NonGausianess of the fluctuating force and its influence on the dynamics of the system

Abstract

We study the dynamics of the nonlinear systems by using the stochastic differential equation. But in this study, we do not discuss the method to solve directly. We are concerned on the transformation SDE``s into the partial differential equations which contain the probability distribution function. Depending on the distribution of the random variables, the type of the partial differential equation is quite different. In particular, we investigate when the case of noise is delta-correlated. This is called delta-correlated process(or white noise process). If its mean value is zero, the fourth cumulants is very important to show the nonGaussianess of the noise distribution. So the fourth cumulant plays an important part in the Kramers-Moyal expansion. The resulting partial differential equations are fourth order and we call this the generalized Fokker-Planck equation. We also examine the random number generating method whose distribution function is nonGaussin. We extend the odd-even method for normal distribution into $Nexp (- \frac{1}{2}x^2 - \frac{1}{4}x^4)$. And we also apply the Ornstein-Uhlenbeck process to generate the colored noise with finite correlation from white one.