q-generalized Bose–Einstein condensation based on Tsallis entropy

22 July 2002

Physics Letters A 300 (2002) 65–70 www.elsevier.com/locate/pla

q-generalized Bose–Einstein condensation based on Tsallis entropy Jincan Chen a,b,∗ , Zhipeng Zhang b , Guozhen Su b , Lixuan Chen a,b , Yaogen Shu b a CCAST (Word Laboratory), P.O. Box 8730, Beijing 100080, PR China b Department of Physics, Xiamen University, Xiamen 361005, PR China

Received 14 February 2002; received in revised form 29 May 2002; accepted 6 June 2002 Communicated by A.R. Bishop

Abstract The thermodynamic properties of a q-generalized ideal boson system with the general energy spectrum ε = aps in a Ddimensional space are investigated, based on the q-generalized statistical distributive function derived by Tsallis entropy. Several important parameters of the system such as the critical temperature that Bose–Einstein condensation (BEC) occurs, the specific heat at constant volume, etc. are derived analytically. It is found that the conditions that BEC may occur and the continuity of the specific heat at critical temperature are the same as those of an ordinary boson system. However, the thermodynamic properties of a q-generalized ideal boson system are closely depend on q and D/s and consequently there are some novel results.  2002 Elsevier Science B.V. All rights reserved. PACS: 05.30; 05.70; 05.20

It has been shown that systems presenting longrange interactions and/or long-duration memory cannot be well described by the Boltzmann–Gibbs statistics [1]. The q-generalized statistical mechanics proposed by Tsallis [2,3] and developed by many researchers is of a powerful tool to deal with such systems. In recent years, it has been used to study the properties of many physical systems [4–11] which are more complex than a standard ideal gas. For example, many authors have investigated the properties of self-gravitating stellar systems [12–14] with q lower than 7/9, low-dimensional dissipative systems [15]

with q < 1, the Lévy flight random diffusion [16, 17] with 5/3 < q < 3, the galaxy model of the generalized Freeman disk [18] with q = −1, the electron plasma two-dimensional turbulence [12] with q = 1/2, cosmic background radiation [19,20] and correlated themes [21], linear response theory [22], the solar neutrinos [23], thermalization of electron–phonon systems [24], etc. A large number of significant results have been obtained. It implies that q can play the role of an effective parameter. It is well known that the q-generalized statistics characterized by a q parameter relies on the so-called Tsallis entropy [2]

* Corresponding author. Mailing address: Department of Physics, Xiamen University, Xiamen 361005, PR China. E-mail address: [email protected] (J. Chen).


q 1 − i pi , Sq = −k 1−q

0375-9601/02/$ – see front matter  2002 Elsevier Science B.V. All rights reserved. PII: S 0 3 7 5 - 9 6 0 1 ( 0 2 ) 0 0 7 8 1 - 8

(1)

66

J. Chen et al. / Physics Letters A 300 (2002) 65–70

where q ∈ R, pi is the probability, k is the Boltzmann
constant, and Sq recovers the standard form −k pi ln pi in the limit q → 1. Within an approximation called factorization approach, the Tsallis entropy may be applied to derive the following beautiful formalism of the generalized Bose–Einstein (BE) distribution [25–28] nq =

1 , [1 + (q − 1)β(ε − µ)]1/(q−1) − 1

(2)

where nq is the average occupation number at a state with energy ε, β = 1/(kT ), T is the absolute temperature, and µ is the chemical potential. When q = 1, Eq. (2) becomes the conventional BE distribution. The generalized BE distribution nq should be nonnegative. This results in the following constrained conditions:   µ
ε
1 (1 − q)β + µ, (3) when q < 1; µ
ε,

outside of FZDA, not only for q → 1, but for the values of q very different from 1. As a matter of fact, most of the practical systems are far away from the FZDA, and consequently it is very significant to use the generalized BE distribution to study the properties of some systems. Now, let us begin to investigate the thermodynamic properties of a D-dimensional q-generalized ideal boson system with the general energy spectrum ε = aps

