For grand partition ensemble, is it true that the introduction of chemical potential allows us to have the sum of number of the particles in each state to be the total number of particles ("On average")? and this fluctuation in the constraint of $N$ will shrink as the number of particles you got is large.
Next question is about Boson gas, why would we have to consider the chemical potential to be negative or zero? Is there anything to do with the reference to the ground state energy level?
What would it mean if we were to integrate the energy of those states below ground states?
NB For last question, I mean when I were to find the total energy of this boson gas system, which I know below a critical temperature Tb, the Bose Einstein condensation becomes important, and we would do the separate sum thing as usual (see first Pic) , Now I wonder why when we do the sum of the energy we integrate the states above ground state , does it mean all the particles at ground state have zero energy?? and we only have to concern those above the ground state?