On the application of truncated generalized Fokker-Planck equations

Abstract

The Pawula theorem states that the generalized Fokker-Planck equation with finite derivatives greater than two leads to a contradiction to the positivity of the distribution function. Though negative values are inconsistent from a logical point of view, we show that such “distribution functions” with negative values can be very useful from a practical point of view. For a Poisson-process, where the exact solution is known, we compare the solution of the second order Fokker-Planck equation to the solutions of Fokker-Planck equations of finite order. It turns out that for certain parameters the approximations of the distribution function and the moments are much better for some higher order and that the magnitude of negative values may be very small in the relevant region of variables.