This isn't homework but I'm interested. So I have the langevin equation dy/dt = -dV/dy +η(t) where V(y) = -by^3/3 + ζy how can I turn this into a fokker-planck equation? What I'm getting is x' = -u(bx^2+ζ) + η(t) Which I don't know how to solve in closed form. Any ideas/suggestions? Thanks!
Yikes... I have some information, but it's not clear how to relate the specific to your question. I have the Langevin equation as m dv/dt = F + f, where f is the randomly fluctuating force. The Fokker-Planck equation is based on the continuity equation and is too messy to write here, but is a time evolution of a probability density. The long-time limit is known as the Smoluchowski equation. The two equations (Langevin vs. Fokker-Planck/Smoluchowski) differ in a few respects- primarily the relevant timescale, but also the difference in viewing a process as 'diffusive' or as direct modeling of fluctuations in particle velocity. My reference for this is Brenner and Edwards, "Macrotransport Processes". They begin by defining a timescale that allows for an average velocity, but short enough to allow for fluctuations. This leads to an integral expression for the probability density, and after a few short (but very dense) pages, they show the two approaches yield identical expressions for the diffusivity. They reference Masters, "Time-scale separations and the validity of the Smoluchowski, Fokker-Planck and Langevin equations as applied to concentrated particle suspensions" Mol. Phys. 57, 303-317 (1986). There may also be some useful material in Balescu's "Statistical Dynamics: Matter out of Equilibrium"- it seems to be an open area of research (by considering Levy distributions instead of Gaussian distributions, for example)