Marco Bianucci — ISMAR-CNR La Spezia #
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 talk, 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 <i> non linear </i>),
and the <i>average</i> linear response to external perturbations of the
irrelevant part.
We do not use <i> ad hoc</i> statistical assumptions to introduce as <i>given a priori</i>
phenomenological equilibrium or transport coefficients.
Respect to previous approaches of the problem, here we stay in a more
broad and formal context where the system of interest could be
dissipative and the interaction between the relevant and the irrelevant parts could be non Hamiltonian.
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.