Investigation of Bose condensation in ideal Bose gas trapped under generic power law potential in d dimension
aa r X i v : . [ c ond - m a t . qu a n t - g a s ] D ec Investigation of Bose condensation in ideal Bose gas trapped under genericpower law potential in d dimension Mir Mehedi Faruk a,b , Md. Sazzad Hossain c , Md. Muktadir Rahman a Department of Theoretical Physics, University of Dhaka, Dhaka-1000 a Theoretical Physics, Blackett Laboratory, Imperial College, London SW7 2AZ, United Kingdom b Department of Nuclear Engineering, University of Dhaka, Dhaka-1000 c .Email: [email protected], [email protected] 20, 2018 Abstract
The changes in characteristics of Bose condensation of ideal Bose gas due to an external generic power law potential U = P di =1 c i | x i a i | n i are studied carefully. Detailed calculation of Kim et al. (S. H. Kim, C. K. Kim and K. Nahm, JPhys. Condens. Matter 11 10269 (1999).) yielded the hierarchy of condensation transitions with changing fractionaldimensionality. In this manuscript, some theorems regarding specific heat at constant volume C V are presented.Careful examination of these theorems reveal the existence of hidden hierarchy of the condensation transition intrapped systems as well. Many authors have investigated the thermodynamic properties of Bose gas[1, 2, 4, 6, 7, 8, 9, 10, 11, 12, 13], particularly af-ter it was possible to create Bose Einstein Condensation (BEC) in magnetically trapped alkali-metal gases[20, 21, 22]. Theconstrained role of external potential does change the characteristics of quantum gases[14, 15, 16, 17, 18, 19], providing anexciting opportunity to study the quantum mechanical effects. It is seen in the studies that the thermodynamic behaviorof non-relativistic quantum gases are governed by polylogarithm functions both in the case of trapped[10, 14, 15, 16] andfree[23, 24, 25, 26] quantum gases. It is also reported that the polylogarithm functions can give a single unified pictureof bosons and fermions for free[23] and trapped[28] quantum gases as well. Both in the case of Bose and Fermi gases,different structural properties of polylogarithms[9, 15, 27, 29] are related to different statistical effects. The trappingpotential radically changes different thermodynamic properties of quantum gases[15, 14, 27, 28]. Also the behavior ofthese quantities do change with dimensionality. Hence, it will be very intriguing to study the properties of Bose gasin detail before and after the critical temperature in arbitrary dimension with generic trapping potential of the form, U = P di =1 c i | x i a i | n i . In this report, we give a special emphasis on specific heat at constant volume C V as well as itsderivatives, which are the salient features to understand BEC[30].In a previous study with free Bose gas in arbitrary dimension, Kim et. al. [30] reported that the dimensionality con-tribution is a dominant factor to specify the behavior of BEC[30]. In general, there exists no discontinuity in C V at T = T C for free Bose gas in three dimensional space. However, its derivative is discontinuous where the magnitude of thediscontinuity is finite ( . NkT C ). Nevertheless, this is not true for free Bose gas in any arbitrary dimension. For instance,when d > C V is itself discontinuous at T = T C [5] for free Bose gas. In the case of trapped Bose gas with harmonicpotential[5, 14], the scenario completely changes as C V becomes discontinuous even in d = 3. In order to understand thecritical behavior of free Bose system more closely, Kim et. al. performed a calculation to check the discontinuity of l -thderivative of C V at T = T C , which yields the “class” of the C V function changes with dimensionality. This calculationshows that there exists a hierarchy of condensation transitions with changing fractional dimensionality. It is well knownthat, there is no BEC for free Bose gas in d ≤ et. al. found C V to be smooth function in the whole temperaturerange in this situation. If the dimensionality ranges from 2 < d ≤
