  We derive explicit expressions for the volume elements of both the minimal
and maximal monotone metrics over the (n^{2} - 1)-dimensional convex set of n x
n density matrices for the cases n = 3 and 4. We make further progress for the
specific n = 3 maximal-monotone case, by taking the limit of a certain ratio of
integration results, obtained using an orthogonal set of eight coordinates. By
doing so, we find remarkably simple marginal probability distributions based on
the corresponding volume element, which we then use for thermodynamic purposes.
We, thus, find a spin-1 analogue of the Langevin function. In the fully general
n = 4 situation, however, we are impeded in making similar progress by the
inability to diagonalize a 3 x 3 Hermitian matrix and thereby obtain an
orthogonal set of coordinates to use in the requisite integrations.
