  We study four distinct families of Gibbs canonical distributions defined on
the standard complex, quaternionic, real and classical (nonquantum) two-level
systems. The structure function or density of states for any two-level system
is a simple power (1, 3, 0 or -1) of the length of its polarization vector,
while the magnitude of the energy of the system, in all four cases, is the
negative of the logarithm of the determinant of the corresponding
two-dimensional density matrix. Functional relationships (proportional to
ratios of gamma functions) are found between the average polarizations with
respect to the Gibbs distributions and the effective polarization temperature
parameters. In the standard complex case, this yields an interesting
alternative, meeting certain probabilistic requirements recently set forth by
Lavenda, to the more conventional (hyperbolic tangent) Brillouin function of
paramagnetism (which, Lavenda argues, fails to meet such specifications).