3, the discontinuity of l -th derivative of C V at T = T C depends upon the sub-interval we choose in this range (see Table 1 of Ref. [30]), while the first derivative of C V remainsdiscontinuous for d >
3. In this report, we have proposed a new theorem, concerning the l -th derivative of C V for Bose gastrapped under generic power law potential, which coincides with the calculation of Kim et. al. , in the limit all n i → ∞ .Therefore one can can reproduce the previous calculation as a special case of the new theorem. This new theorem helps1s to understand the hidden hierarchy of condensation transition, existing in trapped systems as well. Most importantly,one can find situations even in an integer dimension where the l -th derivative of C V of trapped Bose gas is continuous,with appropriate trapping potential.In this work, the grand potential is determined at first, from which all the thermodynamic quantities such as inter-nal energy E , entropy S , Helmholtz free energy F and C V are calculated. In order to scrutinize them closely, thefugacity (chemical potential) is evaluated numerically in arbitrary dimension with any trapping potential as a functionof reduced temperature τ = TT C (see appendix) following Kelly’s work[31]. Then we propose the theorems regrading C V and its derivative. Beside this, the latent heat of condensation transition is studied in detail from the Clausius-Clapayronequation[6], from which we have deduced the required latent heat for Bose condensation in arbitrary dimension with anytrapping potential. All the calculated quantities reduce to text book result of free Bose gas when all n i → ∞ [5].The report is organized in the following way. The grand potential and the other thermodynamic quantities are pre-sented in section 2 and 3, respectively. In section 4 we deduce C V and the significant theorems regarding this. Section5 is devoted to investigate the latent heat of condensation for trapped system. Results are discussed in section 6. Thereport is concluded in section 7. Considering the ideal Bose gas in a confining external potential in a d-dimensional space with Hamiltonian, ǫ ( p, x i ) = bp l + d X i =1 c i | x i a i | n i (1)Where, p is the momentum and x i is the i th component of coordinate of a particle and b, l, a i , c i , n i are all positiveconstants. Here, c i , a i and n i determine the depth and confinement power of the potential. Using l = 2, b = m onecan get the energy spectrum of the Hamiltonian used in the literature [5, 6, 11, 15, 9]. Note that, x i < a i . For the freesystem all n i −→ ∞ . As | x i a i | <
1, the potential term goes to zero when all n i −→ ∞ .The resulting density of states with this Hamiltonian is[9], ρ ( E ) = C ( m, V ′ d ) E χ − (2)where, C ( m, V ′ d ) is a constant depending on effective volume V ′ d and mass m [9, 14]The grand potential for the Bose system, q = − X ǫ ln (1 − zexp ( − βǫ )) (3)where k , µ and z = exp( βµ ) being the Boltzmann Constant, the chemical potential and fugacity respectively and β = kT .Replacing the sum by integration we get, q = q − Z ∞ ρ ( ǫ ) ln (1 − zexp ( − βǫ )) (4)which yields,[9, 14] q = q + g V ′ d λ ′ d g χ +1 ( z ) (5)2here, g l ( z ) = Z ∞ dx x l − z − e x − ∞ X j =1 z j j l (6) q = − ln (1 − z ) (7) V ′ d = V d d Y i =1 ( kTc i ) /n i Γ( 1 n i + 1) (8) λ ′ = hb l π ( kT ) l [ d/ d/l + 1 ] /d (9) χ = dl + d X i =1 n i (10)And, V ′ d and g χ ( z ) is known as effective volume and Bose function respectively. Now, as z →
