Density shift of Bose gas due to the Casimir effect and mean field potential
aa r X i v : . [ c ond - m a t . s t a t - m ec h ] M a y Density shift of Bose gas due to the Casimir effect and mean field potential
G. M. Bhuiyan a) Department of Theoretical Physics, University of Dhaka, Dhaka-1000,Bangladesh
The shift of density of Bose gas due to the mean field potential (MFP) and theCasimir effect is systematically investigated in the d -dimensional configurationspace from the point of thermodynamic consideration. We show that, for d = 3,the shift of density arises completely due to the Casimir effect and, the MFPremains totally ineffective regardless of the state, condensate or non-condensate.But for dimension d > n c / n c ≈ -∆ T c / T c .It is important to note that, the MFP causes a shift of density for d > d -dimension, Bose-Einstein condensation. a) [email protected] . INTRODUCTION Since the demonstration by Einstein, a great many attempts have been made tounderstand the Bos-Einstein condensation(BEC) of the ideal Bose gas. Most of themare focused on the bulk properties including condensation through the thermodynamicroute, occupation of the states and their fluctuations[1-5]. The effect of interparticleinteraction on the critical density and critical temperature of the Bose-Einstein con-densation has attracted many theoretical efforts after the pioneering work by Lee andYang[6], but it gets enhanced momentum after the observation of BEC in undercoldatomic gases[7-9] in 1995.The Casimir force due to the fluctuation of density of massless and massive bosonsplaced between two parallel slabs is well studied[10-25]. But we are not aware of anystudy focusing on the density shifts or critical temperature shift of Bose gas due to theCasimir interaction. In the present work, we study the effect of the Casimir interactionon the bulk density shift across the critical point of the Bose-Eienstein phase transition.We also show an interesting result that the mean field potential (MFP) does not affect inany way the density of Bose gas in three dimension, but does affect at higher dimensions.It is worth noting that, in some previous works[26,27] existing in literature the effect ofdimensionality on BEC has been studied, but how the shift of density or critical densityand crtical temperature are affected by the dimensionality along with the MFP is yetto be explored.The critical density or critical temperature shift in BEC due to interactions or trapshas been extensively explored by many authors [27-35]. But results found for theamount of shift vary within certain extent, and become even contradictory in the signof the shift. For example, in [28,30] it is predicted that the sign of ∆ n c is positive forrepulsive interaction in dilute homogeneous Bose gas, whereas other works includingmeasurement[29] and simulation studies[30,32] predict a negative sign. The correspond-ing sign for ∆ T c is just opposite, that is negative in the former cases and positive inlatter ones, because the two shifts are related as ∆ T c / T c → - ∆ n c / n c . Interestingly, thework by Giorgini et al. [25] shows that the density of excited bosons increases due to2epulsive interaction in an isotropic harmonic trap and, consequently, the critical tem-perature goes down unlike homogeneous Bose gas. In the present study, we consider anideal Bose gas system in a mean field potential and placed within two parallel walls. Itis shown in an experiment [36] that the influence of interaction between particles on theBose-Einstein condensation transition temperature is only of few percent. This resultfairly justifies the validity of our approach for a qualitative description of density shiftof Bose gas.The magnitude