(5)

and a degree of degeneracy g, where a and s are two positive constants, respectively. For a non-relativistic particle system, s = 2 and a = 1/(2m), where m is the mass of a particle. For a ultra-relativistic particle system, s = 1 and a = c, where c is the light speed. For the sake of convenience, Eq. (2) is rewritten as nq =

∞ 

j/(1−q) j q−1 zq 1 + (q − 1)zq βε ,

(6)

j =1

(4)

when q > 1. It was pointed out [29] that “in dealing with the associated grand partition function, the authors of Ref. [26] face a serious obstacle: non-extensivity impedes the customary procedure of factorizing the partition function into single-particle factors. They circumvent this difficulty by appealing to the dilute nature of the quantum gas in order to factorize the partition function anyway.” Thus, the generalized BE distribution derived in Refs. [25,26] can be regards only as an approximation. For some values of the temperature and the chemical potential, the results obtained with the approximate distribution appreciably deviate from the exact ones [29]. The authors of Ref. [10] further calculated the difference between the approximate distribution and the exact one. They found that “the difference is almost equal to zero for relatively ‘low temperature’ (T < 1010 K for the total number of particles N = 105 and T < 1020 K for N = 1015 ) and very high temperature (T > 1012 K for N = 105 and T > 1022 K for N = 1015 ). Between these temperatures, the difference is significant. This temperature interval can be called forbidden zone of the dilute approximation (FZDA).” In spite of the existence of a FZDA, the dilute gas approximation for the generalized BE distribution function is evidently a good one

where  1/(1−q) zq = 1 + (1 − q)βµ

(7)

is the q-generalized fugacity. Using Eqs. (3)–(6), one can calculate the total number of particles in a q-boson system, which is given by  g N = N0 + D nq d D r d D p h gLD = N0 + D gq,D/s (zq ), (8) λs where N0 is the particle number in the ground state, h is the Planck constant, LD is the volume of a Ddimensional system,  

1/s Γ (D/2 + 1) 1/D ahs λs = (9) Γ (D/s + 1) π s/2 kT is the generalized thermal wavelength [30] of the qbosons, which is independent of q, and  j−(q−1)n
zq  Γ [j/(q−1)−n]  ∞ (q > 1), j =1 (q−1)n Γ [j/(q−1)] gq,n (zq ) = j+(1−q)n
 Γ [j/(1−q)+1]  ∞ zq j =1 (1−q)n Γ [j/(1−q)+n+1] (q < 1), (10) may be called as the q-generalized Bose integral. In order to guarantee the integral to be larger than

J. Chen et al. / Physics Letters A 300 (2002) 65–70

zero, q < (n + 1)/n must be satisfied. It is seen from Eq. (10) that there exist the following relations: lim gq,n (zq ) =

q→1

∞ j  z = gn (z) jn

(11)

j =1

and ∂gq,n (zq ) 1 = gq,n−1 (zq ), ∂zq zq where gn (z) is the Bose integral. Because

∂N = 0, ∂T LD

(12)

(13)

we also obtain another useful relation

∂gq,n (zq ) D gq,n−1 (zq ) gq,D/s (zq ) =− ∂T s T gq,D/s−1 (zq ) D N,L (14) from Eq. (10). It may be seen from Eqs. (10) and (7) that gq,n (zq ) is a monotonically increasing function of zq and the maximum value of zq is equal to 1 when µ = 0. Thus, using Eq. (8), we can define the critical temperature Tq,c of Bose–Einstein condensation (BEC) of the qbosons similarly to in an ordinary boson system where q = 1 [31]. When T  Tq,c , the number of q-bosons in the excited states is approximately equal to the total number of q-bosons in the system, i.e., Ne = N =

gLD gLD g (z ) = ζq (D/s), q,D/s q λD λD s s,c

(15)

where λs,c is the generalized wavelength of the qbosons when T = Tq,c , and 
Γ [j/(q−1)−n] 1  ∞ (q > 1), j =1 (q−1)n Γ [j/(q−1)] ζq (n) =
∞ Γ [j/(1−q)+1] 1  j =1 (1−q)n Γ [j/(1−q)+n+1] (q < 1), (16) may be defined as the q-generalized Riemann Zeta function. It can be seen from Eq. (16) that the relation between the function ζq (n) and the Riemann Zeta function ζ (n) is lim ζq (n) = ζ (n).