1, the Bose function g χ ( z )approaches ζ ( χ ), for χ > g χ ( z ) due to Robinson is [5, 30] g χ ( e − η ) = Γ(1 − χ ) η − χ + ∞ X i =0 ζ ( χ − i ) η i (11) The number of particles can be evaluated from the grand potential[9, 14], N = z ( ∂q∂z ) β,V ⇒ N − N = g V ′ d λ ′ d g χ ( z ) (12)The above equation suggests the only relevant quantity that determines BEC to take place is χ . In case of BEC as T → T C , z →
1. So, BEC would take place when, χ = dl + d X i n i > T c = 1 k [ N c h d b d/l Q di =1 c /n i i gC n Γ( dl + 1) V d Q di =1 Γ( n i + 1) ζ ( χ ) ] χ (14)For massive bosons (with l = 2 ), one can find the BEC criterion, d d X i n i > P di n i → d > N N = 1 − ( TT c ) χ (17)The above equation produces the exact ground state fraction for free system when all n i → ∞ [5]. Again, obtaining3 U / N K B T n = n = n = n = n = n = (a) S / N K B n = 1 n = 2 n = 3 n = 5 n = 7 n = (b) n= n= n= n= n= n= F / N K B T (c) N / N n = 1 n = 2 n = 3 n = 4 n = 5 n = (d) Figure 1: Fig (a), (b), (c) and (d) respectively represents internal energy, entropy, Helmholtz free energy and groundstate fraction of Bose system, with different trapping potential in d = 3 . n = ∞ denotes free Bose system.internal energy E [9, 14] from the grand potential, E = − ( ∂q∂β ) z,V = ( N kT χ g χ +1 ( z ) g χ ( z ) , T > T c N kT χ ζ ( χ +1) ζ ( χ ) ( TT c ) χ , T ≤ T c (18)Now, the entropy S [9, 14], below and greater than the critical temperature S = kT ( ∂q∂T ) z,V − N k ln z + kq = ( N k [ v ′ d λ ′ d ( χ + 1) g χ +1 ( z ) − ln z ] , T > T c ( N − N ) k v ′ d λ ′ d ( χ + 1) ζ ( χ + 1) , T ≤ T c (19)And, finally the expression of Helmholtz free energy, A = − kT q + N kT ln zAN kT = ( − g χ +1 ( z ) g χ ( z ) + ln z , T > T c − νλ d ζ ( χ + 1) , T ≤ T c (20)4 Heat capacity at constant volume C V Heat capacity at constant volume C v below and above T c [9, 14], C V = T ( ∂S∂T ) N,V = ( N k [ χ ( χ + 1) g χ +1 ( z ) g χ ( z ) − χ g χ ( z ) g χ − ( z ) ] , T > T c N kχ ( χ + 1) ζ ( χ +1) ζ ( χ ) ( TT C ) χ , T ≤ T c (21)Another important quantity ∂C v ∂T below and above T C ,1 N k ∂C v ∂T = ( T h χ ( χ + 1) g χ +1 ( z ) g χ ( z ) − χ g χ ( z ) g χ − ( z ) − χ g χ ( z ) g χ − ( z ) g χ − ( z ) i , T > T c T χ ( χ + 1) ζ ( χ +1) ζ ( χ ) ( TT C ) χ , T ≤ T c (22) C V / N K B n = n = n = n = n = (a) d=3 n = 1 n = 2 n = 3 n = 5 n = 7 C V / N K B (b) d=2 Figure 2: Fig (a) and (b), represents C V of Bose system, with different trapping potential in d = 3 and d = 2, respectively. Theorem 4.1 : Let an ideal Bose gas in an external potential, U = P di =1 c i | x i a i | n i ,(i) if the Bose gas does condense ( χ > ), heat capacity C V will be discontinuous at T = T c if and only if, χ = dl + d X i =1 n i > T c as∆ C V | T = Tc = C V | T − c − C V | T + c = N kχ ζ ( χ ) ζ ( χ − Proof : 5s T → T c , z → η →
0, where η = − ln z . For T → T + c , C V ( T + C ) = N k [ χ ( χ + 1) ν ′ λ ′ D g χ +1 ( z ) | z → − χ g χ ( z ) g χ − ( z ) | z → ]= N k [ χ ( χ + 1) ν ′ λ ′ D ζ ( χ + 1) − χ ζ ( χ ) g χ − ( z ) | z → ] (23)As the denominator of the second term of the right hand side contains g χ − ( z ), we can not simply substitute it with zetafunction as z → C V ( T + C ) = N k [ χ ( χ + 1) ν ′ λ ′ D ζ ( χ + 1) − χ ζ ( χ )Γ(2 − χ ) η − χ | η → ] (24)On the other hand C V ( T − C ) = N k [ χ ( χ + 1) ν ′ λ ′ D ζ ( χ + 1) (25)Taking the difference between C V ( T + C ) and C V ( T − C ), we get,∆ C V | T = Tc = χ ζ ( χ )Γ(2 − χ ) η − χ | η → (26)Which dictates, C | T = Tc will be non zero for χ >
2. So, C V will be discontinuous when χ > χ should be greater than two for ∆ C V at T = T C to be non-zero, we can re-write equation (23), by substitut-ing g χ − ( z ) by zeta function. C V ( T + C ) = N k [ χ ( χ + 1) ν ′ λ ′ D ζ ( χ + 1) − χ ζ ( χ ) ζ ( χ −
1) ] (27)Now, from Eq. (25) and (27) we can write.∆ C V | T = Tc = C V | T − c − C V | T + c = N kχ ζ ( χ ) ζ ( χ − C V at T = T C for d >
4, producingthe same result as Ziff et al.
