Lorentzian Entropies and Olbert's κ - Distribution

Abstract

This note derives the various forms of entropy of a systems subject to Olbert distributions (generalized Lorentzian probability distributions known as κ-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 a function of probability, and we 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 a function of probability, an expression that is fundamental to statistical mechanics and, here, to its Olbertian version. Lorentzian systems through internal collective interactions cause correlations that add to the entropy. Fermi systems do not obey Olbert statistics, while Bose systems might do so at temperatures that are sufficiently far from zero.

Olbert’s Kappa Fermi and Bose Distributions

“…Tsallis' theory calls it nonextensive. Recently, the analytical form of the classical (nonquantum) kappa entropy has been constructed [33], which shows that it is super-extensive while being different from its thermo-statistical cousin, indicating that the fields of application in physics and statistics presumably refer to different domains.…”

Olbertian Partition Function in Scalar Field Theory

“…with real κ > 0 and s some fixed number (for a recent account of the Olbert distribution and entropy, see, for instance [20]). This distribution applies to finite temperature β ≪ ∞ as we have discussed in previous work.…”

“…To some degree, they are related to Tsallis' thermostatistics [19]. We recently [20] investigated their connection to Olbert's entropy. 2, Here, we are interested in the role they might play in field theory which is the continuous version of the partition function [21][22][23][24][25][26].…”

Olbert's kappa Fermi and Bose distributions

“…Tsallis' theory calls it non-extensive. Recently the analytical form of the Olbert entropy has been constructed (Treumann & Baumjohann, 2020) which shows that it is superextensive while different from its thermo-statistical cousin indicating that the fields of application in physics and statistics presumably refer to different domains.…”