Download keshavarzi,_How the Nonextensivity Parameter Affects Energy

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]:
𝑘
𝑞
𝑆 = 𝑞−1 (1 − ∑𝑤
𝑖 𝑃𝑖 )
(1)
where 𝑃𝑖 is the probability function of finding the system in i’th 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/√N 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:
𝑤
𝑤
𝑞
∑ 𝜀𝑖 𝑃𝑖 − ̅̅̅
𝐸𝑞 = 0
;
(2)
𝑍𝑞
1
= ∑[1 − (1 − 𝑞)𝛽𝜀𝑖 ](1−𝑞)
1
𝑖
(2)
𝑤
𝑤
∑(𝜀𝑖 − ̅̅
𝐸̅̅𝑞 )𝑃𝑖 𝑞 = 0
;
𝑖
1
(1−𝑞)
´
̅̅̅̅
𝑍(4)
𝑞 = ∑[1 − (1 − 𝑞)𝛽 (𝜀𝑖 − 𝐸𝑞 )]
𝑖
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]:
̅̅̅̅̅
̅̅̅2̅ − 𝐸 2
𝛥𝐸 2 = (𝐸̅ − 𝐸)2 = 𝐸
(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, √(𝐸̅ − 𝐸)2 ≠ √̅̅
𝐸̅2̅ − 𝐸
2
and (𝐸̅ − 𝐸)2 can be calculated as follows:
𝑤
̅̅̅̅̅
̅̅̅2̅ − 2𝐸 2 + 𝐸 2 ∑ 𝑃𝑖 𝑞 ≠ 𝐸
̅̅̅2̅ − 𝐸 2
𝛥𝐸 2 = 𝐸
(4)
𝑖=0
In Eq. (4) inequality is due to 〈1〉𝑞 ≠ 1. In fact, in the second version of the Tsallis statistics
we have [5]:
4
w
̅̅̅q
∑ Pi q = Zq1−q + (1 − q)βE
(5)
i
Therefore, the following result will be obtained for energy fluctuation:
2
̅̅̅̅̅
̅̅̅𝑞 ]
𝛥𝐸 2 = ̅𝐸̅̅2̅ + 𝐸 [𝑍𝑞 1−𝑞 − 2 + (1 − 𝑞)𝛽𝐸
(6)
̅̅̅̅̅
𝐸𝑞 2 in the second version of the Tsallis statistical mechanics is defined as:
𝑤
𝑤
̅̅̅̅̅
𝐸𝑞 2 = ∑ 𝜀𝑖 2
𝑖
1
1 𝜕
𝑃𝑖 𝑞 = − 𝑞
[∑ 𝜀𝑖 [1 − (1 − 𝑞)𝛽𝜀𝑖 ](1−𝑞) ]
𝑍𝑞 𝜕𝛽
(7)
𝑖
the term in the bracket may be written as 𝑍𝑞 ∑𝑤
𝑖 𝜀𝑖 𝑃𝑖 . Therefore, Eq. (7) can be rearranged
as:
𝑤
̅̅̅̅̅
̅̅̅𝑞 (∑ 𝜀𝑖 𝑃𝑖 ) −
𝐸𝑞 2 = [𝐸
1
𝑤
1
𝜕
∑ 𝜀𝑖 𝑃𝑖 ]
𝑍𝑞 (𝑞−1) 𝜕𝛽 𝑖
(8)