q→1

(17)

Eqs. (7), (10), and (15) show clearly that for a qgeneralized ideal boson system with given the particle

67

number, volume, temperature, and energy spectrum, the values of q affect directly the generalized fugacity, but do not affect the q-generalized Bose integral gq,n (zq ). Although the q-generalized Bose integral includes q, it does not vary with q, and consequently, we have gq,n (zq ) = gq−r,n (zq−r ) = gn (z)

(18)

for a given system mentioned above. Using Eq. (15), we can obtain the critical temperature of BEC of a qboson system as s/D  1 ahs N Γ (D/2 + 1) . (19) Tq,c = s/2 π k gLD Γ (D/s + 1) ζq (D/s) When q = 1,

 s/D N Γ (D/2 + 1) 1 ahs T1,c = s/2 π k gLD Γ (D/s + 1) ζ(D/s) ≡ Tc ,

(20)

which is just the critical temperature of an ordinary boson system. When q = 1, we obtain   Tq,c ζ(D/s) s/D = . (21) Tc ζq (D/s) It is seen from Eq. (19) (a detailed derivation is given in Appendix A) that only when D/s > 1, can BEC of a q-boson system occur. The condition is the same as that of an ordinary boson system. Using Eq. (21), we can plot the Tq,c /Tc versus q curves, as shown in Fig. 1. It is seen from Fig. 1 that Tq,c /Tc decreases with the increase of q, so that Tq,c > Tc when q < 1 and Tq,c < Tc when q > 1. This shows that for a q-boson system of q < 1, BEC occurs more easily than an ordinary boson system. Fig. 1 also shows that Tq,c /Tc increases with the increase of D/s when q < 1 and Tq,c /Tc decreases with the increase of D/s when q > 1. When T < Tq,c , one part of the q-bosons condenses in the ground state. The number of q-bosons in the exited states is equal to

gLD T D/s , Ne = D ζq (D/s) = N (22) λs Tq,c and consequently, the fraction of condensation in the ground state is given by

T D/s N0 =1− . (23) N Tq,c

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D D 1+ Nk = s s   gq,D/s+1(zq ) gq,D/s (zq ) D × (28) − gq,D/s (zq ) D + s gq,D/s−1 (zq ) and

∂E(T > Tq,c ) ∂T LD



D T D/s D 1+ Nk = s s Tq,c ζq (D/s + 1) . × ζq (D/s)

Cv (T > Tq,c ) =

Fig. 1. The Tq,c /Tc versus q curves for some different values of D/s.

We now continue to calculate the total energy of the system. As described in Ref. [32], there may be three different choices for the total energy of the system. It is important to note that in the process of deriving the generalized BE distribution [26], the second choice in Ref. [32] has been used. With the help of the results obtained in Ref. [26], the expression of the total energy of the system may be expressed as  g E = D nq ε d D r d D p. (24) h From Eq. (24), one can calculate that when T > Tq,c , E=

gq,D/s+1 (zq ) D NkT ; s gq,D/s (zq )

(25)

when T
Tq,c , D gLD kT ζq (D/s + 1) s λD s

D T D/s ζq (D/s + 1) = NkT . s Tq,c ζq (D/s)

E=

(26)

Using Eqs. (25) and (26) and the specific heat at constant volume

∂E Cv = (27) , ∂T N,LD one can derive Cv (T > Tq,c )

∂E(T > Tq,c ) = ∂T N,LD

(29)

