Interpolation of stochastic and deterministic reduced dynamics

There are roughly speaking the two extreme types of dynamical density functional theories: one is deterministic without noise and another is fully stochastic with noise. We attempt to interpolate these extreme limiting theories for general reduced description of time evolution by making use of a series of projection operators which generalize the so-called Kawasaki-Gunton operator. This helps to clarify, though in a formal way, the controversies or confusions surrounding the current dynamical density functional theories.