Novel Perturbation Expansion for the Langevin Equation

Carl Bender, Fred Cooper, Greg Kilcup, L. M. Simmons, Jr., and Pinaki Roy

90-023

We discuss the randomly driven system $dx/dt = -W(x) + f(t)$, where $f(t)$ is a Gaussian random function or stirring force with $ = 2 $\delta$(t-t')$, and $W(x)$ is of the form $gx^{1+2\delta}$. The parameter $\delta$ is a measure of the nonlinearity of the equation. We show how to obtain the correlation functions $_f$ as a power series in $\delta$. We obtain three terms in the $\delta$ expansion and show how to use Padé approximants to analytically continue the answer in the variable $\delta$. By using scaling relations we show how to get a uniform approximation to the equal-time correlation functions valid for all $g$ and $\delta$.