thermodynamics - Why do we use grand canonical ensemble for BEC description?

The system we consider has constant $N$, $V$ and $T$ (the number of particles, volume and temperature) This is just the thermodynamic variables for the canonical ensemble, why we use fugacity $z$ or chemical potential $\mu$ there?

Answer

If you choose to set $N$, $V$ and $T$, then you are choosing the Canonical Ensemble, so $N$ is fixed. In this approach, the chemical potential $\mu$ is just a parameter that you've to determine by imposing that the total number of particles is $N$.

Viceversa, if you choose the grandcanonical ensemble, you fix $\mu$ and so the number of particles $N$, fluctuates around an average value $\langle N \rangle$. I hope that I've well understood your problem and answered properly.