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How the Nonextensivity Parameter Affects Energy Fluctuations Ezat Keshavarzi *,1, Abbas Helmi1, and Mohammad Kamalvand2 1 Physical chemistry group, Chemistry Department, Isfahan University of Technology, Isfahan, Iran, 8415683111 2 Department of Chemistry, Faculty of Science, Yazd University, Yazd, Iran, 89195741 * Corresponding author: E-mail: [email protected] Fax: +98-311-391-2350 Tel: +98-311-391-3281 Abstract In this article, the effect of the nonextensivity parameter on the energy fluctuations of nonextensive systems has been studied in two different versions of the Tsallis statistical mechanics. While a general expression for the energy fluctuations in the second version has been reported, the ideal gas and harmonic oscillator have been studied in the second and fourth versions of the Tsallis statistical mechanics. Our results in the fourth version show that the relative energy fluctuations are strongly affected by the nonextensivity parameter via the number of accessible states. In fact, in the case of subextensive systems, the nonextensivity parameter causes lower accessible states in comparison with the extensive systems and therefore, smaller relative energy fluctuations are expectable. But for super-extensive systems the relative energy fluctuations are larger than extensive systems because of the more accessible states which are available for a system. Our studies show that the un-normalized nature of second version causes very large relative energy fluctuations and in some cases limit its application. 1 Keywords: Energy fluctuations; nonextensive systems; Harmonic oscillator; ideal gas; Tsallis statistical mechanics 1. Introduction In the recent years the generalized Tsallis statistical mechanics has been of interest for many researchers in different fields such as physics, chemistry, astrophysics, and biology, as well as engineering. They have applied it to predict the phenomena in different nonextensive systems for which Boltzmann-Gibbs statistics fails [1-8]. In this statistical mechanics the entropy is defined as [9]: π π π = πβ1 (1 β βπ€ π ππ ) (1) where ππ is the probability function of finding the system in iβth state, k is a positive constant, and q is the entropic index related to the degree of the nonextensivity. Various expressions for probability function, and consequently for partition function may be obtained via using different energy constraints, in normalized and un-normalized forms, by extremizing the entropy function in the canonical ensemble. So far four different energy constraints have been used which only one of them (known as 2nd version) is un-normalized [5,9-10]. The energy fluctuations of simple model systems, such as free particle and the harmonic oscillator, are strongly dependent on the version, beside of the kind of statistical mechanics, in which the system is investigated. Therefore in the literature, there are several studies on the energy fluctuations for ideal gas and harmonic oscillator in different versions of Tsallis statistics [11-13]. For example, an expression for energy fluctuation in the first version of Tsallis statistical mechanics has been reported in 2003[13]. The relative energy fluctuation in this version consists of two terms, the related heat capacity term and the square of the mean energy. In this work, it has been reported that the relative energy fluctuation for a large system is small with respect to the Boltzmann-Gibbs statistics [13]. For the third 2 version, Liyan and Jiulin [12] showed that the energy fluctuation for an ideal gas with a large number of particles when q<1, is thoroughly negligible, and hence the