of the Casimir effect depends on the boundary conditions (bc) to beused. We note here that, the sums for the periodic, Neumann and Dirichlet boundaryconditions starts at −∞ , −∞ to + ∞ . On the other hand, the operator relationsfor Neumann and Dirichlet bc are P ∞ → ( P + ∞−∞ +1) and P ∞ → ( P + ∞−∞ − d is made by using the one particle density of states, insection 2. The general expression for the density shift is also presented and analysedfor different dimensions in the same section. Section 3 is devoted to present the resultsand discussion. This paper is concluded in section 4. II. THEORETICAL EVALUATIONA. The grand potential energy of the Bose gas in MFP and in d -dimensional configuration space The repulsive pair interaction between a pair of identical massive bosons can bedescribed within the mean field theory as a/V , where a is a positive constant and V is3he volume containing the Bose gas. Let us now look at a particular boson moving inthe N boson system in the mean field due to the rest of ( N −
1) particles. Obviously,the average potential energy experienced by the tagged boson is aV ( N − ≈ aV N . Theone boson Hamiltonian will, therefore, be H = p m + an (1)where p is the momentum of the boson and m is its mass; n denotes the number densityof the bosons ( N/V ).Let us assume that, the bosons are enclosed in a d -dimensional volume V ( d ) = L d , L being the edge of the rectangular box and L → ∞ . The spacing between energy levelswill therefore be very small, so the summation over the states can be approximatelyreplaced by integration. Therefore, the bulk density of states in the phase space withspatial d -dimension is γ ( ǫ ) = 1(2 π ~ ) d Z d d r Z d d p δ ( ǫ − p m − a n )= V ( d ) ( m π ~ ) d d ) ( ǫ − a n ) d − (2) ǫ in equation (2) is the energy eigen value of the boson. The density of states in the( d -1)-dimensional surface is γ ( ǫ ) = V ( d − ( m π ~ ) d − d − ) ( ǫ − a n ) d − . (3)We now assume that the Bose gas in the mean field potential is placed within twoparallel slabs such that the d -dimensional volume V ( d ) = L d − D , where D is the sep-aration distance of the slabs. As D is finite, the spacing of the energy levels will belarge along the d − th direction. So, the summation over energy levels along the d − thdirection cannot be approximated by integration. The grand potential energy, using4he Neumann boundary condition ( k d = πD l ; l = 0 , , , ..... ), may be expressed as φ D ( T, µ ) k B T = ∞ X l =0 Z ∞ ln (1 − z exp( − βπ ~ mD l ) exp( − β ǫ )) γ ( ǫ ) dǫ = − V ( d − λ d − ∞ X l =0 ∞ X r =1 z r r d +12 × exp( − πλ l r D ) exp( − βanr ) ( d − ) Γ( d − , − βanr ) ) (4)where z = exp( βµ ), β is inverse temperature times Boltzmann constant, µ the chemicalpotential, Γ( d − , − βanr ) the lower incomplete gamma function and λ = h/ √ πmk B T , h being the Planck’s constant. Now separating the l = 0 term and using the Jacobiidentity relation as in [18] we have ∞ X l =0 e − παl = 1 + ∞ X l =1 e − παl = ( 12 √ α + 12 ) + 1 √ α ∞ X l =1 e − π l /α (5)Substituting equation(5) into (4) one can write, φ D ( T, µ ) k B T = − V ( d − λ d − "(cid:18) Dλ (cid:19) ∞ X r =1 z r r d +22 + 12 ∞ X r =1 z r r d +12 + 2 (cid:18) Dλ (cid:19) ∞ X r =1 ∞ X l =1 z r r d +22 × e − π ( Dλ ) l /r i e − βanr ( d − ) Γ( d − , − βanr ) ) (6)It appears from equation (6) that, the first and the third terms on right hand side arerelated to the bulk energy density, whereas second term provides the surface energydensity. According to Ref. 18 and 19 the third term is also recognized as the Casimirinteraction energy. 