Unified statistical thermodynamics of quantum gases trapped under generic power law potential in d dimension and equivalence in d=1
aa r X i v : . [ c ond - m a t . qu a n t - g a s ] J un Unified statistical thermodynamics of quantum gases trapped under genericpower law potential in d dimension and equivalence in d = 1 Mir Mehedi FarukDepartment of Theoretical Physics, University of Dhaka, Dhaka-1000Theoretical Physics, Blackett Laboratory, Imperial College, London SW7 2AZ, United KingdomEmail: [email protected], [email protected] 10, 2018
Abstract
A unified description for the Bose and Fermi gases trapped in an external generic power law potential U = P di =1 c i | x i a i | n i is presented using the grandpotential of the system in d dimensional space. The thermodynamic quan-tities of the quantum gases are derived from the grand potential. An equivalence between the trapped Bose andFermi gases is constructed in one dimension ( d = 1) using the Landen relation. It is also found that the establishedequivalence between the ideal free Bose and Fermi gases in d = 2 (M. H. Lee, Phys. Rev. E 55, 1518 (1997)) is lostwhen external potential is applied. The two types of quantum gases manifest different thermodynamic behaviour due to inherent difference of their statisticaldistribution[1, 2]. Fermi gas, which obeys Pauli exclusion principle exhibit distinct characteristic such as zero point energyand pressure[1, 2] whether Bose gas condensates[1, 2], not obeying this principle. The thermodynamic properties of theBose and Fermi gases are determined by Bose and Fermi function[1, 3] which have different mathematical structures. Butan unified formulation for quantum gases was presented recently and Lee[4, 5, 6] established a remarkable equivalencebetween ideal free Bose and Fermi gases in d = 2. The equivalence is based on a certain invariance of the polylogarithmsunder Euler transformation[7] of the fugacities. After the inspiring work of May[8] considerable attactions are drawn tostudy further the equivalence between quantum gases. Point to note, plenty of study are made to investigate the thermo-dynamic properties of quantum gases under trapping potential[9, 10, 11, 12] after it was possible to create Bose-Einsteincondensate in magnetically trapped Alkali gases[13, 14, 15]. It was demonstrated in recent papers that, trapping potentialcan change the characteristics of quantum gases. For instance, although there is no Bose condensation for ideal Bose gasin d < d < d dimensional space. The thermodynamic quantities are then derived from the grand potential. From the general expres-sions of the calculated thermodynamic quantities we have investigated closely the case with d = 1 and n = 2 (harmonicpotential) and found an equivalence can be obtained between the thermodynamic quantities of Bose and Fermi gases. It isalso seen, the established equivalence for ideal free quantum gases in d = 2 disappears when a external potential is applied.The report is organized in the following way. The grand potential of quantum gases under generic power law poten-tial is calculated in section 2. In section 3 we have presented the thermodynamic quatities in an unified way for bothtypes of quantum gases. The useful landen relations are explored in section 4. And section 5 is devoted to explore theequivalence in d = 1 with harmonically trapped quantum gases. A discussion on the equivalence of free and trappedquantum gases is presented in section 6. The report is concluded in section 7.1 Grand potential
