Thermodynamics of boson and fermion systems with fractal distribution functions

Abstract

Starting with the fractal inspired distribution functions for Maxwell-Boltzmann, Bose-Einstein and Fermi systems, as reported by F. Büyükkiliç and D. Demirhan, we obtain the corresponding probability distributions and study their thermodynamic behavior. We compare our results with those corresponding to ideal gases (q = 1), and Bose-Einstein and Fermi systems with quantum group symmetry. In particular, we show that the hamiltonian that gives the Bose-Einstein generalized distribution function can be interpreted as a q-deformation of the ideal gas hamiltonian. PACS number(s):05.30.-d

Correlation functions in the factorization approach of nonextensive quantum statistics

“…According to our previous work [9], the parameter q cannot be any real number but its values are restricted to those such that 1/(q − 1) is an integer. In addition, the thermodynamic functions are well defined only in the interval 1 ≤ q ≤ 1.5.…”

“…However, a comparison of the heat capacity and entropy functions, in Ref. [9], for systems with a particle distribution function as given by Eq. (8) with those for Bose and Fermi gases described by quantum group invariant hamiltonians [11] shows that nonextensivity is unrelated to quantum group invariance.…”

“…[9] we made a study of the basic thermodynamics that result from the particle distribution functions in Eq. (8) for classical and quantum gases.…”

Remarks on (1−q) expansion and factorization approximation in the Tsallis nonextensive statistical mechanics

“…Motivated by these difficulties some approximated methods, such as (1 − q) expansion [11,14,21,22], factorization approximation [23], perturbative expansion [24], generalized Bogoliubov inequality [24,25], and semi-classical expansion [26], have been developed. The two first methods mentioned above have been employed largely in the analysis of black body radiation [11,13,14] and in other many independent particle systems [27,28,29,30,31,32], but without a careful analysis of the validity of these methods. Thus, a detailed discussion of the (1 − q) expansion and the factorization approximation plays a special role in this scenario.…”

“…These investigations are basically employed in the discussion of aspects related to nonextensive phenomena, such as, Lévy-type anomalous superdiffusion [5], Euler turbulence[6], self-gravitating systems and related themes [6,7,8,9,10], cosmic background radiation [11,12,13,14] [11,13,14] and in other many independent particle systems [27,28,29,30,31,32], but without a careful analysis of the validity of these methods. Thus, a detailed discussion of the (1 − q) expansion and the factorization approximation plays a special role in this scenario.…”