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/|q-1|, with kappa, q elements of {R} both the Kappa-Bose and Kappa-Fermi partition functions are obtained in quite a straightforward way, from which the Bose and Fermi distributions follow for kappa ~ ∞. These are subject to the well-known restrictions that they can be used only at temperatures far from zero. They thus, as shown earlier, have little value for quantum physics, which is reasonable, because physical kappa-systems imply strong correlations. These are absent at zero temperature where appart from stochastics all dynamical interactions are frozen. In the classical large temperature limit one obtains physically reasonable 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. Whether it makes any physical sense remains an open question. It is not investigated here. The choice of Bessel functions is motivated solely by their convergence properties and not by reference to any physical demands.

Further Information
  author = {Treumann, R. A. and Baumjohann, W.},
  doi = {10.5194/angeo-34-557-2016},
  journal = {Annales Geophysicae},
  language = {en},
  note = {arXiv:1604.05295 [cond-mat.stat-mech], 8 pages, no figures},
  pages = {557-564},
  title = {Generalised partition functions: Inferences on phase space distributions},
  url = {},
  volume = {34},
  year = {2016},
%O Journal Article
%A Treumann, R. A.
%A Baumjohann, W.
%R 10.5194/angeo-34-557-2016
%J Annales Geophysicae
%G en
%O arXiv:1604.05295 [cond-mat.stat-mech], 8 pages, no figures
%P 557-564
%T Generalised partition functions: Inferences on phase space distributions
%V 34
%D 2016