Considering an ideal quantum system trapped in a generic power law potential in d dimensional space with a singleparticle Hamiltonian, ǫ ( p, x i ) = bp l + d X i =1 c i | x i a i | n i (1)Where, b, l, a i , c i , n i are all postive constants, p is the momentum and x i is the i th component of coordinate of a particle.Here, c i , a i , n i determines the depth and confinement power of the potential and l being the kinematic parameter. Now,the well known formula of density of states [9, 16], ρ ( ǫ ) = Z Z d d rd d p (2 π ℏ ) d δ ( ǫ − ǫ ( p, r )) (2)So, from the above equation density of states is[9, 16], ρ ( ǫ ) = B Γ( dl + 1)Γ( χ ) ǫ χ − (3)where, B = V d C d h d a d/l d Y i =1 Γ( n i + 1) c ni i (4)Here, C d = π d Γ( d/ , V d = 2 d Q di =1 a i is the volume of an d -dimensional rectangular whose i -th side has length 2 a i .Γ( l ) = R ∞ dxx l − e − x is the gamma function and χ = dl + P di =1 1 n i .The grand potential of quantum gases can be written as[1], q = 1 a X ǫ ln (1 + azexp ( − βǫ )) (5) β = kT , where k being the Boltzmann Constant and z = exp( βµ ) is the fugacity, where µ being the chemical potential.a is equal to -1 for Fermi system and +1 for Bose system. In experiments with trapped gases, thermal energies far exceedthe level spacing[14]. So, using the Thomas-Fermi semiclassical approximation[27] and re-writing the previous equation, q = q + 1 a Z ∞ ln(1 + az exp( − βǫ )) ρ ( ǫ ) dǫ (6)Here, q = a ln(1 + az ). Now finally the grand potential stands as, q = (cid:26) q + B Γ( dl + 1)( kT ) χ f χ +1 ( z ) , Fermi system q + B Γ( dl + 1)( kT ) χ g χ +1 ( z ) , Bose system (7)Here, g l ( z ) and f l ( z ) are Bose and Fermi function respectively. Defined as g l ( z ) = Z ∞ x l − z − e x − ∞ X j =1 z l j l (8) f l ( z ) = Z ∞ x l − z − e x + 1 = ∞ X j =1 ( − j z l j l (9)Now, Bose and Fermi functions can be written in terms of Polylogarithmic functions, Li q ( t ) = g q ( t ) (10) Li q ( − t ) = − f q ( t ) (11)2here, Li q ( m ) is the polylog of q and m . If q ≥ Li q ( m ) is analytic everywhere. It is a real valued function if m ∈ R and −∞ < m <
1. A useful integral representation of polylog is Li q ( m ) = 1Γ( q ) Z m [ln( mη )] q − dη − η , (12)for Re ( m ) <
1. To write the grand potential compactly, defining a quantity σ as, σ = (cid:26) − z , Fermi system z ,
Bose system (13)So, re writing the grand potential, q = q + sgn ( σ ) B Γ( dl + 1)( kT ) χ Li χ +1 ( σ ) (14) The number of particles N can be obtained, N = z ( ∂q∂z ) β,V ⇒ N − N = N e = sgn ( σ ) V ′ d λ ′ χ Li χ ( σ ) (15) ⇒ ρ = N e V ′ d = sgn ( σ ) 1 λ ′ d Li χ ( σ ) (16)Where V ′ d and λ ′ are defined as [16] V ′ d = V d d Y i =1 ( kTc i ) /n i Γ( 1 n i + 1) (17) λ ′ = hb l π ( kT ) l [ d/ d/l + 1 ] /d (18)It is noteworthy, lim n i →∞ V ′ d = V d (19)lim n i →∞ χ = dl (20)lim l → ,b → m λ ′ = λ = h (2 πmkT ) / (21)Now, the other thermodynamic quantities in case of trapped system can be calculated from grand potential as below, U = − ( ∂q∂β ) z,V ′ d = N kT χ Li χ +1 ( σ ) Li χ ( σ ) (22) S = kT ( ∂q∂T ) z,V ′ d − N k ln z + kq = N k ( χ + 1) Li χ +1 ( σ ) Li χ ( σ ) − ln | σ | (23) P = 1 β ( ∂q∂V ′ d ) β,z = N kT V ′ d Li χ +1 ( σ ) Li χ ( σ ) (24) C V = T ( ∂S∂T ) N,V ′ d = N k [ χ ( χ + 1) Li χ +1 ( σ ) Li χ ( σ ) − χ Li χ ( σ ) Li χ − ( σ ) ] (25) κ T = − V ′ d ( ∂V ′ d ∂P ′ ) N,T = V ′ d N kT Li χ − ( σ ) Li χ ( σ ) (26)3he above expressions compactly represent the thermodynamic quantities related to trapped Bose[16, 9] and Fermigas[28, 29] In case of free system (all n i −→ ∞ ) the above quantities reduce to, ρ = sgn ( σ ) 1 λ d Li d ( σ ) (27) U = N kT dl Li dl +1 ( σ ) Li dl ( σ ) (28) S = N k ( dl + 1) Li dl +1 ( σ ) Li dl ( σ ) − log | σ | (29) P = N kT V d Li dl +1 ( σ ) Li dl ( σ ) (30) C V = N k [ dl ( dl + 1) Li dl +1 ( σ ) Li dl ( σ ) − ( dl ) Li dl ( σ ) Li dl − ( σ ) ] (31) κ T = V d N kT Li dl − ( σ ) Li dl ( σ ) (32)So, choosing l = 2 in case of non-relativistic quantum gas, the Eq. (27)-(32) reduces to those in Ref.