  The thermodynamics for a system with given temperature, density, and volume
is described by the Canonical ensemble. The thermodynamics for a corresponding
system with the same temperature, volume, and average density is described by
the Grand Canonical ensemble. In general a chosen thermodynamic potential
(e.g., free energy) is different in the two cases. Their relationship is
considered here as a function of the system size. Exact expressions relating
the fundamental potential for each (free energy and pressure, respectively) are
identified for arbitrary system size. A formal asymptotic analysis for large
system size gives the expected equivalence, but without any characterization of
the intermediate size dependence. More detailed evaluation is provided for the
simple case of a homogeneous, non-interacting Fermi gas. In this case, the
origin of size dependence arises from only two length scales, the average
inter-particle distance and quantum length scale (thermal deBroglie or Fermi
length). The free energies per particle calculated from each ensemble are
compared for particle numbers $2\le N\le 64$ for a range of temperatures above
and below the Fermi temperature. The relevance of these results for
applications of density functional theory is discussed briefly.
