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 ${\ensuremath{\tau}}_{c}$, we find the Markoff process that is a good approximation to the Langevin process for $\ensuremath{\Delta}t>{\ensuremath{\tau}}_{c}$. 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.