Density functional theory for small systems: Hard spheres in a closed spherical cavity

We present a new approach to determining the equilibrium structure of a closed (canonical) inhomogeneous fluid which combines and canonical density functional theory and a series expansion of the distribution functions in powers of (1/(N) over bar), where (N) over bar is the average number of particles. For hard spheres in a hard spherical cavity, comparison with canonical Monte Carlo results shows that the method is rather accurate, even for small values of (N) over bar. In certain (high packing) situations the density profile develops a pronounced peak in the center of the cavity. Accounting properly for such peaks provides a severe test of any density functional approximation.