Some Fundamental Properties of the Integer-States in Open-System,   Ensemble Energy-Density Functional Theories
  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. 
Steven M. Valone