# Two level system permutations

1. Nov 16, 2016

### Kara386

1. The problem statement, all variables and given/known data
Sounds like a physics problem but I'm sure of the physics, stuck on the maths. At high T a two level system has $\frac{N}{2}$ particles in each level. If entropy is given by $S = k\ln(\Omega)$, where $\Omega$ is the number of ways of getting $\frac{N}{2}$ particles per level, show the high temperature limit is $Nk\ln(2)$.

2. Relevant equations

3. The attempt at a solution
To the best of my knowledge, $Omega = \frac{N!}{(\frac{N}{2})!(\frac{N}{2})!}$. Taking $ln$ of this and using the Stirling approximation:

$N\ln(N) - N - [\frac{N}{2}\ln(\frac{N}{2}) - \frac{N}{2}] - [\frac{N}{2}\ln(\frac{N}{2})-\frac{N}{2}]$
$= N\ln(N) - N\ln(\frac{N}{2})$
$= -Nln(2)$
I've gone wrong with a minus sign somewhere but I really can't see where! Thanks for any help!

2. Nov 16, 2016

### Staff: Mentor

How do you write $-N \ln(N/2)$ as a sum of two logarithms?

3. Nov 16, 2016

### Kara386

Oh yes, there's the missing minus sign. Thank you! :)