ensemble equivalence is also achieved for different ensembles in the Tsallis statistical mechanics. They have also reported that the relative energy fluctuation is related to 1/N in nonextensive statistics instead of 1/βN in extensive statistics [12]. The investigation of the energy fluctuations for ideal gas in nonextensive (Tsallis) and extensive (Boltzmann-Gibbs) statistics also reveals that the correlation which is induced by the nonextensivity of Tsallis entropy, has an important role in energy fluctuations [11]. It has been also claimed [11] that the introduction of the correlations between particles energy leads to the smaller energy fluctuations. But it should be noted that all of these studies [11-13] have been done on subextensive systems, where q is smaller than unity. Additionally, there isn't any investigation on the energy fluctuations in the second and fourth versions. While our aim in this article is to investigate the energy fluctuations in the second and fourth versions, but the main question is whether the existence of the nonextensivity parameter in the Tsallis statistical mechanics always causes smaller fluctuations? In fact, how can nonextensivity parameter affect the energy fluctuations? To do so, at first we have obtained the general expression for energy fluctuations in the second version of Tsallis statistics. The ideal gas and harmonic oscillator, as simple textbook examples, have been investigated in the second version for q smaller and larger than unity. Also, the fourth version of Tsallis statistical mechanics, as the most recent normalized version, has been selected to investigate the role of the nonextensivity parameter in the energy fluctuation. The remainder of the paper has been organized as follows: in Sec. 2 the energy fluctuation in second version of the Tsallis statistics is derived, in Sec 3 the energy fluctuation in 2th and 4nd versions of the Tsallis statistical mechanics for an ideal gas and a quantum harmonic oscillator are discussed and compared with the energy fluctuation in the Boltzmann-Gibbs statistical mechanics and finally in Sec. 4 some conclusions are reported. 3 2. The energy fluctuations in the second version of the Tsallis statistical mechanics As it has been already mentioned, the different energy constrains, un-normalized and normalized, in the second and fourth versions have been used to obtain the probability function in canonical ensemble in the Tsallis statistical mechanics. Therefore, the different partition functions in these two versions have been obtained corresponding to these different constraints as: π€ π€ π β ππ ππ β Μ Μ Μ πΈπ = 0 ; (2) ππ 1 = β[1 β (1 β π)π½ππ ](1βπ) 1 π (2) π€ π€ β(ππ β Μ Μ πΈΜ Μ π )ππ π = 0 ; π 1 (1βπ) ´ Μ Μ Μ Μ π(4) π = β[1 β (1 β π)π½ (ππ β πΈπ )] π where superscripts denote the number of the statistics versions. As it is well known, all the average quantities in the second version are un-normalized. Therefore in this version, a different behavior in some characteristic of a system expects. For example, we know that the energy variance in normalized statistical mechanics in the canonical ensemble is defined as [14]: Μ Μ Μ Μ Μ Μ Μ Μ 2Μ β πΈ 2 π₯πΈ 2 = (πΈΜ β πΈ)2 = πΈ (3) In fact Eq. (3) may be applied for the Boltzmann-Gibbs statistics and also all normalized versions of Tsallis statistics (1st, 3rd and 4th versions). But in the second version the square root of the energy variance is not equal to the energy fluctuation, β(πΈΜ β πΈ)2 β βΜ Μ πΈΜ 2Μ β πΈ 2 and (πΈΜ β πΈ)2 can be calculated as follows: π€ Μ Μ Μ Μ Μ Μ Μ Μ 2Μ β 2πΈ 2 + πΈ 2 β ππ π β πΈ Μ Μ Μ 2Μ β πΈ 2 π₯πΈ 2 = πΈ (4) π=0 In Eq. (4) inequality is due to β©1βͺπ β 1. In fact, in the second