On the correspondence between a large class of dynamical systems and stochastic processes described by the generalized Fokker-Planck equation with state-dependent diffusion and drift coefficients

In this paper using a projection approach and defining the adjoint-Lie time evolution of differential operators, that generalizes the ordinary time evolution of functions, we obtain a Fokker-Planck equation for the distribution function of a part of interest of a large class of dynamical systems. The main assumptions are the weak interaction between the part of interest and the rest of the system (typically non linear) and the average linear response to external perturbations of the irrelevant part. We do not use ad hoc statistical assumptions to introduce as given a priori phenomenological equilibrium or transport coefficients. The drift terms induced by the interaction with the irrelevant part is obtained with a procedure that is reminiscent of that developed some years ago by Bianucci and Grigolini (see for example (Bianucci et al 1995 Phys. Rev. E 51 3002)) to derive in a 'genuine' way thermodynamics and statistical mechanics of macroscopic variables of interest starting from microscopic dynamics. However here we stay in a more broad and formal context where the system of interest could be dissipative and the interaction between the two systems could be non Hamiltonian, thus the approach of the cited paper can not be used to obtain the diffusion part of the Fokker-Planck equation. To face the problem of dealing with the series of differential operators stemming from the projection approach applied to this general case, we introduce the formalism of the Lie derivative and the corresponding adjoint-Lie time evolution of differential operators. In this theoretical framework we are able to obtain well defined analytic functions both for the drift and the diffusion coefficients of the Fokker-Planck equation. We think that the basic elements of Lie algebra introduced in our projection approach can be useful to achieve even more general and more formally elegant results than those here presented. Thus we consider this paper as a first step of this formal path to statistical mechanics of complex systems.