematic adiabatic elimination of fast variables from a many-dimensional Fokker-Planck equation

The Chapman-Enskog method for the adiabatic elimination of fast variables is applied to a general Fokker-Planck equation linear in the fast variables. This equation is the counterpart of the generalized Haken-Zwanzig model, a system of coupled Langevin equations often encountered in quantum optics and in the theory of non-equilibrium phase transitions. After a few equilibration times for the fast variables the time dependence of smooth solutions of this Fokker-Planck equation is completely governed by the reduced distribution function of the slow variables, which obeys a closed evolution equation. We obtain an explicit perturbation series for the generator of this reduced evolution. The system considered here is the most general system for which the Chapman-Enskog hierarchy can be solved explicitly by exploiting an analogy between the unperturbed operator and the Hamiltonian for coupled harmonic oscillators.