  We present and compare two families of ensembles of random density matrices.
The first, static ensemble, is obtained foliating an unbiased ensemble of
density matrices. As criterion we use fixed purity as the simplest example of a
useful convex function. The second, dynamic ensemble, is inspired in random
matrix models for decoherence where one evolves a separable pure state with a
random Hamiltonian until a given value of purity in the central system is
achieved. Several families of Hamiltonians, adequate for different physical
situations, are studied. We focus on a two qubit central system, and obtain
exact expressions for the static case. The ensemble displays a peak around
Werner-like states, modulated by nodes on the degeneracies of the density
matrices. For moderate and strong interactions good agreement between the
static and the dynamic ensembles is found. Even in a model where one qubit does
not interact with the environment excellent agreement is found, but only if
there is maximal entanglement with the interacting one. The discussion is
started recalling similar considerations for scattering theory. At the end, we
comment on the reach of the results for other convex functions of the density
matrix, and exemplify the situation with the von Neumann entropy.
