How Can the Langevin Equation Be Transformed into a Fokker-Planck Equation?

[JVanUW]

Homework Statement

This isn't actually homework but I'm really interested in finding the solution.

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?

Homework Equations

x' = v(x) +η(t)

v(x)= -udV/dx

The Attempt at a Solution

Using the format of the langevin equation x' = v(x) +η(t), I get

x' = -u(bx^2+ζ) + η(t) (( v(x)= -udV/dx ))

Which I don't know how to solve in closed form.

Any ideas/suggestions?

Thanks!

 

[clamtrox]

Your differential equation is in the "standard form"
$$dy = -b y^2 dt + dB$$

For a generic SDE, we have
$$dx = b(t,x) dt + a(t,x) dB$$
and the corresponding Fokker-Planck equation is
$$\frac{\partial f(t,x)}{\partial t} = - \frac{\partial}{\partial x} (b(t,x) f(t,x)) + \frac{1}{2} \frac{\partial^2}{\partial x^2} (a^2(t,x) f(t,x))$$