- Email: [email protected]
Thermodynamics of a two-parameter deformed quantum group boson gas
7 January 2002
Physics Letters A 292 (2002) 251–255 www.elsevier.com/locate/pla
Thermodynamics of a two-parameter deformed quantum group boson gas Abdullah Algin Physics Department, Osmangazi University, Me¸selik, Eski¸sehir, Turkey Received 6 June 2001; received in revised form 7 November 2001; accepted 7 November 2001 Communicated by C.R. Doering
Abstract A two-parameter deformed quantum group boson gas with SU q1 /q2 (2) symmetry is discussed and for high temperatures, its thermodynamical properties obtained by means of a SU q1 /q2 (2)-invariant bosonic Hamiltonian are investigated in terms of the deformation parameters q1 and q2 . The usual boson gas results can be recovered in the limit q1 = q2 = 1. 2002 Published by Elsevier Science B.V.
1. Introduction In the past decade, quantum groups and their associated algebras [1–5], which are generalizations of ordinary Lie groups and Lie algebras with particular deformation parameters have produced many developments in the framework of statistical mechanics as well as other areas of physics and mathematics. For instance, possible relations between quantum groups and generalized statistical mechanics have been investigated [6–8]. Meanwhile, a great deal of effort has been spent in order to obtain some physical interpretation on the deformation parameter q by considering a canonical ensemble of q-oscillators [9,10], which are q-deformations of the bosonic harmonic oscillator algebra. Thermodynamical and statistical properties of a gas of deformed bosons as well as fermions have been extensively investigated [11–15]. Interestingly, it was shown in Ref. [16] that the high temperature properties
E-mail address: [email protected] (A. Algin).
of the bosonic and fermionic gases having the symmetry of the quantum group SU q (N) depend radically on the deformation parameter q. Although two-parameter extensions of such investigations have been studied in the literature [17–19], the complete history of the generalization of the conventional physical quantities of the bosons and fermions coming out by deformations is still under active consideration. Inspired by the work of Ubriaco [16], the present Letter is mainly aimed to give a two parameter generalization related to the thermodynamical properties of a quantum group boson gas described by the symmetry of the quantum group SU q (2), where q = q1 /q2 in the high temperature (T ) limit, namely for z = eβµ 1, where β = 1/kB T , kB is Boltzmann’s constant, and µ is the chemical potential. This Letter is organized as follows. We first review some basic definitions and properties concerning the quantum group SU q (N) bosons and particularly the SU q (2) one. Then we introduce our model described by a Hamiltonian in terms of the SU q1 /q2 (2) generators. This leads to a calculation of the thermodynamical properties of the model such as the average num-
0375-9601/02/$ – see front matter 2002 Published by Elsevier Science B.V. PII: S 0 3 7 5 - 9 6 0 1 ( 0 1 ) 0 0 7 9 1 - 5
252
A. Algin / Physics Letters A 292 (2002) 251–255
ber of particles and the pressure obtained via the grand partition function. Finally, we present our conclusions.
it can be found from Eq. (7) [22] that ab = qba, cd = qdc,
2. Review on the quantum group SUq (N ) bosons The conventional boson oscillators satisfy the following commutation relations: φi φj+ − φj+ φi = δij , φi φj − φj φi = 0, φi+ φi = Ni ,
