  Ensemble averages are an approximation technique for connecting macroscopic
and microscopic properties of a system. For systems open with respect to
exchange of particles with a bath, the microscopic states are those with
integer numbers of particles. When a property of the open system is represented
as an ensemble average over these microscopic states, self-consistency dictates
several implications for the properties of both for open-system energy density
functionals and the integer-state functionals describing the microscopic
states. The first is that each integer-state energy density functional is a
functional of the original type discovered by Levy. Another is that the
dependence of the open-system functional on the ensemble density is linear
whereas the dependence of the integer-state functional is decidedly nonlinear.
Finally, the derivative discontinuity behavior with respect to particle number
of some open-system density functionals appears to be connected to the
long-range behavior of the effective external potentials of the integer-state
functionals governing the interactions among subsystems in the ensemble.
