Note on BEC in Nonextensive Statistical Mechanics

Note on BEC in Nonextensive Statistical Mechanics

Kwok Sau Fa¹ and E. K. Lenzi²

1Departamento de Física, Universidade Estadual de Maringá,

Av. Colombo 5790, 87020-900 Maringá-PR, Brazil

2Centro Brasileiro de Pesquisas Físicas, R. Xavier Sigaud 150,

22290-180 Rio de Janeiro, Brazil

Received on 17 October, 2000. Revised version received on 8 January, 2001

The generalized Bose-Einstein distribution, within the dilute gas assumption, in the nonextensive Tsallis statistics is worked without approximation for the Bose-Einstein condensation (BEC). The results obtained are compared with the recent results presented in Int. J. Mod. Phys. B 14, 405 (2000) by L. Salasnich. Furthermore, in order to promote a complete analysis for the BEC in the nonextensive scenario we also find exact expression within the normalized constraints in a harmonic trap.

I Introduction

In this last decade, we have witnessed a growing interest in the Tsallis statistics [1, 2]. The starting point of Tsallis statistics is based on the nonextensive entropy S_q = k(1 - p)/(q - 1), where k is a positive constant and q Î R. In the limit q® 1 the usual Boltzman-Gibbs entropy is recovered. It has been applied in many situations such as, Lévy-type anomalous superdiffusion [3], Euler turbulence [4], self-gravitating and correlated systems politrop, anomalous relaxation through electron-phonon interaction koponen, ferrofluid-like systems [7], and among others [8]. In particular, the Bose-Einstein and Fermi-Dirac distributions have been extensively analyzed in the Tsallis framework [6, 9-16]. Further, we can mention the specific heat obtained for ⁴He [17] which is in agreement with the experimental results reported in [18]. For the most of these cases the analyses involving these generalized Bose-Eisntein and Fermi-Dirac distribution have been restricted for free systems, i.e., the interaction between the particles is absent. However, important questions about the complete solution for the Bose-Einstein distribution and its implications for the thermodynamics of these physical systems are still present in this formalism. In this context, it may not be out of place to mention here the generalized Bose-Einstein distribution in the dilute gas assumption [13], given by,

and its expansion at first order in (q - 1)

The equation (2) has been applied widely in the literature [6, 12, 13, 14, 15] (see [8] and references therein) for obtaining thermodynamical quantities that emerge from it. In this way, a complete solution involving Eq.(1) can be useful for quantitative and qualitative analysis of systems in the nonextensive context. Furthermore, the development based on Eq.(1) can also be useful in the description of the systems with (multi)fractal structure (similar situation in metals [19] has been analyzed in [16]), systems with long-time memories, and among others. Another important point is about the constraints and the relation between the Lagrange parameter b and temperature in the formulations of Tsallis statistics[2].

Thus, we propose, in this work to find the complete solution for the BEC transition temperature considering D dimension and q > 0, within the dilute gas assumption. Then, we compare it with the approximate results presented in the literature, and in particular the Salasnich's work[12], by varying the values of q and D, and as a consequence to check the validity of the approximation (2). In addition, we show that the results obtained by using Eq.(2) are not accurate for most situations when compared with our ones. Also, to promote a future discussion involving the constraints, Lagrange parameters, temperature and experimental data, we obtain the Bose-Einstein distribution by taking into account the normalized constraints[2].

In order to facilitate our analysis, let us recall some results evaluated in [12] with Eq.(2). The quantities evaluated in [12] are: the BEC transition temperature, the condensed fraction and the energy per particle to the homogeneous gas, the gas in a harmonic trap and the relativistic homogeneous gas in D-dimensional space. For non-interacting bosons, the total number of particle is evaluated by the following expression

where r() is the density of states. In a D-dimensional box of volume V, the densities of states for homogeneous gas and gas in harmonic trap are given respectively by r() = {(V/)[m/(2p²)]^D/2}/G(D/2) and r() = {[/()]^D }/(G(D) ), where is the geometric average of the trap frequencies and G(x) is the gamma function.

To calculate the BEC transition temperature T_q, with m = 0, we substitute Eq.(2) and r() into (3), and we obtain

for the homogeneous gas, and

for a gas in a harmonic trap. z(x) is the Riemann z-function. The energy is calculated by the following expression

The results are

for the homogeneous gas, and

for a gas in a harmonic trap.