From Eqs. (28) and (29), we can obtain the curves of Cv varying with T /Tq,c , as shown in Figs. 2(a) and (b), where D/s = 1.5 (a) and D/s = 3 (b). It is important to note that some significant results may be deduced from the curves in Fig. 2. (i) The specific heat at all temperatures increases with the increase of q. (ii) When D/s = 1.5, the specific heat at Tq,c is continuous; when D/s = 3 the specific heat at Tq,c is discontinuous. (iii) The specific heat of a qboson system is different from that of an ordinary boson system even at high temperatures. (iv) For the systems of q > 1, Cv first decreases and then increases with T when T is higher than Tq,c , so there is a minimal value for Cv . This result has been observed experimentally [28]. (v) For the systems of q < 1, Cv decreases monotonously with T when T is higher than Tq,c . (vi) When q = 1, the characteristics of the specific heat of an ordinary boson system are obtained directly. It can be derived from Eqs. (28) and (29) that at the critical temperature of BEC,  −  + ∆Cv = Cv Tq,c − Cv Tq,c 2 ζq (D/s) D = (30) . Nk s ζq (D/s − 1) By using the result in Appendix A and Eq. (30), it may be proven that if D/s
2, ζq (D/s − 1) will diverge and ∆Cv (Tq,c ) = 0, so that Cv will continue at Tq,c . Otherwise, there is a gap of Cv at Tq,c . The condition is also the same as that of the ordinary boson system. It can be further proven from Eq. (30) that for the q-boson systems of D/s > 2, the gap of Cv at Tq,c increases with the increase of q.

J. Chen et al. / Physics Letters A 300 (2002) 65–70

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values of q and D/s. These results are suitable for a qboson system not only in a three-dimensional space but also in an other dimensional space.

Appendix A In order to give the convergent condition of ζq (n), Eq. (16) is rewritten as 
∞ 1 j =1 (q−1)n ∆j (q > 1), ζq (n) =
∞ (A.1) 1 (q < 1), j =1 (1−q)n Xj where   ∆j =  Xj =

Γ [j/(q−1)−n] Γ [j/(q−1)] Γ [j/(1−q)+1] Γ [j/(1−q)+n+1]

(q > 1), (A.2) (q < 1).

(1) q > 1. For a very large value of x, the Stirling’s formula gives √ Γ (x) ≈ 2πx x−1/2e−x . (A.3) According to Eq. (A.3), we have [j/(q − 1) − n][j/(q−1)−n]−1/2e−[j/(q−1)−n] [j/(q − 1)]j/(q−1)−1/2 e−j/(q−1) j/(q−1)  −1/2  n n 1− = 1− j/(q − 1) j/(q − 1) 1 × en (A.4) [j/(q − 1) − n]n

∆j ≈

Fig. 2. The specific heat as a function of T /Tq,c for some different values of q. (a) D/s = 3/2, (b) D/s = 3.

The results obtained above have shown clearly that after introducing some significant physical parameters such as the generalized Bose integral, the generalized Riemann Zeta function, the generalized thermal wavelength, and so on, we can successfully derive the analytic expressions of some thermodynamic parameters of a q-generalized boson system with the general energy spectrum ε = aps . These important parameters include the total number of particles, critical temperature, fraction of condensation in the ground state, total energy, specific heat at constant volume, and so on. It is found that the thermodynamic properties of a qgeneralized boson system are closely dependent on the

for a very large value of j . Because j/(q−1)  n ≈ e−n , 1− j/(q − 1)  −1/2 n 1− ≈ 1, j/(q − 1) 1 1 ≈ , [j/(q − 1) − n]n [j/(q − 1)]n we get 1 1 ∆j ≈ n . n (q − 1) j

(A.5)

It is clearly seen from Eq. (A.5) that ζq (n) is convergent when n > 1 and divergent when n
1. (2) q < 1. Similarly, it can be proven that the convergent condition of ζq (n) when q < 1 is the same as that of ζq (n) when q > 1.

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