The Solution of the Two-Dimensional Gross-Pitaevskii Equation Using Lattice-Boltzmann and He s Semi-Inverse Method

Transcription

1 Internatonal Journal of Mathematcal Analyss Vol., 7, no., HIKARI Ltd, The Soluton of the Two-Dmensonal Gross-Ptaevsk Equaton Usng Lattce-Boltzmann and He s Sem-Inverse Method F. Fonseca Unversdad Naconal de Colomba Grupo de Cenca de Materales y Superfces Departamento de Físca Bogotá-Colomba Copyrght c 7 F. Fonseca. Ths artcle s dstrbuted under the Creatve Commons Attrbuton Lcense, whch permts unrestrkd use, dstrbuton, and reproducton n any medum, provded the orgnal work s properly cted. Abstract We solve the two-dmensonal tme ndependent Gross-Ptaevsk equaton usng the lattce-boltzmann technque for a dq9 lattce velocty scheme. Also, applyng the He s sem nverse method we fnd soltary wave soluton. We get stable densty profles. Keywords: d Bose-Ensten condensates, lattce-boltzmann, He s sem nverse method Introducton The Gross-Ptaevsk equaton, (GEq), descrbes the Bose-Condensates phenomena, whch s a very mportant nvestgaton feld n condensed matter physcs, at low temperature []. Ths nonlnear dfferental equaton s used, e.g., n nonlnear Optcs, quantum flud dynamcs, and the magnetc trap of Rubdum atoms at very low temperatures []-[3]. Research n pure and appled mathematcs has been done for GEq. For nstance, usng tme-splttng spectral [4], fnte-dfference scheme [5], doman truncaton technques [6]. Also, the use of the lattce-boltzmann (LB) technque, s wdely use n Physcs. For

2 7 F. Fonseca example, LB has been appled from research felds as dverse as magnetohydrodynamcs [7], tll the Posson equaton [8]. Also, the tme dependent GEq had been solved usng LB, obtanng good results, [9] (and references theren). Ths work solves the Gross-Ptaevsk equaton usng the lattce Boltzmann and He s sem nverse method. In secton ()-(3), we present the lattce-boltzmann technque and the moment dstrbuton functons appled to dervaton of the GEq. In secton (4), we get the GEq. Then, n secton (5), we obtan the equlbrum dstrbuton functons usng dq9 scheme. In secton (6), we use the He s sem nverse method. And at last, n secton (7), we present results and conclusons. The lattce Boltzmann model The lattce-boltzmann equaton s: f ( r + vδt, t + δt) f ( r, t) = Ω ( r, t) () Where Ω( r, t) s the collson term, and usng the B.G.K approxmaton []: Ω ( r, t) = [ fj ( r, t) f eq j ( r, v) ] () τ Expandng the one-partcle dstrbuton functon n a Taylor seres, we have: + ( f( r + δ r, t + δt) f ( r, t) = δt t + δx x + δy ) f (3) y ( δt t + δx x + δy ) + δxδt + δyδt y xt yt + δxδy f xy Consderng δ r = vδt, δx = δtv x and δy = δtv y, and replacng n eq.(4) f( r + vδt, t + δt) f ( r, t) = δt [ ] [ ] t + v f + δt t + v f (4) Dong a perturbatve expanson at frst order n the spatal dervatves, second order to the tme dervatve, and second order to the partcle dstrbuton functon, we have: x = ɛ x, y = ɛ, y t = ɛ + ɛ, f = f + ɛf + ɛ f (5) t t Assumng = x + y j and replacng eqs. (5) n eq. (4)

3 Soluton of the two-dmensonal Gross-Ptaevsk equaton 7 (( τ (f + ɛf + ɛ f f eq ) = (δt ɛ + ɛ ) + v [ɛ t t ] ) (6) [( + δt ɛ + ɛ ) + v [ ɛ t t ] ] ) ( ) f + ɛf + ɛ f The terms at order ɛ n eq. (6) and assumng f = f eq, we have: [( ) ] [ ] ɛf (f τ = ɛδt + v t ) The terms at order ɛ n eq (6), we get: (7) τ ɛ f + δt Replacng eq. (7) n eq. (8), we get: = ɛ δt f [ ] + ɛ (δt + v t t f (8) ( ) ( ) + v t (f δt + v t ) ) τ [ ] [ ] f ( ) = δt f t + δt + v t f ( ) τ (9) Summng eq. (7) to eq. (9), we have: τδt (ɛf + ɛ f ) = ( + v t [ ) ɛf + ɛ f 3 Moments of the dstrbuton ( )] + ɛ ( ) f τ t () Dong the next defntons for the statstcal meoments of f, we have: (ρ µρ ρ ρ) = f () ; u = v f () () Π () α,β = v,α v,β f () () f (k) =, (k ) (3) Where ρ and u are the densty feld an velocty feld of the physcal system. Besdes, we assume the dstrbuton functon f satsfy the probablty conservaton condton wth the equlbrum dstrbuton f eq, such that f = f eq

4 7 F. Fonseca 4 The Gross-Ptaevsk equaton Takng the summaton about () n eqs. () and () (v,α ), we get: (ρ µρ ρ ρ) t The tensor Π () s defned a dagonal matrx ( ) Π () ρ = t ρ t Replacng eq. (6) n eq. (5), we obtan: + u =. (4) u t + Π() =, (5) (6) Interchangng dervatves u t ( ρ t ) = (7) Applyng dvergence n eq. (8), we get: u t (ρ) = (8) u t (ρ) = (9) Fgure : The lattce veloctes dq9 scheme. Usng eq. (4) n eq. (9), we obtan: (ρ µρ ρ ρ) t t (ρ) = ()

