Nonextensive quantum H-theorem

Abstract

A proof of the quantum H-theorem taking into account nonextensive effects on the quantum entropy S Q q is shown. The positiveness of the time variation of S Q q combined with a duality transformation implies that the nonextensive parameter q lies in the interval [0,2]. It is also shown that the equilibrium states are described by quantum q-power law extensions of the Fermi-Dirac and Bose-Einstein distributions. Such results reduce to the standard ones in the extensive limit, thereby showing that the nonextensive entropic framework can be harmonized with the quantum distributions contained in the quantum statistics theory.

Generalized ensemble theory with non-extensive statistics

“…Furthermore, as far as the non-extensive quantum statistical mechanics is concerned, the generalized Bose-Einstein distribution for bosons and Fermi-Dirac distribution for fermions (referred to as q-BED and q-FDD hereafter) have been used [8][9][10][11][12][13][14][15]. However, in literature there is a confusion…”

“…In the work of [8][9][10][11][12][13], the particle distribution functions in the non-extensive quantum statistics are shown as:…”

Nonextensive kinetic theory and H-theorem in general relativity

“…Even the nonrelativistic distribution is also obtained as a limiting case of expression (31). Physically, it is also very compelling that the heart of the H-theorem, that is, the fact that the entropy source is non-negative, as required by the second law of thermodynamics, constrains the nonextensive q-parameter on the same interval [0, 2] irrespective of the physical regime [6][7][8].…”

“…In principle, the Boltzmann "Stossahlansatz", as well as the own measure of entropy will be modified [6][7][8]. The new hypothesis combined with changes in the entropy flux and collisional term, encode all the statistical properties of the nonextensive microscopic description of a dilute quantum gas.…”

“…Special cases like the special relativistic, classical and quantum limits [6][7][8] are all covered here. Even the nonrelativistic distribution is also obtained as a limiting case of expression (31).…”