Anisotropic Juttner (relativistic Boltzmann) distribution

A rigorous derivation of the Juttner (covariant Boltzmann) distribution is provided for anisotropic pressure (or temperature) tensors. It was in similar form anticipated first by Gladd (1983). Its manifestly covariant version follows straightforwardly from its scalar property

Olbertian Partition Function in Scalar Field Theory

The Olbertian partition function is reformulated in terms of continuous (Abelian) fields described by the Landau–Ginzburg action, respectively, Hamiltonian. In order to make some progress, the Gaussian approximation to the partition function is transformed into the Olbertian prior to adding the quartic Landau–Ginzburg term in the Hamiltonian. The final result is provided in the form of an expansion suitable for application of diagrammatic techniques once the nature of the field is given, that is, once the field equations are written down such that the interactions can be formulated

Auroral Kilometric Radiation and Electron Pairing

We suggest that pairing of bouncing medium-energy electrons in the auroral upward current region close to the mirror points may play a role in driving the electron cyclotron maser instability to generate an escaping narrow band fine structure in the auroral kilometric radiation. We treat this mechanism in the gyrotron approximation, for simplicity using the extreme case of a weakly relativistic Dirac distribution instead the more realistic anisotropic Juttner distribution. Promising estimates of bandwidth, frequency drift and spatial location are given

Generalised partition functions: inferences on phase space distributions

It is demonstrated that the statistical mechanical partition function can be used to construct various different forms of phase space distributions. This indicates that its structure is not restricted to the Gibbs-Boltzmann factor prescription which is based on counting statistics. With the widely used replacement of the Boltzmann factor by a generalised Lorentzian (also known as the q-deformed exponential function, where kappa = 1/vertical bar q - 1 vertical bar, with kappa, q is an element of R) both the kappa-Bose and kappa-Fermi partition functions are obtained in quite a straightforward way, from which the conventional Bose and Fermi distributions follow for kappa -> infinity. For kappa not equal infinity 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 apart 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

Lorentzian Entropies and Olbert's κ - Distribution

This note derives the various forms of entropy of a 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 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