  This note derives the various forms of entropy of systems subject to Olbert
distributions (generalized Lorentzian probability distributions known as
$\kappa$-distributions) which are frequently observed particularly in high
temperature plasmas. The general expression of the partition function in such
systems is given as well in a form similar to the Boltzmann-Gibbs probability
distribution, including a possible exponential high energy truncation. We find
the representation of the mean energy as function of probability, and provide
the implicit form of Olbert (Lorentzian) entropy as well as its high
temperature limit. The relation to phase space density of states is obtained.
We then find the entropy as function of probability, an expression which is
fundamental to statistical mechanics and here to its Olbertian version.
Lorentzian systems through internal collective interactions cause correlations
which add to the entropy. Fermi systems do not obey Olbert statistics, while
Bose systems might at temperatures sufficiently far from zero.
