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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.

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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 $$ x_i = \frac{N_i}{N} = \frac{N_i}{\sum_i N_i} $$

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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.

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