Bose-Einstein and Fermi-Dirac distributions in nonextensive quantum statistics: An exact approach

Bose-Einstein (BE) and Fermi-Dirac (FD) distributions in nonextensive quantum statistics have been discussed with the use of exact integral representations for the grand canonical partition function [Rajagopal, Mendes and Lenzi, Phys. Rev. Lett. 80, 3907 (1998)]. Integrals along real axis in the case of q > 1.0 are modified by an appropriate change of variable, which makes numerical calculations feasible, q denoting the entropic index. The q dependence of coefficients in the generalized Sommerfeld expansion has been calculated. Model calculations have been made with a uniform density of states for electrons and with the Debye model for phonons. It has been shown that the linear-T electronic specific heat and the T 3 phonon specific heat at low temperatures are much increased with increasing q from q = 1.0 while they are decreased with decreasing q from unity. It is pointed out that the factorization approximation, which has been applied to many subjects in the nonextensive quantum systems, is not accurate: in particular its FD distribution yields inappropriate results for q < 1.0. Based on the exact results, we have proposed the interpolation approximation to BE and FD distributions, which yields results in agreement with the exact ones in the limits of q → 1.0, and zero and high temperatures. Applications of our approximate q-BE distribution to the black-body radiation and the Bose-Einstein condensation are also discussed. PACS No.: 05.30.-d, 05.70.Ce