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From Langevin to Fokker-Planck

  1. Dec 9, 2005 #1
    Hi.
    I'm studying quantum Brownian motion right now and I need to see that the (classical) Langevin equation for a Brownian particle is equivalent to the Fokker-Planck equation for the phase-space distribution function of the same particle.
    Does anyone know where I can find such a derivation? I've been looking all over the internet for it but usually they start with a Langevin equation containging only first derivatives (that is the have excluded the possible outer potential felt by the particle).
    I need the derivation for the case where i have a Langevin equation of the type:
    m \ddot x(t) + \gamma \dot x(t) + V(x(t)) = F(t)
    (written in LaTeX syntax).
    I would be very grateful for help!
    Alternatively, since I have the derivation for the Langevin equation starting with a Lagrangian for a particle interacting linearly with a bath of harmonic oscillators (initially in thermal equilibrium), I could also accept a derviation of the Fokker-Planck equation starting with the same assumptions.
     
  2. jcsd
  3. Dec 9, 2005 #2

    Physics Monkey

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    For such stochastic methods, a very good book is "Handbook of Stochastic Methods" by C. Gardiner. Written by an expert in the field, this book is very physical and to the point unlike many other excessively mathematical treatments of stochastic processes.

    If you're interested, I can step you through the derivation here, but you should get that book for sure.
     
  4. Dec 10, 2005 #3
    Okej!
    Thank you very much. I'll have a look at that book. Found it at the university library. If there is something I get stuck with I might ask you again :)
     
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