i, j = 1, 2, . . . , N,
(1)
φi+
are the bosonic annihilation and crewhere φi and ation operators, respectively, and Ni is the boson number operator. These relations are obviously invariant under the action of the undeformed group SU(N). The quantum group analogues of the above equations are defined by the commutation relations [5,16] Φj Φ¯ i = δij + qRkij l Φ¯ l Φk , Φl Φk = q
−1
Rj ikl Φj Φi ,
i, j = 1, 2, . . . , N,
where the N 2 × N 2 matrix R [20] is Rj ikl = δj k δil 1 + (q − 1)δij + q − q −1 δik δj l θ (j − i),
(2) (3)
(4)
and the function θ (j − i) is defined by 1 if j > i, θ j −i = (5) 0 otherwise. Eqs. (2) and (3) are covariant under the linear transformation
Φ = T Φ,
(6)
where the matrix T ∈ SUq (N). The SU q (N) transformation matrix T and the R-matrix satisfy the relations [21] RT1 T2 = T2 T1 R,
(7)
R12 R13 R23 = R23 R13 R12 ,
(8)
where T1 = T ⊗ 1, T2 = 1 ⊗ T ∈ V ⊗ V and (R23 )ij k,i j k = δii Rj k,j k ∈ V ⊗ V ⊗ V . For a unitary quantum group matrix a b , T= c d
bc = cb,
ac = qca, bd = qdb, ad − da = q − q −1 bc,
Detq (T ) = ad − qbc = 1,
(9)
with the unitary conditions a¯ = d, b¯ = qc, q ∈ R. For N = 2, the commutation relations generating the quantum group SU q (2) bosons can be obtained [5] by Eqs. (2) and (3) as follows: Φ1 Φ¯ 1 − q 2 Φ¯ 1 Φ1 = 1, Φ2 Φ¯ 2 − q 2 Φ¯ 2 Φ2 = 1 + q 2 − 1 Φ¯ 1 Φ1 ,
(10)
Φ1 Φ2 = qΦ2 Φ1 , Φ1 Φ¯ 2 = q Φ¯ 2 Φ1 ,
(12)
(11) (13)
which they become the usual boson algebra in the limit q = 1. However, we exploit a different quantum group SU q (2) bosons, where q = q1 /q2 . The notion of these deformed oscillators was first introduced in Ref. [23] during a realization of the most general quantum group invariant oscillator algebra. The commutation relations generating the quantum group SU q1 /q2 (2) bosons are defined by Φ1 Φ¯ 1 − q12 Φ¯ 1 Φ1 = q22N , Φ2 Φ¯ 2 − q12 Φ¯ 2 Φ2 = q22N + q12 − q22 Φ¯ 1 Φ1 , Φ1 Φ2 = q1 q2−1 Φ2 Φ1 , Φ¯ 1 Φ¯ 2 = q2 q1−1 Φ¯ 2 Φ¯ 1 , Φ1 Φ¯ 2 = q1 q2 Φ¯ 2 Φ1 ,
(14) (15) (16) (17) (18)
where N is the total boson number operator and q1 , q2 ∈ R. Hereafter, we consider 0 < q1 < ∞ and 0 < q2 < ∞. The limit q2 = 1 corresponds to the usual quantum group SU q1 (2) as defined by Eqs. (10)–(13). Furthermore, a differential calculus on the quantum hyperplane corresponding to the quantum group SU q (2) was discussed by Wess and Zumino [20] and a two-parameter generalization of the Wess and Zumino differential calculus was constructed in [24].
3. SUq1 /q2 (2) boson model Let us consider the following Hamiltonian in terms of SU q1 /q2 (2) generators for two kinds of bosons with
A. Algin / Physics Letters A 292 (2002) 251–255
the same energy: εk (M1,k + M2,k ), H=
4. Thermodynamic properties (19)
k
where the operators M1,k and M2,k are defined by M1,k = Φ¯ 1,k Φ1,k ,
M2,k = Φ¯ 2,k Φ2,k ,
(20)
εk is the spectrum of energy, k = 0, 1, 2, . . . , and obviously [Φ¯ i,k , Φj,k ] = 0 for k = k . These operators also satisfy the following relations for a given k: M2 Φ1 − q1−2 Φ1 M2 = 0, M1 Φ2 − q2−2 Φ2 M1
= 0.
(21)
The normalized states of the above Hamiltonian can be built by applying successively the operators Φ¯ on the vacuum state |0, 0 for a given k as 1 Φ¯ m1 Φ¯ m2 |0, 0, |m1 , m2 = √ [m1 ]![m2 ]! 1 2
(22)
where the Fibonacci basic number [m], which is a generalization of the usual q-numbers is in the form [23] [m] =
q12m − q22m q12 − q22
.
