  We initiate the recently proposed $\boldsymbol{w}$-ensemble one-particle
reduced density matrix functional theory ($\boldsymbol{w}$-RDMFT) by deriving
the first functional approximations and illustrate how excitation energies can
be calculated in practice. For this endeavour, we first study the symmetric
Hubbard dimer, constituting the building block of the Hubbard model, for which
we execute the Levy-Lieb constrained search. Second, due to the particular
suitability of $\boldsymbol{w}$-RDMFT for describing Bose-Einstein condensates,
we demonstrate three conceptually different approaches for deriving the
universal functional in a homogeneous Bose gas for arbitrary pair interaction
in the Bogoliubov regime. Remarkably, in both systems the gradient of the
functional is found to diverge repulsively at the boundary of the functional's
domain, extending the recently discovered Bose-Einstein condensation force to
excited states. Our findings highlight the physical relevance of the
generalized exclusion principle for fermionic and bosonic mixed states and the
curse of universality in functional theories.