version of the Tsallis statistics we have [5]: 4 w Μ Μ Μ q β Pi q = Zq1βq + (1 β q)Ξ²E (5) i Therefore, the following result will be obtained for energy fluctuation: 2 Μ Μ Μ Μ Μ Μ Μ Μ π ] π₯πΈ 2 = Μ πΈΜ Μ 2Μ + πΈ [ππ 1βπ β 2 + (1 β π)π½πΈ (6) Μ Μ Μ Μ Μ πΈπ 2 in the second version of the Tsallis statistical mechanics is defined as: π€ π€ Μ Μ Μ Μ Μ πΈπ 2 = β ππ 2 π 1 1 π ππ π = β π [β ππ [1 β (1 β π)π½ππ ](1βπ) ] ππ ππ½ (7) π the term in the bracket may be written as ππ βπ€ π ππ ππ . Therefore, Eq. (7) can be rearranged as: π€ Μ Μ Μ Μ Μ Μ Μ Μ π (β ππ ππ ) β πΈπ 2 = [πΈ 1 π€ 1 π β ππ ππ ] ππ (πβ1) ππ½ π (8) Using the definition of the probability function, Eq. (2), in this version it is possible to Μ Μ Μ write βπ€ π ππ ππ verses πΈπ and ππ as: π€ β ππ ππ = 1 π€ 1 ππ π€ π (1βπ) [β ππ ππ β (1 β π)π½ β ππ 2 ππ π ] 1 = 1 ππ (1βπ) 1 Μ Μ Μ Μ Μ 2 ] Μ Μ Μ π β (1 β π)π½πΈ [πΈ π (9) By insertion of Eq. (9) into Eq. (8) we have: 1 Μ Μ Μ Μ Μ Μ Μ Μ Μ Μ 2 ]) Μ Μ Μ π β (1 β π)π½πΈ πΈπ 2 = Μ Μ Μ πΈπ ( (1βπ) [πΈ π ππ Μ Μ Μ Μ Μ 2 ] Μ Μ Μ π β (1 β π)π½πΈ π π [πΈ β ( (πβ1) ( )) (1βπ) ππ½ ππ ππ 1 where for second term in right hand side we have: 5 (10) Μ Μ Μ Μ Μ 2 ] Μ Μ Μ π β (1 β π)π½πΈ π π [πΈ ( ) (1βπ) ππ½ ππ ππ (1βπ) = π Μ Μ Μ Μ Μ Μ Μ Μ 2 ] β [πΈ Μ Μ Μ Μ Μ 2 ] π π (1βπ) Μ Μ Μ π β (1 β π)π½πΈ [πΈ β (1 β π)π½πΈ π π ππ½ π ππ½ π (ππ (1βπ) ) 2 (11) and Μ Μ Μ Μ Μ 2 ] Μ Μ Μ π β (1 β π)π½πΈ π π [πΈ ( ) ππ (πβ1) ππ½ ππ (1βπ) 1 = π Μ Μ Μ Μ Μ 2 ] β 1 [πΈ Μ Μ Μ Μ Μ 2 ] π π (1βπ) (12) Μ Μ Μ π β (1 β π)π½πΈ Μ Μ Μ π β (1 β π)π½πΈ [πΈ π π ππ½ ππ½ π ππ (1βπ) hence 1 Μ Μ Μ Μ Μ Μ Μ Μ Μ Μ 2 ]) β π [πΈ Μ Μ Μ Μ Μ 2 ] Μ Μ Μ π β (1 β π)π½πΈ Μ Μ Μ π β (1 β π)π½πΈ πΈπ 2 = Μ Μ Μ πΈπ ( (1βπ) [πΈ π π ππ½ ππ + 1 Μ Μ Μ Μ Μ 2 ] π π (1βπ) Μ Μ Μ π β (1 β π)π½πΈ [πΈ π ππ½ π ππ (1βπ) (13) and after some simplifications we can have: Μ Μ Μ Μ Μ πΈπ 2 = 2 Μ Μ Μ πΈπ ππ β (1βπ) + where π ππ½ Μ Μ Μ Μ Μ πΈπ 2 = Μ Μ Μ Μ Μ 2 Μ Μ Μ π Μ Μ Μ π (1 β π)π½πΈ ππΈ ππΈ π 2 Μ Μ Μ Μ Μ Μ Μ Μ Μ Μ 2 (1 πΈ β + β π)π½ + (1 β π)πΈ π π (1βπ) ππ½ ππ½ ππ 1 ππ (1βπ) ( π Μ Μ Μ Μ Μ 2 ] Μ Μ Μ π β (1 β π)π½πΈ π (1βπ) ) [πΈ π ππ½ π (14) Μ Μ Μ π and finally: ππ (1βπ) = β(1 β π)πΈ Μ Μ Μ Μ Μ 2 Μ Μ Μ π Μ Μ Μ π Μ Μ Μ π 2 (1 β π)πΈ (1 β π)π½πΈ ππΈ ππΈ π 2 2 Μ Μ Μ Μ Μ Μ Μ Μ Μ Μ β πΈπ β + (1 β π)π½ + (1 β π)πΈπ β ππ½ ππ½ ππ (1βπ) ππ (1βπ) ππ (1βπ) 2 Μ Μ Μ πΈπ Μ Μ Μ π (1 β π)2 π½πΈ Μ Μ Μ Μ Μ + πΈπ 2 (1βπ) ππ (15) 2 π Μ Μ Μ Now we subtract (2 β βπ€ π ππ )πΈq from the both sides of Eq. (15) and we have: 6 Μ Μ Μ Μ Μ Μ Μ π₯πΈπ 2 = Μ Μ Μ Μ Μ 2 Μ Μ Μ π Μ Μ Μ π Μ Μ Μ π 2 (1 β π)πΈ (1 β π)π½πΈ ππΈ ππΈ π 2 2 Μ Μ Μ Μ Μ Μ Μ Μ Μ Μ β πΈπ β + (1 β π)π½ + (1 β π)πΈπ β ππ½ ππ½ ππ (1βπ) ππ (1βπ) ππ (1βπ) 2 Μ Μ Μ πΈπ π€ Μ Μ Μ π (1 β π)2 π½πΈ 2 Μ Μ Μ Μ Μ + πΈπ 2 β (2 β β ππ π ) Μ Μ Μ πΈq (1βπ) ππ π (16) And finally by rearrangement of the Eq. (16) the following equation may be obtained: Μ Μ Μ Μ Μ 2 Μ Μ Μ π ππΈ ππΈ 2 2 π Μ Μ Μ Μ Μ Μ Μ Μ Μ Μ Μ Μ 2 β Μ Μ Μ Μ Μ Μ (π΄ (1 π₯πΈπ 2 = π΄ (πΈ πΈ ) + πΈ β π΅) β + β π)π½ π π π ππ½ ππ½ (17) where A and B coefficients are as follow: π΄ = (1 β π) + Μ Μ Μ π (π 2 β π)π½πΈ ππ (1βπ) Μ Μ Μ π β ππ (1βπ) β , π΅ = (2 + π½(π β 1)πΈ π ππ (1βπ) ) (18) Therefore, the energy variance in second version of the Tsallis statistics is as follows: Μ Μ Μ Μ Μ Μ Μ π₯πΈπ 2 = β Μ Μ Μ Μ Μ 2 π΄ β π΅ Μ Μ Μ π 1 β π ππΈ 1 ππΈ 2 π Μ Μ Μ + π½ + πΈ 1 β π΄ ππ½ 1 β π΄ ππ½ 1βπ΄ π (19) As we have shown in Eq. (19), the expression of the energy fluctuation in this version has been composed of three terms. Comparison of Eq. (19) with those expressions for energy fluctuations in other versions of the Tsallis statistics [11-13] and also with that in the Boltzmann-Gibbs [14], shows that the first term in the right hand side of Eq. (19), the heat capacity's term, is