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Langevin equation to Fokker Planck

  1. Jun 7, 2012 #1
    1. The problem statement, all variables and given/known data

    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?



    2. Relevant equations

    x' = v(x) +η(t)

    v(x)= -udV/dx

    3. 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!
     
  2. jcsd
  3. Jun 7, 2012 #2
    Your differential equation is in the "standard form"
    [tex] dy = -b y^2 dt + dB [/tex]

    For a generic SDE, we have
    [tex]dx = b(t,x) dt + a(t,x) dB[/tex]
    and the corresponding Fokker-Planck equation is
    [tex] \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)) [/tex]
     
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