Non Standard Fluctuation Dissipation Processes in Ocean-Atmosphere Interaction and for General Hamiltonian or Non Hamiltonian Phenomena: Analytical Results

For Hamiltonian systems, the statistical mechanics tools are welldeveloped and well-suited for studying the large scale emerging phenomena from a underlying chaotic and complex dynamics. Among them, the Zwanzig projection approach is one of the most frequently used, and most powerful, leading to important results such as the microscopic foundation of the fluctuation dissipation relation, and the canonical equilibrium density function (see, for example, M. Bianucci et al., Phys. Rev. E 51,3002 (1995)). Actually, the projection approach leads the calculus with differential operators that are usually almost intractable, but that are drastically simplified when the basic dynamics of the system of interest is Hamiltonian-as happens in foundation of thermodynamics problems. In most of the real physical cases, however, the fundamental equations are not Hamiltonian. This happens, for example, in fluid dynamics, where the physics is described by the Navier Stokes equations, or in biology in general, just to cite a couple among many research fields. Here we show how it is still possible to get, in the non-Hamiltonian case, a generalized Fokker Planck equation describing the time large scale statistics of a part of interest of the whole complex system. As an example, we focus our attention on the ocean-atmosphere system, where the multiplicative character of the interaction makes the non-standard feature of the fluctuation dissipation process crucial.