a common term in the energy fluctuation for all systems including extensive and nonextensive, in normalized and un-normalized versions. The second term in the right hand side of Eq. (19) which is related to Μ Μ Μ Μ Μ 2 ππΈ π ππ½ , is appeared in energy variance for nonextensive systems in the second and third versions of Tsallis statistic. This term is originated from the definitions of the average quantity in a weighted form, ππ π . But the 2 third term in the right hand side of Eq. (19) which is related to theΜ Μ Μ Μ πΈπ , is appeared in the first and second versions as it has been already reported by Potiguar and Costa for the first version [13]. 7 It should be noted that Eq. (19) in the limit of qβ1, tends to that corresponding equation in the Boltzmann Gibbs statistics as: 1 Μ Μ Μ Μ Μ Μ Μ 2 )2 (π₯πΈ π Μ Μ Μ πΈπ Μ Μ Μ π 1/2 1 ππΈ 1 =β ( ) = (ππ 2 πΆπ£ )1/2 Μ Μ Μ Μ Μ Μ πΈπ ππ½ πΈπ (20) In the next section, the effect of the nonextensivity parameter on the energy fluctuation, in both normalized and un-normalized versions, has been discussed in detail for generalized ideal gas and harmonic oscillator cases. Also they have been compared with those corresponding values in Boltzmann-Gibbs. 3. Energy fluctuations for generalized ideal gas and harmonic oscillator in 2nd and 4th versions of Tsallis statistics The relative energy fluctuations for ideal gas in the second version of Tsallis statistical mechanics have been calculated using Eq. (19). However, for the case of ideal gas, although analytical partition functions are accessible in some cases [15], all data has been obtained numerically, because Μ Μ Μ Μ Μ πΈπ 2 cannot be obtained directly from the partition function. To obtain the required quantities, the following partition function has been used: π£πππ₯ ππ = β« 0 1 (1βπ) π Ρ(πΈ) [1 β π½(1 β π)( ππ£ 2 )] ππ£ 2 (21) where Ξ¦(E) is the degeneracy for an ideal gas velocity [14] and vmax is the maximum allowed velocity for the ideal gas molecules: 1 2πππ2 Ξ¦(πΈ) = Ξ(π+1)Ξ(3π/2) ( β π£πππ₯ = { β2 3π/2 ) (22) πβ₯1 2 β2π½ππ(1βπ) (23) π<1 In Fig. 1 the relative energy variance (REV), and its three parts, according to Eq. 19, have been plotted versus q. These data have obtained for an Argon molecule which is in a 8 macroscopic enclosure at T=300K. The value of the REV in Boltzmann-Gibbs statistical mechanics, when q=1, has been also marked. According to this Figure, the relative energy fluctuation is usually greater than Boltzmann-Gibbs ones. In fact, only for a very small region around unity, when q is smaller than unity, the relative energy variance is less than the Boltzmann-Gibbs. It should be noted that the REV is used instead of relative energy fluctuation, because sometimes some terms in Eq. (19) are negative. However, the reported REV and relative energy fluctuation are two monotonic functions. The important fact, which may be understood from this figure, is the domination of third term of Eq. (19) in the REV at different values of q except when qβ1- (where the REV is less than Boltzmann-Gibbs). Since the un-normalized character of this version affects the values of this term, we may say that the more REV for ideal gas with respect to the BoltzmannGibbs arises from the unnormalized nature of this version. To understand more about the role of the third term in REV, we have obtained the REV for quantum harmonic oscillator in this version too. To do so, we have used Eq. (19). The partition function in second version for a quantum harmonic oscillator is as follows: ππππ₯ ππ£ππ,π 1 1βπ 1 = β [1 β π½(1 β π) (π + ) βπ£] 2 (24) π=0 where v is natural frequency of the oscillator and nmax can be obtained as follows: β ππππ₯ = { πβ₯1 2βπ½βπ£(πβ1) πππ‘ππππ ( 2π½βπ£(1βπ) ) π<1 (25) It should be noted that for q<1 the partition function can be obtained simply by direct summation over a finite number of states and for 1<q<2 an analytical expression can be obtained for the partition function [16]. However, as in ideal gas case, all quantities in Eq. (19) for harmonic oscillator have been calculated numerically too. In Fig. 2 the REV and its three parts according to Eq. (19) have been plotted for quantum harmonic oscillator and compared with the corresponding value in the Boltzmann-Gibbs 9 statistics for π½βπ£ = 1. Clearly, the REV is less than Boltzmann-Gibbs when q is smaller than unity and will be greater than Boltzmann-Gibbs when q>1. It should be noted that the anomalous behavior of the variation of the REV for harmonic oscillator when qβ0.6-0.7, is related to the identity of number of accessible states, nmax, in this region. In fact, when q is smaller than unity to get the positive probability in this version, maximum quantum number is ππππ₯ = 2βπ½βπ£(πβ1) 2π½βπ£(1βπ) ; therefore, for π½βπ£ = 1, by variation of q from 0.6 to 0.7, nmax varies from 3 to 3.83; thus, the number of accessible states in this region is 3. This limitation leads to discontinuity of the mean energy versus q and results in the irregular behavior of REV for the harmonic oscillator. The other point which may be understood from this figure is that the third term does not play the main role in REV contrary to the pervious case, ideal gas. To understand this fact that when this term has a dominate role in the REV, we have studied the number of accessible states for these two cases, ideal gas and harmonic oscillator. This is because it is well known that the energy fluctuation is strongly affected by the number of accessible states. In Fig. (3-a), the number of accessible states, the three dimensional velocity distribution of an argon atom at 300K as an example of ideal gas has been plotted in Tsallis and Boltzmann Gibss statistics. To do so, we have used the following expression for the velocity distribution for an ideal gas: 1 (1βπ) π [1 β π½(1 β π) ( ππ£2 )] 2 π(π) = ππ (26) where Zq can be obtained from Eq. (21). It should be noted that in the case of Tsallis statistics when q<1, for some velocities the distribution function will be negative which is not acceptable physically and therefore, the distribution function has a cut-off. The cut-off velocity in 2nd version of the Tsallis statistics has been obtained according to Eq. (23). As it is clear from Fig. (3-a), the velocity distribution is strongly dependent on the entropic index, q and the range of accessible velocities increases with increasing q. 10 In Fig. (3-b) the number of accessible states for a quantum harmonic oscillator in 2nd version of the Tsallis statistics has been shown. The cut-off condition for a quantum harmonic oscillator in 2nd version of the Tsallis statistics has been cited in Eq. (25). Again, the number of the accessible states is dependent on q and increases with increasing q, but it should be noted that the number of accessible states for harmonic oscillator is much smaller than that of ideal gas. Since the role of the third term in energy fluctuation for harmonic oscillator is so smaller than that of its role for ideal gas, it seems that the role of the third term in energy fluctuation is strongly dependent on the number of accessible states and also the shape of probability function. As it is clear from the above discussion, the REV in the second version when q>1 for both ideal gas and harmonic oscillator, are more than Boltzmann-Gibbs since in this region, all the three terms in Eq. (19) determine the value of the REV. According