Using the definition of the probability function, Eq. (2), in this version it is possible to
̅̅̅
write ∑𝑤
𝑖 𝜀𝑖 𝑃𝑖 verses 𝐸𝑞 and 𝑍𝑞 as:
𝑤
∑ 𝜀𝑖 𝑃𝑖 =
1
𝑤
1
𝑍𝑞
𝑤
𝑞
(1−𝑞)
[∑ 𝜀𝑖 𝑃𝑖 − (1 − 𝑞)𝛽 ∑ 𝜀𝑖 2 𝑃𝑖 𝑞 ]
1
=
1
𝑍𝑞
(1−𝑞)
1
̅̅̅̅̅2 ]
̅̅̅𝑞 − (1 − 𝑞)𝛽𝐸
[𝐸
𝑞
(9)
By insertion of Eq. (9) into Eq. (8) we have:
1
̅̅̅̅̅
̅̅̅̅̅2 ])
̅̅̅𝑞 − (1 − 𝑞)𝛽𝐸
𝐸𝑞 2 = ̅̅̅
𝐸𝑞 ( (1−𝑞) [𝐸
𝑞
𝑍𝑞
̅̅̅̅̅2 ]
̅̅̅𝑞 − (1 − 𝑞)𝛽𝐸
𝑞
𝜕 [𝐸
− ( (𝑞−1)
(
))
(1−𝑞)
𝜕𝛽
𝑍𝑞
𝑍𝑞
1
where for second term in right hand side we have:
5
(10)
̅̅̅̅̅2 ]
̅̅̅𝑞 − (1 − 𝑞)𝛽𝐸
𝑞
𝜕 [𝐸
(
)
(1−𝑞)
𝜕𝛽
𝑍𝑞
𝑍𝑞 (1−𝑞)
=
𝜕 ̅̅̅
̅̅̅̅̅2 ] − [𝐸
̅̅̅̅̅2 ] 𝜕 𝑍 (1−𝑞)
̅̅̅𝑞 − (1 − 𝑞)𝛽𝐸
[𝐸 − (1 − 𝑞)𝛽𝐸
𝑞
𝑞 𝜕𝛽 𝑞
𝜕𝛽 𝑞
(𝑍𝑞 (1−𝑞) )
2
(11)
and
̅̅̅̅̅2 ]
̅̅̅𝑞 − (1 − 𝑞)𝛽𝐸
𝑞
𝜕 [𝐸
(
)
𝑍𝑞 (𝑞−1) 𝜕𝛽
𝑍𝑞 (1−𝑞)
1
=
𝜕
̅̅̅̅̅2 ] − 1 [𝐸
̅̅̅̅̅2 ] 𝜕 𝑍 (1−𝑞) (12)
̅̅̅𝑞 − (1 − 𝑞)𝛽𝐸
̅̅̅𝑞 − (1 − 𝑞)𝛽𝐸
[𝐸
𝑞
𝑞
𝜕𝛽
𝜕𝛽 𝑞
𝑍𝑞 (1−𝑞)
hence
1
̅̅̅̅̅
̅̅̅̅̅2 ]) − 𝜕 [𝐸
̅̅̅̅̅2 ]
̅̅̅𝑞 − (1 − 𝑞)𝛽𝐸
̅̅̅𝑞 − (1 − 𝑞)𝛽𝐸
𝐸𝑞 2 = ̅̅̅
𝐸𝑞 ( (1−𝑞) [𝐸
𝑞
𝑞
𝜕𝛽
𝑍𝑞
+
1
̅̅̅̅̅2 ] 𝜕 𝑍 (1−𝑞)
̅̅̅𝑞 − (1 − 𝑞)𝛽𝐸
[𝐸
𝑞
𝜕𝛽 𝑞
𝑍𝑞 (1−𝑞)
(13)
and after some simplifications we can have:
̅̅̅̅̅
𝐸𝑞 2 =
2
̅̅̅
𝐸𝑞
𝑍𝑞
−
(1−𝑞)
+
where
𝜕
𝜕𝛽
̅̅̅̅̅
𝐸𝑞 2 =
̅̅̅̅̅2
̅̅̅𝑞
̅̅̅𝑞
(1 − 𝑞)𝛽𝐸
𝜕𝐸
𝜕𝐸
𝑞
2
̅̅̅̅̅
̅̅̅̅̅2
(1
𝐸
−
+
−
𝑞)𝛽
+ (1 − 𝑞)𝐸
𝑞
𝑞
(1−𝑞)
𝜕𝛽
𝜕𝛽
𝑍𝑞
1
𝑍𝑞
(1−𝑞)
(
𝜕
̅̅̅̅̅2 ]
̅̅̅𝑞 − (1 − 𝑞)𝛽𝐸
𝑍 (1−𝑞) ) [𝐸
𝑞
𝜕𝛽 𝑞
(14)
̅̅̅𝑞 and finally:
𝑍𝑞 (1−𝑞) = −(1 − 𝑞)𝐸
̅̅̅̅̅2
̅̅̅𝑞
̅̅̅𝑞
̅̅̅𝑞 2
(1 − 𝑞)𝐸
(1 − 𝑞)𝛽𝐸
𝜕𝐸
𝜕𝐸
𝑞
2
2
̅̅̅̅̅
̅̅̅̅̅
−
𝐸𝑞 −
+ (1 − 𝑞)𝛽
+ (1 − 𝑞)𝐸𝑞 −
𝜕𝛽
𝜕𝛽
𝑍𝑞 (1−𝑞)
𝑍𝑞 (1−𝑞)
𝑍𝑞 (1−𝑞)
2
̅̅̅
𝐸𝑞
̅̅̅𝑞