253
(23)
In this section, we investigate the thermodynamic properties of the bosonic system described by a Hamiltonian in Eq. (26). The grand partition function Z is written ¯
¯
Z = Tr e−βεk (Φ1,k Φ1,k +Φ2,k Φ2,k ) eβµ(N1,k +N2,k ) ,
(27)
where the trace is taken over the states in Eq. (22). From Eqs. (23)–(26), the grand partition function becomes Z=
∞ ∞
e−βεk [n1 +n2 ] eβµ(n1 +n2 ) ,
(28)
k n1 =0 n2 =0
which can be rewritten as Z=
∞ (n + 1)e−βεk [n] zn ,
(29)
k n=0
where z = eβµ is the fugacity. Since we are dealing with the high temperature case, namely the limit z 1, for the three-dimensional momentum space the grand partition function in Eq. (29) can be calculated in the first few terms as follows: ∞
In order to calculate the thermodynamical properties of the Hamiltonian in Eq. (19), we propose a new representation. By virtue of such a representation, we are able to express the SU q1 /q2 (2) bosons in terms of + the ordinary boson operators φi,k and φi,k satisfying Eqs. (1) for a given k as follows: −1 Φ1 = φ1+ [N1 ]q2N2 , (24) Φ¯ 1 = φ1+ q2N2 , + −1 N1 N + 1 [N2 ]q1 , Φ¯ 2 = φ2 q1 . Φ2 = φ2 (25)
4πV ln Z = 3 h
By using Eqs. (23)–(25), we rewrite the above Hamiltonian as H= (26) εk [N1,k + N2,k ],
which leads to an explicit relation by evaluating the integral, √ π 2m 3/2 4πV z ln Z = 3 2 β h √ 2m 3/2 2 + π (31) z ξ(q1 , q2 ) + · · · , β where the function ξ q1 , q2 is given by
1 1 3 ξ(q1 , q2 ) = (32) − √ . 4 (q12 + q22 )3/2 2
k + where Ni,k = φi,k φi,k and the bracket [x] is defined by Eq. (23). Note that when compared with the original Hamiltonian in Eq. (19), this representation leads to an interacting Hamiltonian. As a result, such a model may be interpreted in a way that φ1 and φ2 correspond to two different kinds of bosonic particles which interact with each other but do not interact among themselves.
p2 dp 0
z2 −βεk [2] 6e − 4e−2βεk × 2e−βεk z + 2! 3 z + 24e−βεk [3] − 18e−βεk [2] e−βεk 3!
−3βεk + ··· , + 16e
(30)
254
A. Algin / Physics Letters A 292 (2002) 251–255
The average number of particles N can then be calculated by 1 ∂ ln Z N = (33) , β ∂µ T ,V which leads to √ 4πV π 2m 3/2 N = 3 z 2 β h √ 2m 3/2 2 +2 π z ξ(q1 , q2 ) + · · · . β
(34)
By reverting this equation, the fugacity can be found as 3/2 1 h2 N z≈ 2 2πmkT V 3 2 N 2 h − ξ(q1 , q2 ) (35) . 2πmkT V The pressure can also be calculated by 1 ∂ ln Z P= , β ∂V T ,µ which gives √ 4π π 2m 3/2 P= 3 z h β 2 β √ 2m 3/2 2 + π z ξ(q1 , q2 ) + · · · . β
(36)
(37)
By using the above equations, the equation of state is derived depending on the deformation parameters q1 and q2 as 3/2 h2 N P V = kT N 1 − ξ q1 , q2 2πmkT V +··· .
(38)
On the other hand, the equation of state can also be calculated by means of similar calculations in the two-dimensional space for the SU q1 /q2 (2) boson gas as follows: h2 N + ··· , P A = kT N 1 − ζ (q1 , q2 ) 2πmkT A (39)
Fig. 1. The coefficient ζ (q1 , q2 ) for the interval 0 < q12 + q22 6. The line at q12 + q22 = 3.0 divides the region between ζ (q1 , q2 ) > 0 and ζ (q1 , q2 ) < 0 which corresponds to bosonic and fermion-like behavior, respectively. Also, in the limit q1 or q2 → ∞ the lowest value of the coefficient ζ (q1 , q2 ) is −0.25.
where A is the surface confining the bosonic system and the function ζ(q1 , q2 ) is
1 3 − 1 , ζ(q1 , q2 ) = (40) 4 (q12 + q22 ) which represents a numerical factor in the second virial coefficient through Eq. (39) for the two-parameter deformed boson gas. We should mention for all equations above that the limit q2 = 1 gives the same results derived in Ref. [16] for the boson gas. We now wish to discuss some important limiting cases considered by means of the coefficients ξ(q1 , q2 ) and ζ(q1 , q2 ) in Eqs. (32) and (40). Obviously, for both two and three-dimensional system, the sign of the second virial coefficient given by either Eq. (38) or Eq. (39) depends on the values of the deformation parameters q1 and q2 . For the case of three-dimensional system, at q1 = q2 = 1 the coefficient ξ(1, 1) = 2−7/2 , which corresponds to the numerical factor in the second virial coefficient for a free boson gas with two different kinds of bosonic particles. The ideal gas ξ(q1 , q2 ) = 0 and the free fermion ξ(q1 , q2 ) = −2−7/2 [25] cases are reached at q12 + q22 ≈ 2.62 and q12 + q22 ≈ 4.16, respectively. On the other hand, for the two-dimensional system, at the values q1 = q2 = 1 and q12 + q22 = 6.0 the system behaves as a free boson
A. Algin / Physics Letters A 292 (2002) 251–255
ζ (1, 1) = 2−3 and fermion gas ζ (q1 , q2 ) = −2−3 [25], respectively. The second virial coefficient in Eq. (39) vanishes at q12 + q22 = 3.0 which corresponds to the ideal gas case. In a simple manner, Fig. 1 shows a graph of the coefficient ζ (q1 , q2 ) as a function of a sum of the model parameters q12 and q22 in the twodimensional space. When we compare with the results of the oneparameter deformed SU q (2) boson model [16], the most interesting point is that the present two-parameter deformed boson model has an infinite number of possibilities in getting the second virial coefficient for the fermion gas. Since the two-parameter deformed boson gas behaves as a fermion gas [25] at the value q12 + q22 = 6.0, we may remark that willingness of fermion-like behavior of the present two-parameter boson model increases too much for a wide range of the parameters q1 and q2 . We end up this section by briefly emphasizing that the parameters q1 and q2 serve as interpolating objects between repulsive and attractive behaviors as is shown in Fig. 1.