to Figs. 1 and 2 in this region, this behavior isnβt originated only from the un-normalized picture of this version although this part has an important role. But for q<1, as it is clear from Fig. 1, the larger relative energy fluctuation with respect to the Boltzmann Gibbs, is mainly because of the third term. But for the case of harmonic oscillator, the third term, because of the less accessible states, does not play a significant role in the energy fluctuation and therefore, the REV is less than Boltzmann-Gibbs. To further clarify the role of the nonextensivity parameter in the REV and also to remove the effect of un-normalization form of the average energy, the REV in fourth version in which the energy constrain is normalized will be investigated. In this version of the Tsallis statistics, the partition function can be obtained via ππ = π£πππ₯ Ρ(πΈ) [1 β«0 π 1 (1βπ) 2 β π½(1 β π)( 2 ππ£ β πΈπ )] 11 ππ£ (27) where πΈπ = 3/2πππ is the mean value of the energy for ideal gas and similar to the Boltzmann-Gibbs statistics, its value is only dependent on the temperature. vmax in Eq. (27) should be obtained from the following equation: β πβ₯1 2 π£πππ₯ = { β2π½(1β 3π )ππ(1βπ) 2 (28) π<1 In this manner, for a harmonic oscillator, the following partition function in 4th version of the Tsallis statistics may be achieved: ππππ₯ 1 ππ = β [1 β (1 β π)π½((π + 1/2)βπ£ β Μ Μ Μ πΈπ )](1βπ) (29) π=0 where πΈπ is the mean value of the energy for the harmonic oscillator and nmax can be obtained from the following equation: β ππππ₯ = { πππ‘ππππ ( πβ₯1 Μ Μ Μ Μ 2β(1βπ)π½(βπ+2πΈ π) (1βπ)π½βπ ) π<1 (30) It is noteworthy that all quantities in 4th version of the Tsallis statistics should be calculated iteratively. However, in the case of ideal gas, because the mean energy is independent of the entropic index, q, and calculations donβt need iteration and partition function can be obtained directly. Here, all calculations can be done numerically. The results related to relative energy fluctuation for Argon atom in a macroscopic three dimensional box at T=300K have been shown in Fig. 4 for 4th version of the Tsallis statistics. As it has been shown in this figure, when entropic index q is smaller than unity, relative energy variance is smaller than Boltzmann-Gibbs value, 2/3, and when q is greater than 1, relative energy fluctuation is greater than 2/3. Also, the relative energy fluctuation for a quantum harmonic oscillator in 4th version of the Tsallis statistics has been shown in this figure. As is clear from this figure, the behavior of relative energy variance of harmonic oscillator is quite similar to the ideal gas. Since the form of the probability function in 2nd and 4th versions of 12 the Tsallis statistics is similar, we may say that the relative energy fluctuations are strongly affected by the nonextensivity parameter via the number of accessible states. In fact, in the case of subextensive systems, the nonextensivity parameter causes lower accessible states in comparison with the extensive system and therefore, smaller relative energy fluctuations are expected. But for super-extensive systems the relative energy fluctuations are larger than extensive systems because of the more accessible states which are available for a system The last important object which should be discussed in this field is the relation between the relative energy fluctuation and the number of the system particles, N. In Fig. 5 the relative energy variance for an ideal gas with a different number of particles has been shown in the fourth version. As it has been mentioned in introduction [12], that the energy fluctuation in 3th version of the Tsallis statistics is related to 1/N instead