Theorem 4.2 : Let an ideal Bose gas in an external potential, U = P di =1 c i | x i a i | n i ,(i) the jump of the the first derivative of C V at T = T C will be finite for χ = , it will be infinite for χ > andno jump for χ < (ii) And the finite jump of the first derivative of C V , at T = T c is ∆ ∂C V ∂T | T = Tc = ∂C V ∂T | T − c − ∂C V ∂T | T + c = 27 N k T c [ ζ ( 32 )] Γ( )Γ( ) Proof :From equation (21) as T −→ T c , we obtain,1 N k ∂C v ∂T = ( T c h χ ( χ + 1) g χ +1 ( z ) g χ ( z ) − χ g χ ( z ) g χ − ( z ) − χ g χ ( z ) g χ − ( z ) g χ − ( z ) i , T → T + c T c χ ( χ + 1) ζ ( χ +1) ζ ( χ ) , T → T − c (28)6ow taking the difference, ∆ ∂C V ∂T | T = Tc = N kT c h χ g χ ( z ) g χ − ( z ) + χ g χ ( z ) g χ − ( z ) g χ − ( z ) i | z → (29)Using the representation due to Robinson,∆ ∂C V ∂T | T = Tc = N kT c h χ ζ ( χ )Γ(2 − χ ) η − χ + χ Γ(3 − χ )[Γ(2 − χ )] [ ζ ( χ )] η − χ ] | η → (30)Therefore the above equation suggests, ∆ ∂C V ∂T | T = Tc will be nonzero when χ ≥ / χ > the second term obviously diverges towards infinity and making the whole term infinite. But as χ = , the first termvanishes as η → χ = inEq. (30) yields ∆ ∂C V ∂T | T = Tc = ( 32 ) Γ(3 − )[Γ(2 − )] [ ζ ( 32 )] ] = 27 N k T c [ ζ ( 32 )] Γ( )Γ( ) (31)Which completes the second part of the proof. In case of free massive (all n i → ∞ ) bosons in d = 3 Eq. (31) yields,the magnitude of the discontinuity of first derivative of C V at T = T C is 3 . NkT C , reproducing the exact same result ofPathria[5]. Theorem 4.3 : Let an ideal Bose gas in an external potential, U = P di =1 c i | x i a i | n i , The between l -th derivative of heatcapacities at T = T C is, ∆ l ( T C ) = lim T → T C [( ∂∂T ) l C − v ( T ) − ( ∂∂T ) l C + v ( T )] = lim η −→ l X j =1 a lj η j +2 − ( j +1) χ Proof:
We prove the above equation by the method of induction.For l = 1, ∆ ( T c ) = lim η → a η − χ (32)This is clearly true from equation (13) and the known result for D = 3.Let’s assume the equation holds for any positive integer l . Then,∆ l ( T c ) = lim η → l X j =1 a lj η j +2 − ( j +1) χ (33)Now, in case of l + 1, ∆ l +1 ( T c ) = lim η → l X j =1 a lj ( j + 2 − ( j + 1) χ ) χζ ( χ ) T c Γ(2 − χ ) η j +3 − ( j +2) χ = lim η → k +1 X j =2 a l +1 ,j − ( j + 1 − jχ ) χζ ( χ ) T c Γ(2 − χ ) η j +2 − ( j +1) χ = lim η → l +1 X j =2 a l +1 ,j η j +2 − ( j +1) χ (34) a lj satisfies the recurrence relation. a l +1 ,j = ( j + 1 − jχ ) χζ ( χ ) T c Γ(2 − χ ) a l +1 ,j − (35)(36)7here, j = 2 , , , ..., k + 1.Therefore, (a) and (b) enable us, for any positive integer l , to write∆ l ( T c ) = lim η −→ l X j =1 a lj η j +2 − ( j +1) χ (37)This above equation coincides with Kim et. al. in case of free massive boson. L / K B T d n = 1 n = 2 n = 3 n = Figure 3: Latent heat of Bose condensationJust as, any first order phase transition pressure is governed by Clausius-Clapeyron equation, in the transitionline[6, 11]. Like the BEC for free Bose gas at d = 3, BEC for trapped Bose gas do also exhibit first order phasetransition as they obey the Clausius-Clapayron equation[6]. The Clausius-Clapeyron equation which is derived fromMaxwell relations take the form, dPdT = ∆ s ∆ v = LT ∆ v (38)where, L , ∆ s and ∆ v are the latent heat, change in entropy and change in volume respectively. The effective pressure inphase transition line is, P ( T ) = kTλ ′ d g χ +1 (1) (39)Differentiating with respect to T leads, dP dT = 1 T v g [( χ + 1) kT g χ +1 (1) g χ (1) ] (40)When two phases coexist the non condensed phase has specific volume v g whether the condensed phase has specificvolume has specific volume 0, concluding ∆ v = v g . So, the latent heat of transition per particle in case of trapped Bosonis L = ( χ + 1) ζ ( χ + 1) ζ ( χ ) kT (41)8o, in the case of free massive boson the latent heat per particle in three dimensional space is, L = 52 kT ζ (5 / ζ (3 /
2) (42)same as the text book results[6].
