The grand canonical partition function for a Bose-Einstein gas is $$ Z_{\text{grand bos}} = \exp \left( \sum_{j=0}^{\infty} -\ln \left( 1-e^{\beta(\mu-\epsilon_j)} \right)g_j \right) $$ where $\beta$ is the reciprocal temperature, $\mu$ is the chemical potential, $\epsilon_j$ is the energy of state $j$, and $g_j$ is the degeneracy of state $j$. I can replace the exp function with its Taylor series to obtain $$ Z_{\text{grand bos}} = \sum_{N=0} \dfrac{1}{N!} \left( \sum_{j=0}^{\infty} -\ln \left( 1-e^{\beta(\mu-\epsilon_j)} \right)g_j \right)^N $$ This form of the partition function is very similar to the grand canonical partition of a classical gas $$ Z_{\text{grand classical}} = \sum_{N=0} \dfrac{1}{N!} \left( \sum_{j=0}^{\infty} e^{\beta(\mu-\epsilon_j)}g_j \right)^N $$ where $N$ represents the number of particles for different instances of the same gas. If I know how many particles are in my gas, but I do not know what its chemical potential is, I can remove the sum over $N$ to obtain the classical local ensemble $$ Z_{\text{local classical}} = \dfrac{1}{N!} \left( \sum_{j=0}^{\infty} e^{\beta(-\epsilon_j)}g_j \right)^N $$ where the presence of $\mu$ is redundant as a homogeneous scalar. With this classical logic I ask the question, can I also interpret $N$ in the Bose-Einstein grand partition function as the number of particles for a given instance of the gas and express its local ensemble equivalent as $$ Z_{\text{local bos}} = \dfrac{1}{N!} \left( \sum_{j=0}^{\infty} -\ln \left( 1-e^{\beta(\mu-\epsilon_j)} \right)g_j \right)^N $$ or does there not exist such a form?


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