An equation for chemical potential from the Boltzmann entropy equation I am trying to derive an equation for chemical potential from Boltzmann entropy and this is what I have come up with so far:
$$S = k\ln{\left(\frac{N!}{N_\mathrm{up}!\,N_\mathrm{down}!}\right)} \label{eq:1} \tag{1}$$
and the derivation of chemical potential from the 1st law of thermodynamics,
$$\mu = -T\left(\frac{\partial S}{\partial N}\right)_{UV} \tag{2}$$
I have used Stirling's approximation and properties of logs and have come up with an answer, but after searching for my answer it is not online anywhere, so it feels like I might be going in the wrong direction.
From my computation of entropy which I found to be,
$$S = k_b\left( N\cdot\ln{N} - \sum(N_i\cdot\ln{N_i}) \right) \tag{3}$$
I got chemical potential to be
$$\mu = -Tk_b\left( \ln{N} +1 - \sum{(\ln{N_i} + 1)} \right) \tag{4}$$
Is this a derivation for entropy that anyone has ever seen before? Is it even possible to take the equation $\eqref{eq:1}$ and derive an equation for chemical potential?
Any answer or push in the right direction would be greatly appreciated.
 A: Start with 
$$
   S(N_1,N_2) = k \log\frac{(N_1+N_2)!}{N_1! N_2!} 
$$
Write the chemical potential as
\begin{align}
   \mu &= - kT\left(\frac{\partial S}{\partial N_1}\right)_{N_2}\\
       &= -kT \frac{S(N_1+1,N_2)-S(N_1,N_2)}{(N_1+1)-N_1} \\
       & = -kT \log 
      \frac{(N_1+N_2+1)!}{(N_1+N_2)!}
      \frac{N_1!}{(N_1+1)!}
      \frac{N_2!}{N_2!}\\
   & = -kT \frac{N_1+N_2+1}{N_1+1}
\end{align}
Make the approximation
$$
   \frac{N_1+N_2+1}{N_1+1} \approx \frac{N_1+N_2}{N_1} = \frac{1}{x_1}
$$
and write the final result as
$$
\boxed{
\vphantom\int
   \mu = kT \log x_i
}
$$
with 
$$
   x_i = \frac{N_i}{N} = \frac{N_i}{\sum_i N_i}
$$
A: So the main conceptual problem that you are facing I think is that if say I am looking at a thermodynamic system of water molecules and salt molecules, then I expect there to be a chemical potential for water molecules and also a separate one for salt molecules. It doesn't make much sense for me to define a chemical potential for molecules overall.
In your case, you are writing this in the first part to suggest to me that you've got two different things, $N_{\text{up,down}}$ that presumably should each have a chemical potential. But then you seem to be calculating the overall chemical potential instead, which is not at all obvious to me.
Given these labels “up, down,” I guess maybe you have a system of spins? Or are they just “above” some divider vs below it? Basically I guess I am asking what this system is going to be in contact with, just as you start to ask about the temperature of a system when you are considering it sharing energy with its surroundings. A chemical potential is like sharing energy, except you're sharing particles of some sort or another.
If you do have particles, then there's a difference between being in contact with a reservoir of exclusively spin-up particles that you might want to add while holding $N_\text{down}$ constant, versus say a more “spintronics” application where $N$ remains constant but you are going to be in contact with a source of angular momentum such that $N_\text{up}-N_\text{down}$ can change, you will need a different notion of chemical potential for the two cases.
So it's worth understanding first how you think of these populations before you evaluate their respective chemical potentials.