In this section we discuss the influence of trapping potential on thermodynamic quantities. Beside this we also explainthe theorems and their implications as well. In the figures, we have used n = n = .. = n i = .. = n d = n (isotropicpotential), but the formulas described in the above sections are more general.In Fig (1), we have described the influence of different power law potentials on thermodynamic quantities such asinternal energy E , entropy S , free energy F and ground state fraction of Bose gas in three dimensional space. In thecase of internal energy, a strict nonlinear behavior is observed when T < T C and this behavior is more noticeable whenwe decrease the value of n . But in principle, the effect of power law potential is seen in both below and above T C . Samephenomena is also observed in the case of S and F . Entropy remains constant in the condensed phase, with a specificchoice of n . But the entropy increases while the value of free energy gets lower with trapping potential. Now turning ourattention towards ground state fraction, Eq. (15) and (17) dictates, | dN dT | T = T c | > NT C . Thus, this relation depicts a verysignificant characteristic at the onset of BEC, that the N − T curves are always convex for Bose system in condensedphase. But their curvatures are different depending on trapping potentials as shown in figure 1 (d).Fig 2 (a), (b) illustrates the C V of Bose system with different power law potentials in three and two dimensional space,respectively. When d = 3, there is no discontinuity in free system for C V at T = T C . But according to theorem 4.1 inthree dimensional space, C V becomes discontinuous when n <
6, for isotropic trapping potential, which is visible in fig.2 (a). In the case of d = 2, theorem A dictates that C V becomes discontinuous when n < C V at T = T C . Now, let us turn our attention towards latent heat. Fig 3 illustrates the behavior of latent heat oftrapped Bose gas, with changing dimensionality. As BEC is a 1st order phase transition[6], latent heat is associated withthis phase transition. So, zero latent heat denotes no phase transition i.e. no condensed phase. As there is no phasetransition in d < d < d = 2 for any trapping potential unless P i =1 , n i = 0. Fig. 3 is in accordance with thisfact as latent heat is seen to be non-zero with any trapping potential. In the case of d = 1 we get from Eq. (13) thatBEC will take place if n <
2. It is seen in the Fig. 3 that latent heat is non-zero when n = 1 indicating the existence ofcondensed phase in one dimensional space.Table 1: The hierarchy of the condensation transition of ideal Bose gas trapped under generic power law potential. Theresult of table are in agreement with Kim et al. for free system. But in the case of trapped system the ”class” of functionssignificantly change depending on the values of n i χ ( ∂/∂T ) C v ( ∂/∂T ) C v ( ∂/∂T ) C v ( ∂/∂T ) C v ..... Class χ = d C ≤ χ < c d C ≤ χ < c c d C ... l +2 l +1 ≤ χ < l +1 l c c c ...(d) C l − ... χ = 1 c c c c c C ∞ et. al. shows how thecondensed phases can be different, depending on the dimensionality for free Bose gas. The dimensional dependence ofdiscontinuity of the l th derivative of C V indicates hidden hierarchy of the condensation transition with changing frac-tional dimensionality. Theorem 4.3 generalizes this result for trapped systems indicating a similar hierarchy where wefind the class of l th derivative of C v depends on χ . We now elaborate how this theorem classifies the class of l th derivativeof C v and present it in table 1.(i) From Eq. (37), when l = 1, we see the difference between 1st derivative before and after T c ,∆ = a η − χ | η → In order to be discontinuous we need ∆ = 0, which will be true, when 3 − χ ≤ χ ≥ . Furthermore, if χ > , the1st derivative is infinitely discontinuous and χ = denotes finite discontinuity.(ii) Again, for l = 2, ∆ = a η − χ + a η − χ | η → Therefore, for the 2nd derivative to be discontinuous we need, ≤ χ < .(iii) Similarly, for the l th derivative to be discontinuous, the necessary condition is l +2 l +1 ≤ χ < l +1 l (iv) Careful observation of Eq. (37) reveals ∆ l is independent of j for χ = 1, which indicates ∆ = 0 for η → χ = 1 all derivatives of C v are continuous. Using all these information, one can find out the hierarchy of thecondensation transition (see table 1). z n = n = n = n = n = (a) d=3 z n = n = n = n = d=2 (b) Figure 4: Fugacity as a function of τ = TT C , with different power law potentials.In this section we demonstrate how fugacity can be expressed as a funtion of τ = TT C following kelly’s work. Thenumber of particles in the excited state near critical temperature, N e = 1 λ dc ζ ( χ )10nd, the total number of particles can be written as, N = 1 λ d g χ ( z )In the noncondensed phase, one can approximate N e ≃ N . So, in that case we get from the above two equations, τ d/ g χ ( z ) ζ ( χ ) = 1Solving the above equation numerically, using mathematica we get our desired result (see figure 4)Another very important relation used in deriving the different thermodynamic quantities are,( ∂z∂T ) V = − χ zT g x ( z ) g x − ( z ) (43) The changes in characteristics of Bose condensation of ideal Bose gas due to an external generic power law potentialare studied from the grand potential. The presented theorems turn out to be very important for trapped Bose systems(non-relativistic). But it will be interesting to generalize these theorems for relativistic Bose gas.
