Partition Function and Degeneracy

[stevendaryl]

There doesn't seem to be a forum that is specifically about statistical mechanics, so I'm posting this question here. I apologize for the long-winded introduction, but I think it's needed to provide context for my question:

If you have a discrete collection of single-particle energy levels $\epsilon_i$, then the grand canonical partition function (for noninteracting particles) is defined by:

$\mathcal{Z}(\mu, \beta) = \sum_i \sum_{N_i} exp(N_i \beta(\mu - \epsilon_i))$
$= \sum_i \sum_{N_i} exp(\beta(\mu - \epsilon_i))^{N_i}$

where $N_i$ is the occupancy number: the number of particles in state $i$. To get Bose statistics, the allowable values for $N_i$ are $N_i = 0, 1, 2, ...$, leading to

$\mathcal{Z}(\mu, \beta) = \sum_i \dfrac{1}{1 - exp(\beta(\mu - \epsilon_i))}$

(because $1+x+x^2 + ... = \frac{1}{1-x}$)

For Fermi statistics, the only possible values for $N_i$ are $N_i = 0, 1$, leading to:

$\mathcal{Z}(\mu, \beta) = \sum_i (1 + exp(\beta(\mu - \epsilon_i)))$

Here's the question: Suppose that the energy levels for an electron are independent of spin direction. That means that for every single-particle state, there is a second state with the same energy and opposite spin state. Then it seems to me that there are two different ways to take this degeneracy into account:

(1) Replace $\sum_i ...$ by $\sum_i g_i ...$, where $g_i$ is the degeneracy of energy level $i$, and where the index $i$ ranges only over states with distinct energies. In this case, $g_i = 2$, so the result is just to multiple $\mathcal{Z}$ by 2.

(2) Modify the allowable occupancy number $N_i$ to range from $0$ to $g_i$, rather than just 0 or 1.

These two approaches give different answers, but I don't understand, physically, why. It seems that the only thing that should be important is how many electrons can have energy $\epsilon_i$.

 

[mfb]

The second approach would require the states with the same number of electrons in such a state to be indistinguishable, but "electron up" and "electron down" are different things.

 

[stevendaryl]

Thank you.