Canonical versus grand canonical occupation numbers for simple systems

An ideal gas ofN indistinguishable particles is described by a canonical ensemble (c.e.) and also by a grand canonical ensemble (g.c.e.) which hasN as themean total number of particles, the temperature and volume being the same in both cases. Exact mean occupation numbersn _j(N) are found if the system has only two states 1 and 2 of energiesE ₂E ₁. This should apply to quantum wells and similar simple systems. For systems which have captured one particle, the theory gives the simplest answers, and one find a maximum discrepancy of 17% between the two ensembles for the fermion case. It occurs whenE ₂–E ₁53 meV at room temperature. ForN=1 the mean occupation number for the c.e. is identical for fermions and for bosons, being in both cases given byn ₂(1)={exp(E ₂-E ₁)/kT]+1}^-1,n ₁(1)=1-n ₂(1) For largeN one reverts to the usual situation and the discrepancy between the ensembles becomes small.