(adsbygoogle = window.adsbygoogle || []).push({}); "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 positionsx=mawithman integer in the range-L < m < L, and with time stamp (tau). The probability when at positionm'of a step to the rightm' -> m' + 1is

P_ = 0.5 ( 1 + m / L)

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

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 thata^2 / (tau) -> 2DandLa^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.

**Physics Forums | Science Articles, Homework Help, Discussion**

Dismiss Notice

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Homework Help: Ehrenfest Urn Problem Kramers-Moyal Coefficients and Planck-Fokker Equation

Can you offer guidance or do you also need help?

Draft saved
Draft deleted

**Physics Forums | Science Articles, Homework Help, Discussion**