5. Conclusion In this Letter, we have investigated the thermodynamical properties of a two-parameter deformed quantum group boson gas SU q1 /q2 (2) for the high temperature case. For instance, the average number of particles and the pressure are derived and subsequently the equation of state is obtained. We have also studied the simplest quantum group SU q (2) and discussed the defining commutation relations for a two-parameter deformed SU q1 /q2 (2)-covariant boson algebra. The model presented here generalizes the one-parameter deformed quantum group boson gas results developed recently in [16]. However, the limit q1 = q2 = 1 gives the ordinary boson gas properties. As a final remark, the low temperature behavior of the model is another face of the work in which the
255
main focus will be on the study of the Bose–Einstein condensation. Acknowledgements This work was started at Feza Gürsey Institute, Istanbul, Turkey. I would like to thank the Institute and its director Prof. I.H. Duru. References [1] V.G. Drinfeld, Proc. ICM 1 (1986) 798. [2] M. Jimbo, Lett. Math. Phys. 11 (1986) 247. [3] L.D. Faddeev, N.Y. Reshetikhin, L.A. Takhtajan, Quantization of Lie groups and Lie algebras, Preprint LOMI (1987). [4] S.L. Woronowicz, Comm. Math. Phys. 111 (1987) 613. [5] W. Pusz, S.L. Woronowicz, Rep. Math. Phys. 27 (1989) 231. [6] C. Tsallis, Phys. Lett. A 195 (1994) 329. [7] S. Abe, Phys. Lett. A 224 (1997) 326. [8] M. Arik, J. Kornfilt, A. Yildiz, Phys. Lett. A 235 (1997) 318. [9] M.A. Martín-Delgado, J. Phys. A 24 (1991) L1285. [10] P.V. Neskovic, B.V. Urosevic, Int. J. Mod. Phys. A 7 (1992) 3379. [11] C.R. Lee, J.P. Yu, Phys. Lett. A 150 (1990) 63. [12] J.A. Tuszynski, J.L. Rubin, J. Meyer, M. Kibler, Phys. Lett. A 175 (1993) 173. [13] M. Chaichian, R.G. Felipe, C. Montonen, J. Phys. A 26 (1993) 4017. [14] M.R. Ubriaco, Phys. Lett. A 219 (1996) 205. [15] M.R. Ubriaco, Mod. Phys. Lett. A 11 (1996) 2325. [16] M.R. Ubriaco, Phys. Rev. E 55 (1997) 291. [17] R. Chakrabarti, R. Jagannathan, J. Phys. A 24 (1991) L711. [18] M. Daoud, M. Kibler, Phys. Lett. A 206 (1995) 13. [19] R.S. Gong, Phys. Lett. A 199 (1995) 81. [20] J. Wess, B. Zumino, Nucl. Phys. B (Proc. Suppl.) 18 (1990) 302. [21] L.A. Takhtajan, Adv. Stud. Pure Math. 19 (1989) 1. [22] S. Vokos, B. Zumino, J. Wess, Symmetry in Nature, Scuola Normale Superiore, Pisa, 1989. [23] M. Arik, E. Demircan, T. Turgut, L. Ekinci, M. Mungan, Z. Phys. C 55 (1992) 89. [24] A. Schirrmacher, J. Wess, B. Zumino, Z. Phys. C 49 (1991) 317. [25] A. Algin, M. Arik, A.S. Arikan, Phys. Rev. E, submitted.