(1 − 𝑞)2 𝛽𝐸
̅̅̅̅̅
+
𝐸𝑞 2
(1−𝑞)
𝑍𝑞
(15)
2
𝑞 ̅̅̅
Now we subtract (2 − ∑𝑤
𝑖 𝑃𝑖 )𝐸q from the both sides of Eq. (15) and we have:
6
̅̅̅̅̅̅̅
𝛥𝐸𝑞 2 =
̅̅̅̅̅2
̅̅̅𝑞
̅̅̅𝑞
̅̅̅𝑞 2
(1 − 𝑞)𝐸
(1 − 𝑞)𝛽𝐸
𝜕𝐸
𝜕𝐸
𝑞
2
2
̅̅̅̅̅
̅̅̅̅̅
−
𝐸𝑞 −
+ (1 − 𝑞)𝛽
+ (1 − 𝑞)𝐸𝑞 −
𝜕𝛽
𝜕𝛽
𝑍𝑞 (1−𝑞)
𝑍𝑞 (1−𝑞)
𝑍𝑞 (1−𝑞)
2
̅̅̅
𝐸𝑞
𝑤
̅̅̅𝑞
(1 − 𝑞)2 𝛽𝐸
2
̅̅̅̅̅
+
𝐸𝑞 2 − (2 − ∑ 𝑃𝑖 𝑞 ) ̅̅̅
𝐸q
(1−𝑞)
𝑍𝑞
𝑖
(16)
And finally by rearrangement of the Eq. (16) the following equation may be obtained:
̅̅̅̅̅2
̅̅̅𝑞
𝜕𝐸
𝜕𝐸
2
2
𝑞
̅̅̅̅̅̅̅
̅̅̅̅̅2 − ̅̅̅
̅̅̅
(𝐴
(1
𝛥𝐸𝑞 2 = 𝐴 (𝐸
𝐸
)
+
𝐸
−
𝐵)
−
+
−
𝑞)𝛽
𝑞
𝑞
𝑞
𝜕𝛽
𝜕𝛽
(17)
where A and B coefficients are as follow:
𝐴 = (1 − 𝑞) +
̅̅̅𝑞
(𝑞 2 − 𝑞)𝛽𝐸
𝑍𝑞 (1−𝑞)
̅̅̅𝑞 − 𝑍𝑞 (1−𝑞) −
, 𝐵 = (2 + 𝛽(𝑞 − 1)𝐸
𝑞
𝑍𝑞 (1−𝑞)
) (18)
Therefore, the energy variance in second version of the Tsallis statistics is as follows:
̅̅̅̅̅̅̅
𝛥𝐸𝑞 2 = −
̅̅̅̅̅2 𝐴 − 𝐵
̅̅̅𝑞 1 − 𝑞 𝜕𝐸
1 𝜕𝐸
2
𝑞
̅̅̅
+
𝛽
+
𝐸
1 − 𝐴 𝜕𝛽 1 − 𝐴 𝜕𝛽
1−𝐴 𝑞
(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
̅̅̅̅̅
2
𝜕𝐸
𝑞
𝜕𝛽
, 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, 𝑃𝑖 𝑞 . But the
2
third term in the right hand side of Eq. (19) which is related to the̅̅̅̅
𝐸𝑞 , 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 q→1, tends to that corresponding equation in
the Boltzmann Gibbs statistics as:
1
̅̅̅̅̅̅̅2 )2
(𝛥𝐸
𝑞
̅̅̅
𝐸𝑞
̅̅̅𝑞 1/2
1 𝜕𝐸
1
=− (
) =
(𝑘𝑇 2 𝐶𝑣 )1/2
̅̅̅
̅̅̅
𝐸𝑞 𝜕𝛽
𝐸𝑞
(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 ̅̅̅̅̅
𝐸𝑞 2 cannot be obtained directly from the partition function.
