  The principle of maximum entropy is used to study the geometric properties of
an ensemble of finite dimensional Hamiltonian systems with known average
energy. These geometric characterization is given in terms of the generalized
diagonal Bloch vectors and the invariants of the special unitary group in $n$
dimensions. As examples, Hamiltonians written in terms of linear and quadratic
generators of the angular momentum algebra are considered with $J= 1$ and
$J=3/2$. For these cases, paths as functions of the temperature are established
in the corresponding simplex representations, as well as the adiabatic
evolution of the interaction strengths of the Hamiltonian models. For the
Lipkin-Meshkov-Glick Hamiltonian the quantum phase diagram is explicitly shown
for different temperature values in parameter space.