Physics Letters A 292 (2002) 251–255 www.elsevier.com/locate/pla
Thermodynamics of a two-parameter deformed quantum group boson gas Abdullah Algin Physics Department, Osmangazi University, Me¸selik, Eski¸sehir, Turkey Received 6 June 2001; received in revised form 7 November 2001; accepted 7 November 2001 Communicated by C.R. Doering
Abstract A two-parameter deformed quantum group boson gas with SU q1 /q2 (2) symmetry is discussed and for high temperatures, its thermodynamical properties obtained by means of a SU q1 /q2 (2)-invariant bosonic Hamiltonian are investigated in terms of the deformation parameters q1 and q2 . The usual boson gas results can be recovered in the limit q1 = q2 = 1. 2002 Published by Elsevier Science B.V.
1. Introduction In the past decade, quantum groups and their associated algebras [1–5], which are generalizations of ordinary Lie groups and Lie algebras with particular deformation parameters have produced many developments in the framework of statistical mechanics as well as other areas of physics and mathematics. For instance, possible relations between quantum groups and generalized statistical mechanics have been investigated [6–8]. Meanwhile, a great deal of effort has been spent in order to obtain some physical interpretation on the deformation parameter q by considering a canonical ensemble of q-oscillators [9,10], which are q-deformations of the bosonic harmonic oscillator algebra. Thermodynamical and statistical properties of a gas of deformed bosons as well as fermions have been extensively investigated [11–15]. Interestingly, it was shown in Ref. [16] that the high temperature properties
E-mail address: [email protected] (A. Algin).
of the bosonic and fermionic gases having the symmetry of the quantum group SU q (N) depend radically on the deformation parameter q. Although two-parameter extensions of such investigations have been studied in the literature [17–19], the complete history of the generalization of the conventional physical quantities of the bosons and fermions coming out by deformations is still under active consideration. Inspired by the work of Ubriaco [16], the present Letter is mainly aimed to give a two parameter generalization related to the thermodynamical properties of a quantum group boson gas described by the symmetry of the quantum group SU q (2), where q = q1 /q2 in the high temperature (T ) limit, namely for z = eβµ 1, where β = 1/kB T , kB is Boltzmann’s constant, and µ is the chemical potential. This Letter is organized as follows. We first review some basic definitions and properties concerning the quantum group SU q (N) bosons and particularly the SU q (2) one. Then we introduce our model described by a Hamiltonian in terms of the SU q1 /q2 (2) generators. This leads to a calculation of the thermodynamical properties of the model such as the average num-
0375-9601/02/$ – see front matter 2002 Published by Elsevier Science B.V. PII: S 0 3 7 5 - 9 6 0 1 ( 0 1 ) 0 0 7 9 1 - 5
252
A. Algin / Physics Letters A 292 (2002) 251–255
ber of particles and the pressure obtained via the grand partition function. Finally, we present our conclusions.
it can be found from Eq. (7) [22] that ab = qba, cd = qdc,
2. Review on the quantum group SUq (N ) bosons The conventional boson oscillators satisfy the following commutation relations: φi φj+ − φj+ φi = δij , φi φj − φj φi = 0, φi+ φi = Ni ,