To obtain the required quantities, the following partition function has been used:
𝑣𝑚𝑎𝑥
𝑍𝑞 = ∫
0
1
(1−𝑞)
𝑁
ф(𝐸) [1 − 𝛽(1 − 𝑞)( 𝑚𝑣 2 )]
𝑑𝑣
2
(21)
where Φ(E) is the degeneracy for an ideal gas velocity [14] and vmax is the maximum
allowed velocity for the ideal gas molecules:
1
2𝜋𝑚𝑎2
Φ(𝐸) = Γ(𝑁+1)Γ(3𝑁/2) (
∞
𝑣𝑚𝑎𝑥 = {
ℎ2
3𝑁/2
)
(22)
𝑞≥1
2
√2𝛽𝑁𝑚(1−𝑞)
(23)
𝑞<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 q→1- (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:
𝑛𝑚𝑎𝑥
𝑍𝑣𝑖𝑏,𝑞
1
1−𝑞
1
= ∑ [1 − 𝛽(1 − 𝑞) (𝑛 + ) ℎ𝑣]
2
(24)
𝑛=0
where v is natural frequency of the oscillator and nmax can be obtained as follows:
∞
𝑛𝑚𝑎𝑥 = {
𝑞≥1
2−𝛽ℎ𝑣(𝑞−1)
𝑖𝑛𝑡𝑒𝑔𝑒𝑟 ( 2𝛽ℎ𝑣(1−𝑞) )
𝑞<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 𝛽ℎ𝑣 = 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 q≈0.6-0.7, 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 𝑛𝑚𝑎𝑥 =
2−𝛽ℎ𝑣(𝑞−1)
2𝛽ℎ𝑣(1−𝑞)
; therefore, for 𝛽ℎ𝑣 = 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
(1−𝑞)
𝑁
[1 − 𝛽(1 − 𝑞) ( 𝑚𝑣2 )]
2
𝑃(𝜈) =
𝑍𝑞
(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 isn’t 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
𝑍𝑞 =
𝑣𝑚𝑎𝑥
ф(𝐸) [1
∫0
𝑁
1
(1−𝑞)
2
− 𝛽(1 − 𝑞)( 2 𝑚𝑣 − 𝐸𝑞 )]
11
𝑑𝑣
(27)
where 𝐸𝑞 = 3/2𝑁𝑘𝑇 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:
∞
𝑞≥1
2
𝑣𝑚𝑎𝑥 = {
√2𝛽(1−
3𝑁
)𝑁𝑚(1−𝑞)
2
(28)
𝑞<1
In this manner, for a harmonic oscillator, the following partition function in 4th version of
the Tsallis statistics may be achieved:
𝑛𝑚𝑎𝑥
1
𝑍𝑞 = ∑ [1 − (1 − 𝑞)𝛽((𝑛 + 1/2)ℎ𝑣 − ̅̅̅
𝐸𝑞 )](1−𝑞)
(29)
𝑛=0
where 𝐸𝑞 is the mean value of the energy for the harmonic oscillator and nmax can be
obtained from the following equation:
∞
𝑛𝑚𝑎𝑥 = {
𝑖𝑛𝑡𝑒𝑔𝑒𝑟 (
𝑞≥1
̅̅̅̅
2−(1−𝑞)𝛽(ℎ𝜈+2𝐸
𝑞)
(1−𝑞)𝛽ℎ𝜈
)
𝑞<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 don’t 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/√𝑁 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, Φ(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
̅̅̅𝑞 2 ) versus q for a
relative energy variance (REV), Y, (all of them have reduced to 𝐸
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
̅̅̅𝑞 2 ) versus q for a
relative energy variance (REV), Y, (all of them have reduced to 𝐸
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).
Fig. 3-b The comparison of the number of accessible states for a quantum harmonic
oscillator in Boltzmann-Gibbs and 2nd version of the Tsallis statistics.
16
Fig. 4 The relative energy variance (REV) for an Argon atom at temperature 300K and a
quantum harmonic oscillator in 4th version of the Tsallis statistics.
Fig. 5 The Relative energy variance (REV) versus q for systems with different number of
Argon atoms at temperature 300K in the 4th version of the Tsallis statistics.
17
References
[1] H. Hasegawa, Progress in Materials Scieence 52 (2007) 333.
[2] S. Tonga, Physica A 305 (2002) 619.
[3] H.Y. Zhang, X. Gu, X. H. Zhang, X. Ye, X. G. Gong, J. Phys. Lett. A 331 (2004) 332.
[4] C. Beck, Physica A 277 (2000) 115.
[5] E.M.F. Curado and C. Tsallis, Physica A 24 (1991) L69; Errate: 24 (1991) 3187; 25
(1992) 1019.
[6] C. Tsallis, R. S. Mendes and A. R. plastiono, Physica A 261 (1998) 534.
[7] S. Martinez, F. Nicolas, F. Pennini, A. Plastino, Physica A 289 (2000) 489.
[8] E.K. Lenzi, R.S. Mendes, Phys. Lett. A 250 (1998) 270.
[9] C. Tsallis, J. Stat. Phys. 52 (1988) 479.
[10] G.L. Ferri, S. Martinez, A. Plastino, J. Stat. Mech. (2005) P04009.
[11] Z.H. Feng, L. Y. Liu, Physica A 389 (2010) 237.
[12] L. Liyan, D. Jiulin, Physica A 387 (2008) 5417.
[13] F.Q. Potiguar, U. M. S. Costa, Physica A 321 (2003) 482.
[14] D.A. McQuarrie, Statistical Mechanics, Harper Collins Pub. 1976.
[15] A.R. Plastino, A. Plastino, C. Tsallis, J. Phys. A: Math Gen. 27 (1994) 5707.
[16] E. Keshavarzi, M. Sabzehzari, M. Eliasi, Physica A 389 (2010) 2733.
[17] S. Abe, S. Martinez, F. Pennini b, A. Plastino, Phys. Lett. A 278 (2001) 249.
18