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 solution to this equation for α=3/2, with k=ε=1 which corresponds to a gas of bosons in a box. The solid black line is the fraction of excited states 1-N/N  for N =10,000 and the dotted black line is the solution for N =1000. The blue lines are the fraction of condensed particles N/N  The red lines plot values of the
negative of the chemical potential μ and the green lines plot the corresponding values of z . The horizontal axis is the normalized temperature τ defined by
It can be seen that each of these parameters become linear in τ in the limit of low temperature and, except for the chemical potential, linear in 1/τ in the limit of high temperature. As the number of particles increases, the condensed and excited fractions tend towards a discontinuity at the critical temperature.
The equation for the number of particles can be written in terms of the normalized temperature as:
For a given N  and τ, this equation can be solved for τ and then a series solution for z  can be found by the method of inversion of series, either in powers of τ or as an asymptotic expansion in inverse powers of τ. From these expansions, we can find the behavior of the gas near T =0 and in the Maxwell–Boltzmann as T  approaches infinity. In particular, we are interested in the limit as N  approaches infinity, which can be easily determined from these expansions.
This approach to modelling small systems may in fact be unrealistic, however, since the variance in the number of particles in the ground state is very large, equal to the number of particles. In contrast, the variance of particle number in a normal gas is only the square-root of the particle number, which is why it can normally be ignored. This high variance is due to the choice of using the grand canonical ensemble for the entire system, including the condensate state.