i, j = 1, 2, . . . , N,
(1)
φi+
are the bosonic annihilation and crewhere φi and ation operators, respectively, and Ni is the boson number operator. These relations are obviously invariant under the action of the undeformed group SU(N). The quantum group analogues of the above equations are defined by the commutation relations [5,16] Φj Φ¯ i = δij + qRkij l Φ¯ l Φk , Φl Φk = q
−1
Rj ikl Φj Φi ,
i, j = 1, 2, . . . , N,
where the N 2 × N 2 matrix R [20] is Rj ikl = δj k δil 1 + (q − 1)δij + q − q −1 δik δj l θ (j − i),
(2) (3)
(4)
and the function θ (j − i) is defined by 1 if j > i, θ j −i = (5) 0 otherwise. Eqs. (2) and (3) are covariant under the linear transformation
Φ = T Φ,
(6)
where the matrix T ∈ SUq (N). The SU q (N) transformation matrix T and the R-matrix satisfy the relations [21] RT1 T2 = T2 T1 R,
(7)
R12 R13 R23 = R23 R13 R12 ,
(8)
where T1 = T ⊗ 1, T2 = 1 ⊗ T ∈ V ⊗ V and (R23 )ij k,i j k = δii Rj k,j k ∈ V ⊗ V ⊗ V . For a unitary quantum group matrix a b , T= c d
bc = cb,
ac = qca, bd = qdb, ad − da = q − q −1 bc,
Detq (T ) = ad − qbc = 1,
(9)
with the unitary conditions a¯ = d, b¯ = qc, q ∈ R. For N = 2, the commutation relations generating the quantum group SU q (2) bosons can be obtained [5] by Eqs. (2) and (3) as follows: Φ1 Φ¯ 1 − q 2 Φ¯ 1 Φ1 = 1, Φ2 Φ¯ 2 − q 2 Φ¯ 2 Φ2 = 1 + q 2 − 1 Φ¯ 1 Φ1 ,
(10)
Φ1 Φ2 = qΦ2 Φ1 , Φ1 Φ¯ 2 = q Φ¯ 2 Φ1 ,
(12)
(11) (13)
which they become the usual boson algebra in the limit q = 1. However, we exploit a different quantum group SU q (2) bosons, where q = q1 /q2 . The notion of these deformed oscillators was first introduced in Ref. [23] during a realization of the most general quantum group invariant oscillator algebra. The commutation relations generating the quantum group SU q1 /q2 (2) bosons are defined by Φ1 Φ¯ 1 − q12 Φ¯ 1 Φ1 = q22N , Φ2 Φ¯ 2 − q12 Φ¯ 2 Φ2 = q22N + q12 − q22 Φ¯ 1 Φ1 , Φ1 Φ2 = q1 q2−1 Φ2 Φ1 , Φ¯ 1 Φ¯ 2 = q2 q1−1 Φ¯ 2 Φ¯ 1 , Φ1 Φ¯ 2 = q1 q2 Φ¯ 2 Φ1 ,
(14) (15) (16) (17) (18)
where N is the total boson number operator and q1 , q2 ∈ R. Hereafter, we consider 0 < q1 < ∞ and 0 < q2 < ∞. The limit q2 = 1 corresponds to the usual quantum group SU q1 (2) as defined by Eqs. (10)–(13). Furthermore, a differential calculus on the quantum hyperplane corresponding to the quantum group SU q (2) was discussed by Wess and Zumino [20] and a two-parameter generalization of the Wess and Zumino differential calculus was constructed in [24].
3. SUq1 /q2 (2) boson model Let us consider the following Hamiltonian in terms of SU q1 /q2 (2) generators for two kinds of bosons with
A. Algin / Physics Letters A 292 (2002) 251–255
the same energy: εk (M1,k + M2,k ), H=
4. Thermodynamic properties (19)
k
where the operators M1,k and M2,k are defined by M1,k = Φ¯ 1,k Φ1,k ,
M2,k = Φ¯ 2,k Φ2,k ,
(20)
εk is the spectrum of energy, k = 0, 1, 2, . . . , and obviously [Φ¯ i,k , Φj,k ] = 0 for k = k . These operators also satisfy the following relations for a given k: M2 Φ1 − q1−2 Φ1 M2 = 0, M1 Φ2 − q2−2 Φ2 M1
= 0.
(21)
The normalized states of the above Hamiltonian can be built by applying successively the operators Φ¯ on the vacuum state |0, 0 for a given k as 1 Φ¯ m1 Φ¯ m2 |0, 0, |m1 , m2 = √ [m1 ]![m2 ]! 1 2
(22)
where the Fibonacci basic number [m], which is a generalization of the usual q-numbers is in the form [23] [m] =
q12m − q22m q12 − q22
.
