A Generalized Fokker-Planck Equation in the Case of the Volterra Model

Zwanzig and Mori's projection-operator method is used in order to derive a generalized nonlinear Fokker Planck equation for one "relevant" species in the many species conservative Volterra model. The deterministic, autonomous, Markovian equations of motion, when averaged over a suitable ensemble of initial conditions in general give rise to a non-autonomous, non-Markovian stochastic process for the evolution of this relevant species. Moreover, this relevant species may show irreversible damping, although self-interaction terms are absent in the many species model.