[3] for arbitrarydimension. And with d = 3, they reproduce the thermodynamic quantities for free Bose and Fermi gas[1, 2]. The unified formulation shows that the thermodynamic quantities are described by the structural properties of polylogs.Landen[7] found relation between monolog and dilog, which is the key to make the equivalence between ideal free quantumgases[4] as well as trapped gases. If x is a real number and x < x , such that, x = − x − x (33)then one finds, Li ( x ) = − Li ( x ) Li ( x ) + 1 (34) Li ( x ) = − Li ( x ) (35) Li ( x ) = − Li ( x ) −
12 [ Li ( x )] (36)The proof of the above relations are included in Appendix of Ref. [4]. These relations indicate Euler transformation[4]of x to x . d = 1 for trapped gas Note, in both case of free and trapped system the thermodynamic quantities are described by polylogs Li m ( z ). Now thepolylogs are related to each other by landen relations, and the respective variables are related to each other by Eulertransformation. In free system the polylogs describing the thermodynamic system are functions of dimension, while inof trapped system the polylogs describing the thermodynamic system are function of dimension, fugacity and powerlaw exponents. In trapped system the dependence of polylogs on dimension and power law exponents are described by χ = dl + P di =1 1 n i .As l = 2, in case of nonrelativistic massive Boson and choosing d = 1, n = 2 (harmonic potential), χ = 12 + 12 = 1 (37)If the densities are made the same, turning our attention towards, density ρ , with χ = 1, we get from Eq. (27) ρλ = Li ( z B ) = − Li ( − z F ) (38)4here, z B and z F denotes fugacity of Bose and Fermi gas respectively. So, according to Eq. (33) they are related to eachother by Euler transformation. So, we can write following relation z F = z B − z B (39)So, the fugacities are related to each other by Euler transformation, if we put z B = x and z F = − x . Then, we caneasily use the thermodynamic quantities to establish the equivalence. First turning our attention towards internal energy U ( z B ) with d = 1 and n = 2, U ( z B ) = N kT Li ( z B ) Li ( z B ) = N kT Li ( x ) Li ( x ) = N kT − Li ( x ) − ( Li ( x )) − Li ( x ) = N kT [ Li ( x ) Li ( x ) + 12 Li ( x )]= U ( z F ) + N kT Li ( − z F )= U ( z F ) + N kT ρλ
Now point to note, ρ = N e V ′ d and V ′ d ∝ √ T . Also λ ∝ √ T So, obviously the second term is temperature independent. As itturns out, the second term exactly corresponds to ground state energy [28] just in the case of ideal free quantum gases[4].Hence, it can be concluded if the two reduced densities are the same, the fugacities are related by Euler transformationand as a result internal energies of Bose and Fermi gases only differ by the ground state energy of the Fermi gas only.So, denoting ground state energy by U , we can rewrite, U ( z B ) = U ( z F ) − U (40)Since, pressure and energy are related by P V ′ d = Eχ , from the help of Eq. (40), one can get P ( z B ) = P ( z F ) − P (41)Where, P denotes ground state pressure of Fermi gas[16]. Now turning our attention towards entropy, S ( z B ) = N k [2 Li ( z B ) Li ( z B ) − log ( z B )] = N k [2 Li ( x ) Li ( x ) − log ( x )] = N k [2 Li ( x ) + [ Li ( x )] Li ( x ) − log ( − x x )]= N k [2 Li ( x ) Li ( x ) + Li ( x ) − log ( − x ) + log (1 + x ) = N k [2 Li ( x ) Li ( x ) − log ( − x )]= N k [2 Li ( − z F ) Li ( − z F ) − log ( z F )] = S ( z F ) (42)Here, we have used the identity log (1 + x ) = − Li ( x ). Also, it is clear that entropy remain exactly same for two typesof quantum gases in this case. Now, from the equation of specific heat, C V ( z B ) = N k [2 Li ( z B ) Li ( z B ) − Li ( z