The thermodynamics of an ideal Bose gas is best calculated using the grand canonical ensemble. The grand potential for a Bose gas is given by:
where each term in the sum corresponds to a particular single-particle energy level ε; g is the number of states with energy ε; z  is the absolute activity (or "fugacity"), which may also be expressed in terms of the chemical potential μ by defining:
and β defined as:
where k  is Boltzmanns constant and T  is the temperature. All thermodynamic quantities may be derived from the grand potential and we will consider all thermodynamic quantities to be functions of only the three variables z , β (or T ), and V '. All partial derivatives are taken with respect to one of these three variables while the other two are held constant.
The permissible range of z is from negative infinity to +1, as any value beyond this would give an infinite number of particles to states with an energy level of 0 (it is assumed that the energy levels have been offset so that the lowest energy level is 0).