no electrical resistivity at low temperatures. The equivalent model for half-integer particles (like electrons or helium-3 atoms), that follow Fermi–Dirac statistics, is called the Fermi gas (an ensemble of non-interacting fermions). At low enough particle number density and high temperature, both the Fermi gas and the Bose gas behave like a classical ideal gas. Macroscopic limit 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 εi ; gi is the number of states with energy εi ; 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 kB is Boltzmann's 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). Macroscopic limit, result for uncondensed fraction Following the procedure described in the gas in a box article, we can apply the Thomas–Fermi approximation which assumes that the average energy is large compared to the energy difference between levels so that the above sum may be replaced by an integral. This replacement gives the macroscopic grand potential function , which is close to : The degeneracy dg may be expressed for many different situations by the general formula: where α is a constant, Ec is a critical energy, and Γ is the Gamma function. For example, for a massive Bose gas in a box, α=3/2 and the critical energy is given by: where Λ is the thermal wavelength, and f is a degeneracy factor (f=1 for simple spinless bosons). For a massive Bose gas in a harmonic trap we will have α=3 and the critical energy is given by: where V(r)=mω2r2/2 is the harmonic potential. It is