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

Abstract

The validity of (1-q) expansion and factorization approximations are analysed in the framework of Tsallis statistics. We employ exact expressions for classical independent systems (harmonic oscillators) by considering the unnormalized and normalized constrainsts. We show that these approxiamtions can not be accurate in the analysis of systems with many degrees of freedom. I-IntroductionEver since the presentation by Tsallis[1, 2, 3] of a new possible generalization of the statistical mechanics (Tsallis statistics) based on the nonextensive entropy,a large number of investigations has been developed concerning this subject [4]. 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. In this direction, Ref.[33] contains a discussion comparing these approximations in the context of the quantum gases. However, some important aspects of the comparison are still lacking, in particular the importance of the degrees of freedom N . This work is addressed to analyze how precise the (1−q) expansion and the factorization approximation are by considering N arbitrary. To perform this study it is convenient to consider a solvable model in order to carefully understand the degree of accuracy of these approximations. Furthermore, the chosen model must contain important features of other more realistic ones. This is just the case of a set of harmonic oscillators.In this direction, to analyze the (1 − q) expansion and the factorization approximation, we focus our discussion on the classical Tsallis statistics by considering N one dimesional harmonic oscillators, i. e., we employ the Hamiltonian H =

Extensive form of equilibrium nonextensive statistics

“…(1) is supposed for composite systems containing N statistically independent subsystems in order to elicit the nonextensive character by following relation : (2) as one often finds in the literature. Due to this independence, it has been believed by many that exact calculations within NSM should use the additive hamiltonian H 0 = N n=1 H n , where H n is the hamiltonian of n th subsystem [2,[7][8][9][10][11]. However, this hamiltonian is not compatible with neither Eq.…”

Quantum vibrational partition function in the non-extensive Tsallis framework

“…Energy fluctuation in the canonical ensemble and ensemble equivalence in non-extensive statistics were investigated by Liyan and Julian using the generalized ideal gas and the generalized harmonic oscillators [10]. It has also been demonstrated that the (q-1) expansion on the factorization approximation is not useful for a system with N harmonic oscillators when one employs an arbitrary q and a very large N [19]. The sensitivity of the population of state to the value of q has been studied for some two-level model systems by considering the harmonic oscillator model and spin-1/2 [20].…”

Calculation of Generalized Level Densities for Nuclei in Mass Region 20 < A < 50

“…. For the case previous works indicate that in the thermodynamic limit ( ∞ → N ) employing nonextensive statistical mechanics is not suitable to the classical ideal gas [34], the classical systems with N harmonic oscilators [38] and Fermi systems in a general power-law external potential [39]. Therefore, in this work, we consider only the case − .…”