I think that when particles and energy are allowed to freely leave the grand ensemble like that, there'll always be fluctuation in temperature despite the heat bath. But I only see particle and energy fluctuations, and temperature seems to always be considered at constant. So how do I find the T fluctuation, and is it necessary? Thanks.
Temperature is fixed in the grand canonical ensemble. In fact, for the three main ensembles we have:
- Microcanonical ensemble: fixed $N,V,E$
- Canonical ensemble: fixed $N,V,T$
- Grand canonical ensemble: fixed $\mu,V,T$
where $\mu$ is the chemical potential.