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Langevin to Fokkerplanck? Uh oh... 
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#1
Jun712, 06:03 AM

P: 23

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 fokkerplanck 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! 


#2
Jun712, 08:00 AM

Sci Advisor
P: 5,539

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 FokkerPlanck equation is based on the continuity equation and is too messy to write here, but is a time evolution of a probability density. The longtime limit is known as the Smoluchowski equation.
The two equations (Langevin vs. FokkerPlanck/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, "Timescale separations and the validity of the Smoluchowski, FokkerPlanck and Langevin equations as applied to concentrated particle suspensions" Mol. Phys. 57, 303317 (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) 


#3
Jun712, 09:12 AM

P: 23

I'll check it out thanks!



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