5 Soluton of the two-dmensonal Gross-Ptaevsk equaton 73 Takng out the temporal dervatve and assumng zero the nteror term n the parenthess, we get the Gross-Ptaevsk equaton ρ + (ρ ρ ρ) = µρ () 5 The equlbrum dstrbuton functon usng dq9 velocty scheme. The dq9 scheme velocty lattce shown n fg. () s used. The drectons v and weghts w on each cell are defned as: w = { 4 9, f = ; 9, f =,, 3, 4; 36, f = 5, 6, 7, 8 } () Both, drectons v and weghts w, follow the next tensoral relatons: w v,α =, w v,α v,β = 3 δ α,β, The equlbrum functon used s: w v,α v,β v,γ = (3) { f (eq) w [A v = u + B] f > w C otherwse = (4) The A, B y C constants n the equlbrum functon f (eq), are proportonal to the macroscopc quanttes, and they are calculated usng eqs. (-) n combnaton wth eqs. (3), then: B = 3 (ρ) t, A = 3, C = 9 4 (ρ µρ ρ ρ) (ρ) t (5) Then, the equlbrum functon that satsfes the Gross-Ptaevsk equaton s, f (eq) = ( ) 3w v u (ρ) t 9 4 w (ρ µρ ρ ρ) f > (ρ) t and = (6) In order to mplement the dervatve operator of ρ(x, t) used n eq. (6), we appled the dfference dscretzaton scheme of the second dervatve as: ρ ρ(x, t + δt) ρ(x, t) + ρ(x, t δt) = (7) t δt

6 74 F. Fonseca 6 He s sem-nverse method, Soltary wave soluton Accordng to He s sem nverse method [9], we choose the energy functonal [9], that satsfy eqs. (). Then, we get: Or: J(φ) = ( ( ) φ ( φ ) + 4 φ 4 )da (8) J(φ) = (( φ x ) + ( φ y ) φ + 4 φ 4 )dxdy (9) Changng to polar coordnates ξ = x + y ; x = ξ cos (θ); y = ξ sn (θ) (3) Fgure : LB results, usng an ntal profle gven by eq. (3). Then, eq. (3) s: J(φ) = (( φ ξ ) + ξ (φ θ ) φ + 4 φ 4 )ξdξdθ (3) We choose as a soluton:

7 Soluton of the two-dmensonal Gross-Ptaevsk equaton 75 Then, the entre acton s: ρ = a sn(bξ ) exp( bξ ) (3) J(a, b) = a 6 a (µ ) + 3a 8b 5b (33) In order to make J statonary, we calculate: J b = a (µ ) 3a 8b 5b = ; a = 4 (µ ) (34) 3 J = a 6 µ 8b + 6a 5b = ; b = (µ ) 8 (35) Fgures () show the LB results produced by an ntal profle gven by (3). The boundary condtons are zero at all tmes. 7 Conclusons We have solved the Gross-Ptaevsk equaton usng the lattce-boltzmann method and the He s sem nverse method. In addton, we present results for an ntal profle showng stable solutons. Besdes, we gve an explct soluton of Gross-Ptaevsk equaton: ρ(x, y) = a sn(b(x + y )) exp( b(x + y )) (36) As a future work we can extend the method to three dmensons. Acknowledgements. Ths research was supported by Unversdad Naconal de Colomba n Hermes project (35). References [] S. N. Bose, Plancks Gesetz und Lchtquantenhypothese, Zetschrft fuer Physk, 6 (94), German translaton of Boses paper on Plancks law by Abert Ensten.

8 76 F. Fonseca [] M. H. Anderson, J. R. Ensher, M. R. Matthews, C. E. Weman and E. A. Cornell, Observaton of Bose-Ensten condensaton n a dlute atomc vapor, Scence, 69 (995), [3] K. B. Davs, M. O. Mewes, M. R. Andrews, N. J. van Druten, D. S. Durfee, D. M. Kurn, and W. Ketterle, Bose-ensten condensaton n a gas of sodum atoms, Phys. Rev. Lett., 75 (995), [4] W. Bao, S. Jn and P. A. Markowch, On tme-splttng spectral approxmaton for the Schrödnger equatons n the sem-classcal regme, Journal of Comput. Phys., 75 (), [5] M. M. Cermele, M. L. Chofalo, F. Pstella, S. Succ, and M. P. Tos, Numercal soluton of the Gross-Ptaevsk equaton usng an explct fntedfference scheme: An applcaton to trapped Bose-Ensten condensates, Phys. Rev. E, 6 (), no., [6] X. Antonea, C. Besseb and P. Klen, Numercal soluton of tmedependent nonlnear Schrödnger equatons usng doman truncaton technques coupled wth relaxaton scheme, Laser Physcs, (), no. 8, [7] P. J. Dellar, Lattce Boltzmann magnetohydrodynamcs wth currentdependent resstvty, Journal of Computatonal Physcs, 37 (3), [8] Z. Cha and B. Sh, A novel lattce Boltzmann model for the Posson equaton, Appled Mathematcal Modellng, 3 (8), [9] H. Wang and G. Yan, Lattce Boltzmann model for the nteracton of (+)-dmensonal soltons n generalzed GrossPtaevsk equaton, Appled Mathematcal Modellng, 4 (6), [] P. Bathnagar, E. Gross, A. Krook, A Model for Collson Processes n Gases. I. Small Ampltude Processes n Charged and Neutral One- Component Systems, Phys. Rev., 94 (954),

9 Soluton of the two-dmensonal Gross-Ptaevsk equaton 77 [] J-Huan He, A classcal varatonal model for mcropolar elastodynamcs, Internatonal Journal of Nonlnear Scence and Numercal Smulaton, (), no., Receved: January 7, 7; Publshed: January 4, 7