MMF would like to thank Dr. Jens Roder for his hospitality during the author’s visit in ISOLDE, CERN where a majorpart of the work is done. Also the authors would like to thank Mr. Arya Chowdhury for his cordial help to present thework.
References [1] R. H. May, Phys. Rev., A1515, 135, (1964).[2] D. W. Robinson, Commun. Math Phys. 50, 53 (1976).[3] M. Luban and M. Revzen, J Math Phys. 9, 347 (1967).[4] L. J. Landau and I. F. Wilde, Commun. Math Phys. 70, 43 (1979).[5] R. K. Pathria,
Statistical Mechanics , Elsevier, 2004.[6] K. Huang,
Statistical Mechanics , Wiley Eastern Limited, 1991.[7] Z. Yan, Phys. Rev. A 59, 1999.[8] Z. Yan, Phys. Rev. A 61, 2000.[9] Z. Yan, Mingzhe Li, L Chen, C. Chen and J. Chen, J. Phys. A: Math. Gen. 32 (1999) 4069–4078.[10] C. J. Pethick and H. Smith,
Bose-Einstein Condensation In Dilute Gases , Second Edition, Cambridge UniversityPress, 2008 .[11] R. M. Ziff, G. E Uhlenbeck, M. Kac, Phys. Reports 32 169 (1977).[12] R. Beckmann and F. Karch, Phys. Rev. Lett 43 1277 (1979).[13] R. Beckmann, F. Karch and D. E. Miller, Phys. Rev. A 25 561 (1982).[14] M. M. Faruk, arXiv:1502.07054 (To be appeared in Acta Physica Polonica B).[15] L. Salasnich, J. Math. Phys 41, 8016 (2000).[16] A. Jellal and M. Doud, Mod. Phys Lett. B 17 1321 (2003).[17] Mingzhe Li, Zijun Yan, Jincan Chen, Lixuan Chen and Chuanhong Chen, Phys Rev. A, 58 (1998).1118] F. Dalfovo, S. Giorgini, L.P. Pitaevskii, and S. Stringari, Rev. Mod. Phys. 71 (1999).[19] L. Salasnich, A. Parola, and L. Reatto, Phys. Rev. A 59, 2990 (1999).[20] C. C. Bradley, C. A. Sackett, J. J. Tollett and R. G. Hulet, Phys. Rev. Lett. 75, 1687, (1995).[21] M. H. Anderson, J. R. Esher, M. R. Mathews, C. E. Wieman and E. A. Cornell, Science 269, 195, (1995).[22] K. B. Davis, M. O. Mewes, M. R. Andrew, N. J. Van Druten, D. S. Durfee, D. M. Kurn and W. Ketterle, Phys.Rev. Lett., 1687, 75, 1995.[23] Lee, M. H., Phys. Rev. E, 55, 1518-1520 (1997).[24] Lee, M. H., J. Math. Phys. 36, 1217-1230 (1995).[25] Lee, M. H. Acta Phys. Polonica, 40, 1279-1301 (2009).[26] Lee, M. H., Phys. Rev. E, 56, 3909-3912 (1997).[27] M. M. Faruk, arXiv:1504.06050 (To be appeared in Acta Physica Polonica B).[28] M. M. Faruk, arXiv:1506.05423 (To be appeared in Journal of Statistical Physics, Springer).[29] M. M. Faruk, Eur. J. Phys. 36, 2015.[30] S. H. Kim, C. K. Kim and K. Nahm, J Phys. Condens. Matter 11 10269 (1999).[31] James kelly,