Thermal Physics - energy, microstates, and probabilities

1. Sep 17, 2015

Ascendant78

1. The problem statement, all variables and given/known data

2. Relevant equations

3. The attempt at a solution
The first part I'm not worried about, but the second part is worked out in the "relevant equations" section. Honestly, it looks like more magic than a Harry Potter movie going on there to me. I'm at a loss as to what mathematical method/s are being utilized to get to that answer?

2. Sep 19, 2015

stevendaryl

Staff Emeritus
I'm not sure that I understand your concern. From the equations, you can prove exactly that:

$\Omega(E=(r-s) \Delta) = \Omega(E=r \Delta)[\dfrac{r^s}{(N-r)^s}] [\dfrac{ (1-\frac{1}{r}) (1-\frac{2}{r}) ... (1 - \frac{s-1}{r})}{ (1 + \frac{1}{N-r}) (1 + \frac{2}{N-r}) ... (1 + \frac{s}{N-r})}]$

Then the only issue is proving that if $s \ll r$ and $s \ll (N-r)$, then the last factor is approximately 1.