  The statistical mechanical partition function can be used to construct
different forms of phase space distributions not restricted to the
Gibbs-Boltzmann factor. With a generalised Lorentzian both the Kappa-Bose and
Kappa-Fermi partition functions are obtained in straightforward way, from which
the conventional Bose and Fermi distributions follow for $\kappa\to\infty$. For
$\kappa\neq\infty$ these are subject to the restrictions that they can be used
only at temperatures far from zero. They thus, as shown earlier, have little
value for quantum physics. This is reasonable, because physical
$\kappa$-systems imply strong correlations which are absent at zero temperature
where appart from stochastics all dynamical interactions are frozen. In the
classical large temperature limit one obtains physically reasonable
$\kappa$-distributions which depend on energy respectively momentum as well as
on chemical potential. Looking for other functional dependencies, we examine
Bessel functions whether they can be used for obtaining valid distributions.
Again and for the same reason, no Fermi and Bose distributions exist in the low
temperature limit. However, a classical Bessel-Boltzmann distribution can be
constructed which is a Bessel-modified Lorentzian distribution. Whether it
makes any physical sense remains an open question. This is not investigated
here. The choice of Bessel functions is motivated solely by their convergence
properties and not by reference to any physical demands. This result suggests
that the Gibbs-Boltzmann partition function is fundamental not only to
Gibbs-Boltzmann but also to a large class of generalised Lorentzian
distributions as well as to the corresponding nonextensive statistical
mechanics.