5 . Density shift From equation (6) it appears that the bulk energy density in d -dimension is ω ( d ) D = − k B Tλ d " ∞ X r =1 z r r d +22 + 2 ∞ X r =1 ∞ X l =1 z r r d +22 e − π ( Dλ ) l /r × e − βanr ( d − ) Γ( d − , − βanr ) ) (7)The bulk density of bosons may be obtained by differentiating the bulk energy densitywith respect to the chemical potential, µ , n ( d ) D = − ∂ω ( d ) D ∂µ = 1 λ d " ∞ X r =1 z r r d + 2 ∞ X r =1 ∞ X l =1 z r r d e − π ( Dλ ) l /r × e − βanr ( d − ) Γ( d − , − βanr ) ) (8)The lower incomplete gamma function is defined asΓ( q, x ) = Γ( q ) − Z x e − t t q − dt for q > q ) Γ( q, x ) = 1 − q ) Z x e − t t q − dt (10)The above integration is not solvable exactly when q takes on non integral values. Butfor integral values of q it is analytically solvable. So, in this work, we shall solve theintegration for integral values of q only.(i) CASE 1: d = 3, q = ( d − / d = 3, q = ( d − / n (3) D = − ∂ω (3) D ∂µ = 1 λ g ( z ) + 2 λ ∞ X r =1 ∞ X l =1 r e βµr − π ( Dλ ) l /r . (11)The first term on the right hand side of equation(11) is just the free bulk numberdensity, n (3) free , when the distance between two slabs D = ∞ . So the shift in density∆ n (3) D = n (3) D − n (3) free = 2 λ ∞ X r =1 ∞ X l =1 r e βµr − π ( Dλ ) l /r . (12)6ii) CASE 2: d = 5, q = ( d − / ω (5) D = − k B Tλ " g ( z ) + 2 ∞ X r =1 ∞ X l =1 r e βµr − π ( Dλ ) l /r − βan ( g ( z ) + 2 ∞ X r =1 ∞ X l =1 r e βµr − π ( Dλ ) l /r ) . (13)The bulk particle density in five dimensional space, n (5) D = 1 λ " g ( z ) + 2 ∞ X r =1 ∞ X l =1 r e βµr − π ( Dλ ) l /r − βan ( g ( z ) + 2 ∞ X r =1 ∞ X l =1 r e βµr − π ( Dλ ) l /r ) . (14)So, the amount of the density shift∆ n (5) D = 1 λ " ∞ X r =1 ∞ X l =1 r e βµr − π ( Dλ ) l /r − βan ( g ( z ) + 2 ∞ X r =1 ∞ X l =1 r e βµr − π ( Dλ ) l /r ) (15) C. Asymptotic Approximation when µ < : In the asymptotic limit i.e. for
D/λ → ∞ one can show as in Ref.18 that, ∞ X l =1 ∞ X r =1 r d e βµr − π ( Dlλ ) /r ≤ ζ ( d ) e √− πβµ Dλ − ≃ ζ ( d e − √− βπµ Dλ . (16)Equation (12) now reduces as ∆ n (3) D n (3) c ≈ e − √− βπµ Dλ , (17)where n ( d ) c = ζ ( d ) /λ d . Similarly, for d = 5, equation (15) stands as∆ n (5) D n (5) c ≈ e − √− βπµ Dλ − βan " g ( z ) ζ ( ) + 2 ζ ( ) ζ ( ) e − √− βπµ Dλ . (18)7 . Density shift for µ → : For d = 3, the density shift in the condensate (i.e. when µ = 0) can be obtainedfrom equation (12), ∆ n (3) c n (3) c = 2 ζ ( ) ∞ X r =1 ∞ X l =1 r e − π ( Dλ ) l /r (19)Again, from equation (15) it follows for d = 5 that,∆ n (5) c n (5) c = 1 ζ ( ) " ∞ X r =1 ∞ X l =1 r e − π ( Dλ ) l /r − βan ( ζ ( 32 ) + 2 ∞ X r =1 ∞ X l =1 r e − π ( Dλ ) l /r ) (20)Now, the shift of critical temperature may be obtained, in principle, from the followingrelation[28], ∆ T c T c ≈ −
23 ∆ n c n c . (21) III. RESULTS AND DISCUSSION
Results for the density shift for the Bose gas in a MFP and placed between two par-allel slabs is described in this section. Equation (6) gives an exact account for the grandpotential energy for Bose gas in MFP and for the given geometry in a d -dimensionalconfiguration space. The energy apprantly depends on both the distance between slabs, D , and the mean field potential, an . The first and the second terms on the right handside of equation (6) are related to the bulk energy density and the surface energy density,respectively. The third term is related to the bulk energy density but arises due to theCasimir effect. If the MFP is switched off (i.e. set a=0) the third term purely accountsfor the Casimir effect arising from the density fluctuation[10]. However, equation (7)shows the total bulk energy density in d -dimensional configurational space.The negativeof the derivative of the bulk energy density, ω ( d ) D , with respective to the chemical po-tential, µ , yields the bulk number density of the Bose gas for the given situation. This8s shown in equation (8). It is noticed that, equation (8) contains a lower incompletegamma function Γ( q, x ). This is very complex to solve when q has a non-integral value(see Eqn. (10)). In order to make the integration simple and to analyze the effects ofthe Casimir interaction and the MFP analytically, we consider the cases for q = 1 and 2only. As q = ( d − /
