  Gibbsian statistical mechanics is extended into the domain of non-negligible
{though non-specified} correlations in phase space while respecting the
fundamental laws of thermodynamics. The appropriate Gibbsian probability
distribution is derived and the physical temperature identified. Consistent
expressions for the canonical partition function are given. In a first
application, the corresponding Boltzmann, Fermi and Bose-Einstein distributions
are obtained. It is shown that the latter lose their typical quantum
properties, i.e. the degenerate Fermi state and Bose-Einstein condensation.
These distributions apply only to states at finite temperature with
correlations. As a by-product these results \emph{exclude any negative absolute
temperatures} also in the Boltzmann limit.