253
(23)
In this section, we investigate the thermodynamic properties of the bosonic system described by a Hamiltonian in Eq. (26). The grand partition function Z is written ¯
¯
Z = Tr e−βεk (Φ1,k Φ1,k +Φ2,k Φ2,k ) eβµ(N1,k +N2,k ) ,
(27)
where the trace is taken over the states in Eq. (22). From Eqs. (23)–(26), the grand partition function becomes Z=
∞ ∞
e−βεk [n1 +n2 ] eβµ(n1 +n2 ) ,
(28)
k n1 =0 n2 =0
which can be rewritten as Z=
∞ (n + 1)e−βεk [n] zn ,
(29)
k n=0
where z = eβµ is the fugacity. Since we are dealing with the high temperature case, namely the limit z 1, for the three-dimensional momentum space the grand partition function in Eq. (29) can be calculated in the first few terms as follows: ∞
In order to calculate the thermodynamical properties of the Hamiltonian in Eq. (19), we propose a new representation. By virtue of such a representation, we are able to express the SU q1 /q2 (2) bosons in terms of + the ordinary boson operators φi,k and φi,k satisfying Eqs. (1) for a given k as follows: −1 Φ1 = φ1+ [N1 ]q2N2 , (24) Φ¯ 1 = φ1+ q2N2 , + −1 N1 N + 1 [N2 ]q1 , Φ¯ 2 = φ2 q1 . Φ2 = φ2 (25)
4πV ln Z = 3 h
By using Eqs. (23)–(25), we rewrite the above Hamiltonian as H= (26) εk [N1,k + N2,k ],
which leads to an explicit relation by evaluating the integral, √ π 2m 3/2 4πV z ln Z = 3 2 β h √ 2m 3/2 2 + π (31) z ξ(q1 , q2 ) + · · · , β where the function ξ q1 , q2 is given by
1 1 3 ξ(q1 , q2 ) = (32) − √ . 4 (q12 + q22 )3/2 2
k + where Ni,k = φi,k φi,k and the bracket [x] is defined by Eq. (23). Note that when compared with the original Hamiltonian in Eq. (19), this representation leads to an interacting Hamiltonian. As a result, such a model may be interpreted in a way that φ1 and φ2 correspond to two different kinds of bosonic particles which interact with each other but do not interact among themselves.
p2 dp 0
z2 −βεk [2] 6e − 4e−2βεk × 2e−βεk z + 2! 3 z + 24e−βεk [3] − 18e−βεk [2] e−βεk 3!
−3βεk + ··· , + 16e
(30)
254
A. Algin / Physics Letters A 292 (2002) 251–255
The average number of particles N can then be calculated by 1 ∂ ln Z N = (33) , β ∂µ T ,V which leads to √ 4πV π 2m 3/2 N = 3 z 2 β h √ 2m 3/2 2 +2 π z ξ(q1 , q2 ) + · · · . β
(34)
By reverting this equation, the fugacity can be found as 3/2 1 h2 N z≈ 2 2πmkT V 3 2 N 2 h − ξ(q1 , q2 ) (35) . 2πmkT V The pressure can also be calculated by 1 ∂ ln Z P= , β ∂V T ,µ which gives √ 4π π 2m 3/2 P= 3 z h β 2 β √ 2m 3/2 2 + π z ξ(q1 , q2 ) + · · · . β
(36)
(37)
By using the above equations, the equation of state is derived depending on the deformation parameters q1 and q2 as 3/2 h2 N P V = kT N 1 − ξ q1 , q2 2πmkT V +··· .
(38)
On the other hand, the equation of state can also be calculated by means of similar calculations in the two-dimensional space for the SU q1 /q2 (2) boson gas as follows: h2 N + ··· , P A = kT N 1 − ζ (q1 , q2 ) 2πmkT A (39)
Fig. 1. The coefficient ζ (q1 , q2 ) for the interval 0 < q12 + q22 6. The line at q12 + q22 = 3.0 divides the region between ζ (q1 , q2 ) > 0 and ζ (q1 , q2 ) < 0 which corresponds to bosonic and fermion-like behavior, respectively. Also, in the limit q1 or q2 → ∞ the lowest value of the coefficient ζ (q1 , q2 ) is −0.25.
where A is the surface confining the bosonic system and the function ζ(q1 , q2 ) is
1 3 − 1 , ζ(q1 , q2 ) = (40) 4 (q12 + q22 ) which represents a numerical factor in the second virial coefficient through Eq. (39) for the two-parameter deformed boson gas. We should mention for all equations above that the limit q2 = 1 gives the same results derived in Ref. [16] for the boson gas. We now wish to discuss some important limiting cases considered by means of the coefficients ξ(q1 , q2 ) and ζ(q1 , q2 ) in Eqs. (32) and (40). Obviously, for both two and three-dimensional system, the sign of the second virial coefficient given by either Eq. (38) or Eq. (39) depends on the values of the deformation parameters q1 and q2 . For the case of three-dimensional system, at q1 = q2 = 1 the coefficient ξ(1, 1) = 2−7/2 , which corresponds to the numerical factor in the second virial coefficient for a free boson gas with two different kinds of bosonic particles. The ideal gas ξ(q1 , q2 ) = 0 and the free fermion ξ(q1 , q2 ) = −2−7/2 [25] cases are reached at q12 + q22 ≈ 2.62 and q12 + q22 ≈ 4.16, respectively. On the other hand, for the two-dimensional system, at the values q1 = q2 = 1 and q12 + q22 = 6.0 the system behaves as a free boson
A. Algin / Physics Letters A 292 (2002) 251–255
ζ (1, 1) = 2−3 and fermion gas ζ (q1 , q2 ) = −2−3 [25], respectively. The second virial coefficient in Eq. (39) vanishes at q12 + q22 = 3.0 which corresponds to the ideal gas case. In a simple manner, Fig. 1 shows a graph of the coefficient ζ (q1 , q2 ) as a function of a sum of the model parameters q12 and q22 in the twodimensional space. When we compare with the results of the oneparameter deformed SU q (2) boson model [16], the most interesting point is that the present two-parameter deformed boson model has an infinite number of possibilities in getting the second virial coefficient for the fermion gas. Since the two-parameter deformed boson gas behaves as a fermion gas [25] at the value q12 + q22 = 6.0, we may remark that willingness of fermion-like behavior of the present two-parameter boson model increases too much for a wide range of the parameters q1 and q2 . We end up this section by briefly emphasizing that the parameters q1 and q2 serve as interpolating objects between repulsive and attractive behaviors as is shown in Fig. 1.
