  A system of stochastic differential equations describing diffusive phenomena,
which has arbitrary friction depending on both state and distribution is
investigated. The Smoluchowski-Kramers approximation is seen to describe
dynamics in the small mass limit. We obtain the limiting equation and, in
particular, the addition drift terms that appear in the limiting equation are
expressed in terms of the solutions to the Lyapunov matrix equation and
Sylvester matrix equation. Furthermore, we provide the rate of convergence and
extend the system to encompass more general interactions and noise.
