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

[wbrigg]

"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. :-()

Homework Statement

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

Homework 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)]

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.

 

[lippyka]

The justification for the transformation of (a m) / (tau L) -> -D/(2 sigma^2) is based on the fact that in the continuum limit, a^2 / (tau) -> 2D and La^2 -> 2 (sigma)^2. Substituting these values into (a m) / (tau L), we get (-D)/(2 sigma^2), as required.