B ) Li ( z B ) ]= N k [2 Li ( x ) Li ( x ) − Li ( x ) Li ( x ) ] = N k [2 Li ( x ) Li ( x ) − Li ( x ) Li ( x ) ]= N k [2 Li ( x ) + [ Li ( x )] Li ( x ) − Li ( x )[1 + Li ( x )] Li ( x ) ]= N k [2 Li ( x ) Li ( x ) − Li ( x ) Li ( x ) ] = N k [2 Li ( − z F ) Li ( − z F ) − Li ( − z F ) Li ( − z F ) ] = C V ( z F ) (43)This type of result is previously found by May[8] for free quantum gases in two dimensional space. In case of theisothermal compressibilty, κ T ( z B ) = V ′ d N kT Li ( z B ) Li ( z B ) = (1 + z F ) κ T ( z F ) (44)So, the isothermal compressibilty are not equivalent at all temperatures. If z F −→ ∞ (i.e z B −→ κ T ( z F ) ∝ log ( z F ) but κ T ( z B ) ∝ z F log ( z F ) . The latter diverges while the former vanishes. But if z F −→ Q . Rewriting Q ,log Q ( z B ) = V ′ d λ Li ( x B ) = V ′ d λ Li ( x ) = V ′ d λ ( − Li ( x ) −
12 [ Li ( x )] )= V ′ d λ ( − Li ( − z F ) −
12 [ Li ( z F )] )= log Q ( z F ) − V ′ d ρ λ (45)With careful inspection it can be seen from Eq. (16) - (18), the second term ib Eq. (45) is linear in β . So, the grandpartition function of the two systems are related to each other by a term linear in β . As all the thermodynamic quantitiesare basically dervied from grand potential, thus we are able to make such connection for all the thermodynamic quantities.So, when we take first derivative of grand potential with respect to β , the obtained thermodynamic quantity internalenergy of Bose and Fermi system only differ by a constant ( i.e. the ground state energy) which is independent of β . Andwhen we take the second derivative of grand potential with respect to β , the derived thermodynamic quantity specificheat are equal to each other. In this paper, we have seen once again if the fugacities of Bose and Fermi gas are related by Euler transformationan equivalence relation can be establised between the two types of quantum gases. One can check the status of theequivalence relation in d = 2 [4] for trapped quantum gases. Now re-writing the equation of reduced density (Eq. 27)with l = 2, ρλ = Li d + P i ni ( z B ) = − Li d + P i ni ( − z F ) (46)Now, choosing d = 2, the above expression reduces to, ρλ = Li P i =1 , ni ( z B ) = − Li P i =1 , ni ( − z F )From Eq. (35) one can see the Euler transformation type relation between fugacities are possible only for monologs.So, as it stands from the above equation the Euler transformation type relation between fugacities are possible if andonly if P i =1 , n i = 0. Now as, n , n > n −→ ∞ and n −→ ∞ , which isbasically the condition for free system [9, 12, 16, 28]. So, the equivalence relation between the quantum gases is possiblein two dimensional space only for the free system. This phenomenon is due to the fact, that trapped Bose gas actuallycondensates in d = 2 with any trapping potential or in more general, BEC is possible if and only if χ > d = 1, Eq. (46) becomes, ρλ = Li + n ( z B ) = − Li + n ( − z F ) (47)Again Eq. (35) suggests Euler transformation type relation between fugacities are possible in d = 1 if and only if quantumgases are trapped in harmonic potential ( n = 2). From the unified statistical thermodynamics of quantum gases trapped under generic power law potential in d dimension,a case is shown with d = 1 where, Bose and Fermi gases can be treated as equivalent. This is possible only when thequantum gas is trapped in harmonic potential. It will be interesting to check the effect of interaction on this equivalenceas well as to do the whole calculation with relativistic hamiltonian. I would like to thank Fathema Farjana and Dr. Jens Roder for their effort to help me present the manuscript.6 eferences [1] R. K. Pathria,
Statistical Mechanics , Elsevier, 2004.[2] K. Huang,