Lorentzian Entropies and Olbert's $\kappa$-distribution

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.