2, values 1 and 2 for q correspond to d = 3 and d = 5, respectively.For q = 1, Γ( q, x ) / Γ( q ) = e − x = e βanr . So the bulk energy density becomes com-pletely independent of the MFP. As a consequence, the density shift remains totally un-affected by the MFP. The most interesting thing happens when we go for higher dimen-sion that is for d >
3. For d = 5, q = 2, and Γ( q, x ) / Γ( q ) = e − x (1+ x ) = e βanr (1 − βanr ).Equation (13), therefore, shows clearly that, for d > d = 5, theshift in density of the Bose gas depends on both accordingly(see eq. (15)). Again for q =3 and 4, Γ( q, x ) / Γ( q ) = e − x ( x + 2 x + 2) / e − x ( x + 3 x + 6 x + 6) /
6, respec-tively. This is now crystal clear that the MFP contribution remains unvanished in allhigh dimensional cases, d >
D/λ → ∞ ) of the density shift for µ < d = 3 and 5,respectively. Equation (17) illustrates that, the shift in density is directly proprtionalto e − √− πµ Dλ , which decays exponentially with increasing D , and finally vanishes at D → ∞ . This is what we expected because when Casimir effect is zero density shiftdue to it must be zero. As the ∆ n (5) D /n (5) c for d = 5 depends on both the Casimir effectand MFP, equation (18) demands a more careful analysis. It is seen that, if the MFPis switched off the ratio ∆ n (5) D /n (5) c yields the same magnitude as that for d = 3 and,varies with D in a similar way. But if MFP remains on, the ratio does not vanish for D → ∞ . The remainder is βan g ( z ) /ζ ( ) which depends on the MFP. This meansthat, the MFP causes shift of the free bulk density in the non condensate state for d = 5.Equation (19) and equation (20) illustrate the shift of density for d = 3 and 5,respectively, when µ →
0, that is in the condensate. Here, for d = 3, ∆ n (3) c /n (3) c is solelygoverned by the Casimir effect alone. On the other hand, for d = 5, ∆ n (5) c /n (5) c depends9n both the Casimir interaction and the MFP; the former contribution is positive andthe latter contribution is negative. Now, if we accept the general relation between thecritical density shift and the critical temperature shift [28,29], ∆ n c /n c = − ∆ T c /T c , wecan say that the critical temperature shift is negative for d = 3 and could be positive ornegative for d = 5 depending on which dominates, Casimir interaction or MFP. Here,it is worth noting that the sign of the Casimir interaction is geometry dependent [38],so, the sign of the corresponding density shift or critical temperature shift would alteraccordingly. IV. CONCLUSION
The shift of density of Bose gas in MFP, and placed between two parallel slabs issystematically investigated from the point of thermodynamic consideration. From theabove results and discussions the following conclusions may be drawn. For d = 3,the density shift of Bose gas remains unaffected by the MFP regardless of the statecondensate or non-condensate. The Casimir effect solely plays the role in shifting thedensity in this case. But for d > d = 3. Consequently, the corresonding sign of the shift in criticaltemperature of Bose gas would be negative (see equation (21)). This result contradictswith those obtained for repulsive interaction in the homogeneous gas. The attractiveCasimir interaction in the present case clearly explains the cause of being negativesign in ∆ T c /T c . But for d >
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