Classical Noise IV: Langevin Methods

A Langevin theory for linear and nonlinear, stationary and nonstationary, processes is developed and compared with Markoff methods. For short correlation times ${\tau }_c$, we find the Markoff process that is a good approximation to the Langevin process for $\Delta t>\mathrm{}{\tau }_c$. Conversely, given the diffusion coefficients $D_n$ of a Markoff process, we find the moments (to all orders) of the Langevin forces that lead to the same process—exactly for homogeneous noise, approximately for the general nonlinear case. The techniques are illustrated by applications to Fokker-Planck processes, homogeneous noise with linear damping, the one-dimensional impurity band, spin diffusion, and population fluctuations.