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Ehrenfest Urn Problem Kramers-Moyal Coefficients and Planck-Fokker Equation

  1. Oct 21, 2008 #1
    "Ehrenfest Urn" Problem Kramers-Moyal Coefficients and Planck-Fokker Equation

    (For some reason, i can't get latex to work, and the button that used to be in the text box to use it is gone. :-()

    1. The problem statement, all variables and given/known data
    In the "Ehrenfest Urn" Problem, a particle moves randomly in a grid of positions x=ma with m an integer in the range -L < m < L , and with time stamp (tau). The probability when at position m' of a step to the right m' -> m' + 1 is

    P_ = 0.5 ( 1 + m / L)

    and the probability of a step to the left m' -> m' - 1 is

    P_ = 0.5 ( 1 - m / L)

    Evaluate the first Four Kramers-Moyal Coefficients for this process. In the continuum limits

    a -> 0 , (tau) -> 0 , L -> infinity ,

    such that a^2 / (tau) -> 2D and La^2 -> 2 (sigma)^2

    show that the Fokker-Plank equation describing the evoloution of the PDF P(x,t) is

    (all ds are partial)

    dP/dt = (D/(sigma)^2) dP/dx + D (d^2 P)/dx^2

    2. Relevant equations
    KM1 = -am/(tau L)
    KM2 = a^2 / tau
    KM3 = -(a^3) m /(tau L)
    KM4 = a^4 / tau

    KM3+ tend to zero. KM1 and KM2 are the contributing terms.

    dP/dt = lim(tau -> 0) sum[ (from n=1 to infinity) ((-1)^n / n!) (d^n)/(dx^n) [KMn P(x,t)]

    3. The attempt at a solution

    all i need is a justification for (a m) / (tau L) -> -D/(2 sigma^2)


    if you want me to show my working for everything else, i will photgraph it and upload it to imageshack, but that's a lot of hassle for me, and this last step is the bit which is bothering me.
     
    Last edited: Oct 21, 2008
  2. jcsd
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