however, that the macroscopic equation gives an accurate estimate of the number of particles in the excited states, and it is not a bad approximation to simply "tack on" a ground state term to accept the particles that fall out of the continuum: where N0 is the number of particles in the ground state condensate. Thus in the macroscopic limit, when T < Tc, the value of z is pinned to 1 and N0 takes up the remainder of particles. For T > Tc there is the normal behaviour, with N0 = 0. This approach gives the fraction of condensed particles in the macroscopic limit: Limitations of the macroscopic Bose gas model The above standard treatment of a macroscopic Bose gas is straight-forward, but the inclusion of the ground state is somewhat inelegant. Another approach is to include the ground state explicitly (contributing a term in the grand potential, as in the section below), this gives rise to an unrealistic fluctuation catastrophe: the number of particles in any given state follow a geometric distribution, meaning that when condensation happens at T < Tc and most particles are in one state, there is a huge uncertainty in the total number of particles. This is related to the fact that the compressibility becomes unbounded for T < Tc. Calculations can instead be performed in the canonical ensemble, which fixes the total particle number, however the calculations are not as easy. Practically however, the aforementioned theoretical flaw is a minor issue, as the most unrealistic assumption is that of non-interaction between bosons. Experimental realizations of boson gases always have significant interactions, i.e., they are non-ideal gases. The interactions significantly change the physics of how a condensate of bosons behaves: the ground state spreads out, the chemical potential saturates to a positive value even at zero temperature, and the fluctuation problem disappears (the compressibility becomes finite). See the article Bose–Einstein condensate. Approximate behaviour in small Bose gases For smaller, mesoscopic, systems (for example, with only thousands of particles), the ground state term can be more explicitly approximated by adding in an actual discrete level at energy ε=0 in the grand potential: which gives instead . Now, the behaviour is smooth when crossing the critical temperature, and z approaches 1 very closely but does not reach it. This can now be solved down to absolute zero in temperature. Figure 1 shows the results of the