Classical Noise III: Nonlinear Markoff Processes

Our previous treatment of noise in the nonequilibrium steady state is extended to include nonstationary processes, and processes for which the quasilinear approximation is inadequate. By use of backward-equation methods, we show that $M_0(a_0, t, t_0)=〈〉\left(\exp \left[-\int {t_0}^{t}Q\left(a\right(s), t-s) \mathrm{ds}\right]\right)$ subject to $\mathrm{a}\left(t_0\right)={\mathrm{a}}_0$ obeys the differential (integral) equation: $\frac{\partial M_0(a_0, t, t_0)}{\partial t_0}=\left[Q\right(a_0, t-t_0)-\Sigma \stackrel{\infty }{n=1\mathrm{}}{\mathrm{D}}_n(a_0, t_0):{\left(\frac{\partial }{\partial a_0}\right)}^n]M_0,$ where the ${\mathrm{D}}_n$ are the $n\mathrm{th}$-order diffusion coefficients of the $\mathrm{a}\mathrm{}\left(s\right)\mathrm{}$ process, and $Q\left(\mathrm{a}\mathrm{}\right(s), s)$ is an arbitrary function of a and $s$. The choice $D_n=0\mathrm{}$, $n>\mathrm{}2\mathrm{}$, $D_2=D$ $D_1\mathrm{}\left(a\right)=-\Lambda a$ makes $a\left(s\right)\mathrm{}$ an Ornstein-Uhlenbeck (O.U.) process, i.e., white noise that has been filtered through an $\mathrm{RC}$ network with time constant $\frac{1}{\Lambda }$. The choice $Q\left(a\right(s), s)=k(t-s)\mathrm{}{\mathrm{}\left[a\right(s\left)\right]}^2$ squares the output and applies the time smoothing $k(t-s)\mathrm{}$. For $k\left(s\right)=\exp (-2\mathrm{}\beta s)$ [time smoothing through an $\mathrm{RC}$ network with time constant $\left(\frac{1}{2\beta }\right)$], an explicit solution is obtained for the characteristic function $M_0$. For arbitrary positive $k\left(s\right)\mathrm{}$, we show that $M_0$ becomes independent of $a_0$ as $t\to \infty$ if $k\left(\infty \right)=0\mathrm{}$, and $M_0$ becomes stationary if $\Lambda >0\mathrm{}$ and $\int 0^{\infty }k\left(u\right)\mathrm{} \mathrm{du}<\mathrm{}\infty .$