5. Conclusion In this Letter, we have investigated the thermodynamical properties of a two-parameter deformed quantum group boson gas SU q1 /q2 (2) for the high temperature case. For instance, the average number of particles and the pressure are derived and subsequently the equation of state is obtained. We have also studied the simplest quantum group SU q (2) and discussed the defining commutation relations for a two-parameter deformed SU q1 /q2 (2)-covariant boson algebra. The model presented here generalizes the one-parameter deformed quantum group boson gas results developed recently in [16]. However, the limit q1 = q2 = 1 gives the ordinary boson gas properties. As a final remark, the low temperature behavior of the model is another face of the work in which the
255
main focus will be on the study of the Bose–Einstein condensation. Acknowledgements This work was started at Feza Gürsey Institute, Istanbul, Turkey. I would like to thank the Institute and its director Prof. I.H. Duru. References [1] V.G. Drinfeld, Proc. ICM 1 (1986) 798. [2] M. Jimbo, Lett. Math. Phys. 11 (1986) 247. [3] L.D. Faddeev, N.Y. Reshetikhin, L.A. Takhtajan, Quantization of Lie groups and Lie algebras, Preprint LOMI (1987). [4] S.L. Woronowicz, Comm. Math. Phys. 111 (1987) 613. [5] W. Pusz, S.L. Woronowicz, Rep. Math. Phys. 27 (1989) 231. [6] C. Tsallis, Phys. Lett. A 195 (1994) 329. [7] S. Abe, Phys. Lett. A 224 (1997) 326. [8] M. Arik, J. Kornfilt, A. Yildiz, Phys. Lett. A 235 (1997) 318. [9] M.A. Martín-Delgado, J. Phys. A 24 (1991) L1285. [10] P.V. Neskovic, B.V. Urosevic, Int. J. Mod. Phys. A 7 (1992) 3379. [11] C.R. Lee, J.P. Yu, Phys. Lett. A 150 (1990) 63. [12] J.A. Tuszynski, J.L. Rubin, J. Meyer, M. Kibler, Phys. Lett. A 175 (1993) 173. [13] M. Chaichian, R.G. Felipe, C. Montonen, J. Phys. A 26 (1993) 4017. [14] M.R. Ubriaco, Phys. Lett. A 219 (1996) 205. [15] M.R. Ubriaco, Mod. Phys. Lett. A 11 (1996) 2325. [16] M.R. Ubriaco, Phys. Rev. E 55 (1997) 291. [17] R. Chakrabarti, R. Jagannathan, J. Phys. A 24 (1991) L711. [18] M. Daoud, M. Kibler, Phys. Lett. A 206 (1995) 13. [19] R.S. Gong, Phys. Lett. A 199 (1995) 81. [20] J. Wess, B. Zumino, Nucl. Phys. B (Proc. Suppl.) 18 (1990) 302. [21] L.A. Takhtajan, Adv. Stud. Pure Math. 19 (1989) 1. [22] S. Vokos, B. Zumino, J. Wess, Symmetry in Nature, Scuola Normale Superiore, Pisa, 1989. [23] M. Arik, E. Demircan, T. Turgut, L. Ekinci, M. Mungan, Z. Phys. C 55 (1992) 89. [24] A. Schirrmacher, J. Wess, B. Zumino, Z. Phys. C 49 (1991) 317. [25] A. Algin, M. Arik, A.S. Arikan, Phys. Rev. E, submitted.