of 1/βπ in Boltzmann-Gibbs statistics. It means smaller relative energy fluctuations with respect to the Boltzmann Gibbs statistics. According to Fig. 5 this result is correct only for q<1, and as we have shown for q>1, like one particle system, the relative energy fluctuation of a nonextensive system is always greater than extensive one. It is so interesting that for the case of nonextensive one, the slope of variation of relative fluctuation with q increases with the number of particles. This could be due to the increase in the number of accessible states via the dependency of the degeneracy, Ξ¦(E), to the number of system particles. This is noticeable that the entropic index q is related to the particle number N for q>1 and when N tends to infinity, q approaches to unity [17]; therefore, a macroscopic ideal gas with q>>1 is not available mathematically. 4. Conclusion In this article the effect of the non-extensivity parameter on the relative energy fluctuations has been studied in two different versions of the Tsallis statistical mechanics; second and 13 fourth versions. It is clear that the amount of the relative energy fluctuations is strongly dependent on the version in which the systems are considered. Our results in the second version show that un-normalized nature of this version essentially affects the energy fluctuations. In fact, for the cases in which the number of accessible states is so large, the un-normalized picture of this version causes an anomalously large fluctuation. It is obvious the results obtained in this version are not general and cannot extend for the other versions because of its un-normalized definition for average thermodynamic quantities. Therefore to get general results for the energy fluctuations and its comparison with those corresponding values in the extensive systems, the investigation of this subject in a normalized version was necessary. To do so, the fourth version was selected and we believe that the results in this version may be considered as the general results for the non-extensive systems in Tsallis statistical mechanics. Our investigations reveal that the energy fluctuation is strongly dependent on the non-extensivity parameter via the number of the accessible states. In this manner when q is less than unity, the relative energy fluctuation is less than Boltzmann-Gibbs and when q is more than unity, the energy fluctuations will be more. This result is also acceptable for systems with N>1 particles. In fact the energy fluctuations for sub-extensive systems are always lower than extensive ones, while this result will be reversed in comparison with the super-extensive systems with extensive one. In the other words, introduction of the dynamic correlation between particles via the non-extensivity parameter in some cases, q<1, causes less fluctuation while for the other cases, when q>1, it leads to more fluctuations. Acknowledgment The authors acknowledge the financial support provided by Isfahan University of Technology Research Council. 14 Fig. 1 The behavior of the three different terms in the right hand side of Eq. (19) and the Μ Μ Μ π 2 ) versus q for a relative energy variance (REV), Y, (all of them have reduced to πΈ molecule of Argon as an ideal gas at T=300K. Fig. 2 The behavior of the three different terms in the right hand side of Eq. (19) and the Μ Μ Μ π 2 ) versus q for a relative energy variance (REV), Y, (all of them have reduced to πΈ quantum harmonic oscillator with hv/kT=1. 15 Fig. 3-a The comparison of the 3-dimensional velocity distribution of an Argon atom at 300K in extensive and nonextensive statistics (2nd version). 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