Exact Volterra series for mean field dynamics

We derive an exact Volterra series expansion for a mean field of an interacting particle system subject to a potential perturbation, expressing the Volterra expansion kernels in terms of the field's response functions, to any order. Applying this formalism to the mean particle density of a simple fluid, we identify a form reminiscent of dynamical density functional theory, with, however, fundamental differences: A nonlocal mobility kernel appears, and forces derive from a functional of the {\it history} of mean density. The equilibrium density functional is shown to be recovered in the limit of slowly varying perturbation. We identify a freedom in deriving this expansion, which allows different forms of mobility kernels. These developments allow for a systematic improvement of established mean field formalisms.