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

Abstract

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.

Using some results about the Lie evolution of differential operators to obtain the Fokker-Planck equation for non-Hamiltonian dynamical systems of interest

“…10 In Ref. 2, for the specific case we dealt with there, we have shown that this term gives rise exactly to a first order partial derivative; therefore, Eq. (22) results to be a FPE.…”

Analytical probability density function for the statistics of the ENSO phenomenon: Asymmetry and power law tail

“…In this letter, taking advantage of some recent results of Bianucci [2015aBianucci [ , 2015b (B15 hereafter), we shed some light on this issue. In fact, here we do not use a stochastic force to perturb the ROM but a generic deterministic system that formally represents some model mimicking the MJO/WWB interaction with the ENSO system.…”

“…It is proper to observe that according to Bianucci [2015a], the FPE defined by equations (3) and (4) is obtained without hypotheses on the time scale separation between the system of interest (the ROM) and the rest of the system. In fact, in Text S1 in the supporting information, the expressions of the diffusion coefficients are given in terms of the Fourier transform of the normalized autocorrelation function (t) ≡ ⟨ (t) (0)⟩∕⟨ 2 ⟩, exploiting all the possible range of frequencies for the recharged oscillator.…”

Linear or non Linear modeling for ENSO dynamics?

“…Using the terminology of the projection approach [31,38,49] the ROM is viewed as the "system of interest" (or system a), while (ξ, π) represent the booster system (or "rest of the system" or system b), namely a set of general chaotic and fast variables, e.g., the MJO and WWB [50][51][52][53][54], that, perturbing the ROM activate the El Niño/La Niña phenomena and which obey some unspecified equations of motion expressed by the generic functions F (ξ, π) and Q (ξ, π). The value of the parameter determines the intensity of the perturbation to the ROM.…”