For the relativistic gas, the total number of particles is not conserved due to the production of antiparticles, but the difference between the number N of particles and the number N of antiparticles is conserved, i.e., Q = N - = ò dr() [án()ñ_q - á()ñ_q], where á()ñ_q is obtained from án()ñ_q by replacing m® -m. The density of states for the relativistic gas is given by r() = (V2p^D/2)/(2p²)^DG(D/2)) (² - m²c⁴) ^(D-2)/2. For ultrarelativistic region kT >> mc², the critical temperature at which BEC occurs corresponds to |m|=mc². Expanding Q at first order in m yields

The Eqs.(4), (5), (6), (7) and (9) are the results obtained in [12], by using Eq.(2). Similar results for free particles were also obtained in [15]. Now, we calculate again the quantities above by using the distribution (1), instead of (2). To do so, we note that the distribution (1) can be written as a sum, i.e.,

Substituting r() and Eq.(10) into (3) and taking into account the cut-off [2] for q < 1, we obtain the critical temperature as

for the homogeneous gas, and

for a gas in a harmonic trap. Note that Eq. (12) is valid for (q - 1) < 1/D. In addition, the energy is also easily obtained from Eq.(6)

for the homogeneous gas, and

for a gas in a harmonic trap. The Eq. (14) is valid for (q - 1) < 1/(1 + D). We note that our results differ from those obtained by using the distribution (2). To see that closely, we quote T_q with different values of D, for q = 1.1 (see Table 1). We also plot the critical temperature in function of q. For homogeneous gas the convergence of the series of Eq. (11) is slow for D = 3. For convenience, we plot T_q in function of q, for D = 4 (Fig. 1). For a gas in a harmonic trap, we plot T_q in function of q, for D = 3 (Fig. 2). As we can see the values quoted in Table 1, and the two figures (Figs.1 and 2) show clearly that the critical temperatures obtained by using (2) are not accurate when compared with our results. Even for a gas in a harmonic trap with small deviation in the nonextensive parameter q = 1.1 and D = 3, the divergence between our result and Salasnich's result [12] is remarkable. This is not surprise because the expansion made in (2), in the parameter (q - 1), has included the factor b. This last factor is not necessarily small so the expansion of the exponential function at first order of (q - 1) cannot be good in general, especially when |1 - q| and D are increased. We should also stress that the changes in the BEC transition temperature (for q > 1), from the extensive Bose statistics, are greater than those obtained by using the distribution (2). For the relativistic gas, in the ultrarelativistic region kT >> mc², the critical temperature by expanding Q at first order in m is given by

Table 1.
Results for critical temperature of a gas in a harmonic trap system with q = 1.1. The second row shows the results obtained by using the distribution (2) which are given by Eq. (5). While, the last row shows our results obtained by using the distribution (1) which are given by Eq. (12).

Figure. 2
Plots of critical temperature versus the nonextensive parameter q for a gas in a harmonic trap. Here, the three dimension (D = 3) case is considered.

Eq.(15) also differs from that obtained by using the distribution (2). For example: for q = 1.1 and D = 4 we obtain [(4Vp²mc²)/( | Q | (hc) ⁴)]^1/3kT_q = 0.420181 from Eq. (15) and [(4Vp²mc²)/( | Q | (hc)⁴)]^1/3kT_q = 0.448173 from Eq. (9).

For completeness, we show below q-mean value of particle number and internal energy in the normalized constraints [2] by taking into account m = 0

where f(a, g; z) = z^k/(k! G(ak +g)) with a, g > 0 = b/Trr^q, áñ₁µ (1/b)^D and b=1/T₁ where T₁ is the critical temperature. More details about the calculation above can be found in [20], where the free particle case has been analyzed.

In summary, we have obtained the exact standard BEC formula by using the generalized Bose-Einstein distribution, within the dilute gas assumption, in Tsallis statistics. The comparison between our results and those obtained in [12] shows that the approximation Eq.(2) is not accurate for most situations examined here. Thus, the results obtained for the Bose-Einstein condensation [12, 14, 15] and among others, in the light of our results, should be reanalysed. The role of the interactions in the BEC [21, 22] in Tsallis statistics will be addressed in an other opportunity. Furthermore, we have also found the expressions for the BEC within the normalized constraints, in order to promote a possible analysis involving Lagrange parameter, temperature, constraints and experimental data. Finally, we believe that this work can be useful in the analyses of future applications which involve Tsallis statistics and BEC, within the dilute gas assumption.

EKL thanks CNPq and PRONEX (Brazilian agencies) for financial support.

References

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Publication Dates

• Publication in this collection
  05 Mar 2002
• Date of issue
  June 2001

History

• Received
  17 Oct 2000
• Reviewed
  08 Jan 2001