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

Abstract

Finding the generalized Fokker-Planck Equation (FPE) for the reduced probability density function of a subpart of a given complex system is a classical issue of statistical mechanics. Zwanzig projection perturbation approach to this issue leads to the trouble of resumming a series of commutators of differential operators that we show to correspond to solving the Lie evolution of first order differential operators along the unperturbed Liouvillian of the dynamical system of interest. In this paper, we develop in a systematic way the procedure to formally solve this problem. In particular, here we show which the basic assumptions are, concerning the dynamical system of interest, necessary for the Lie evolution to be a group on the space of first order differential operators, and we obtain the coefficients of the so-evolved operators. It is thus demonstrated that if the Liouvillian of the system of interest is not a first order differential operator, in general, the FPE structure breaks down and the master equation contains all the power of the partial derivatives, up to infinity. Therefore, this work shed some light on the trouble of the ubiquitous emergence of both thermodynamics from microscopic systems and regular regression laws at macroscopic scales. However these results are very general and can be applied also in other contexts that are non-Hamiltonian as, for example, geophysical fluid dynamics, where important events, like El Niño, can be considered as large time scale phenomena emerging from the observation of few ocean degrees of freedom of a more complex system, including the interaction with the atmosphere.

Linear or Nonlinear Modeling for ENSO Dynamics?

“…Both approaches have their strengths and weaknesses, and a large literature has been devoted to justify both from a formal and theoretical point of view (for a review about the foundation of the stochastic approach, see [28], and for the projection approach, see, e.g., [29][30][31][32][33]). It is far beyond the scope of the present work to discuss the foundation of these approaches.…”

“…In this way, in fact, a comparison between the FPE predictions and observations should possibly validate the model and allow one to estimate the relative weight of the multiplicative perturbation. In this section, we briefly recall and use some of the results of [17,26,27,32,33] to introduce an explicit FPE for the PDF of the ROM system.…”

“…where {g(x)} b a ≡ g(b) − g(a) and M [1, α, y] is the Kummers (generalized hypergeometric) function of the first kind with the first argument equal to one. We plot Equation (32) in Figure 8 to show the dependence of the mean FPT on T tg for different values of the λ parameter.…”

Linear or non Linear modeling for ENSO dynamics?

“…For that we refer the reader to the results of [55] and (for a more general approach to the Lie evolution of differential operators) to the more recent work of Ref. [57]. Here we report the final result, that is given by Eq.…”

Linear or non Linear modeling for ENSO dynamics?

“…is the Kummers (generalized hypergeometric) function of the first kind with the first argument equal to one. We plot Equation (32) in Figure 8 to show the dependence of the mean FPT on T tg for different values of the λ parameter.…”