We scrutinize an interparticle interaction which can create the type-I generalized Bose-Einstein condensation in spite of other thermodynamic properties of the model coinciding with those for the free boson gas

Papoyan, Vl. V.

Zagrebnov, V. A.

English

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DIAS-STP 89-32.On Generalized Bose-Einstein Condensationin the Almost-Ideal Boson GasVl.V.Papoyan *), V.A.Zagrebnov **)*) Department of Theoretical Physics,Yerevan State UniversityYerevan 375 049, Armenian SSR.**) School of Theoretical Physics, Dublin Institute forAdvanced Studies, 10 Burlington Road, Dublin 4, Ireland.Abstract:We scrutinize an interparticle interaction which can createthe type-I generalized Bose-Einstein condensation in spite of otherthermodynamic properties of the model coinciding with those for the freeboson gas.Permanent address: Laboratory of Theoretical Physics, Joint Institutefor Nuclear Research, Dubna 141 980, USSR1 IntroductionIn a recent paper [1] we investigated the problem of equivalence of thecanonical and grand canonical ensembles in connection with Bose-Einsteincondensation (BEC) This problem is nontrivial already for the freeboson gas, see e.g. [2-4] . That is why in our paper [1] we proposed asequence of exactly soluable models with repulsive interactionsinvolving an increasing number of particles. We observed therestoration of the strong equivalence of ensembles only for the modelwith the strongest interaction (model III of [1] ) in spite of which thestandard BEC persisits: One gets macroscopic occupation of thesingle-particle ground state above the critical density. One of theproposed models is the almost-ideal boson gas (model I of [1]) with theinteraction in the ground state. As we observed for this model, BEC hasa pecularity: the repulsive interaction forces the condensed particlesto the first excited level. In the thermodynamic limit, this levelcoincides with the ground state Therefore, from the naive point ofview, BEC should coincide with that for the free boson gas. But weshall demonstrate that this is true only if the first excited level isnondegenerate. In the opposite case, BEC for model I is in fact thetype-I generalized condensation (GC) in accordance with theclassification proposed in [5,6]. For the readers convenience, werecall the definition:Type-I GC corresponds to the macroscopic occupation of a finite numberof single-particle levels, type-Il GC corresponds to the macroscopicoccupation of an infinite number of single-particle levels; type-Ill GCcorresponds to nonextensive BEC: no levels are macroscopically occupiedon the scale of the volume(2)In this paper, we prove that, in spite of theirthermodynamic equivalence (coincide of their thermodynamic functions)the almost-ideal and the free boson gases have different Kac densitiesTherefore, one can distinguish these two models by their different typesof BEC2. The ModelLet AJR be a region in )-dimensional Euclidean space with volumeV = IAI , smooth boundary A. Let (T) j ëbe the spectrum of the single-particle Hamiltonian T, corresponding toa self-adjoint extension of the operator TA=(—z/Zry), D()C(A)with G-boundary conditions on A Here is the ‘)-dimensionalLaplacian and hi.. is the particle mass. Below we consider ‘nonsticky’boundary conditions:O [7] , and omit the index S . Then,E0(A) 0 and E0(A) 0 for A71R. The thermodynamic limit(t-lim) is implemented by an isotropic dilation of the region A aboutthe origin 0 which is assumed to lie inside A [8]To pass to the many-body problem. We introduce theprobability space2 of terminating sequences cL.={c*.kJk>0 Ofnon-negative integers:Q(N)k’.O N0where kN . The basic dynamic (random)k.ovariables will be the occupation numbers tik: , k = ...(3)(N)The Hamiltonian TA of N free bosons in the region A can be writtenas, TAEJ=ZEkAk[J (1)koThen the Hamiltonian HA of the almost-ideal Bose gas (model I [1] ) hasthe form+ j (2)It corresponds to the switching on of the mean-field repulsiveinteraction between the bosons occupying the single-particle groundstate.Now we can define on the space c the grand canonicalfinite-volume Gibbs states and <>H for thetemperature 0 and the chemical potentialO. It is clearthat they correspond to the product measures. For the model (2) thismeasure has the formexpP [ce] (3)[v(?A) -p0(,))] A, 0where is the finite-volume free boson gas Gibbs measure:exp (L]-NL])]P [C*]= (4)A,0(4)Here N [J = is the total number of particles in thekOconfiguration c) , and°,)= (?v) fZ exp [-([]- NPA() = (v)ft f eocp (A[wJN []]}are the grand canonical pressures for the models (1) and (2) Thethermodynamics potentials in the canonical ensemble (free-energydensity) have the formeA (,f) =-(v { Z N) (6)eThey correspond to the finite-volume Gibbs measures in the canonicalensemble:(N) (N) (N) (N)A ‘A t’S2 (7)Theorem 1. Let ‘)?1 and . Then, the model (2) isthermodynamically equivalent to the free boson gas (1):(a) -& A’9 =(b)0(5)(0)Proof. (a) Let and PA be the finite-volume thermodynamicpotentials of the free boson gas with the single-particle spectrumThen, one gets that)) 0 and theinequalitiesL A’f[2cA (?(-))] , (8)where = and p = . Now, taking into account the(0) (0)monotonicity of the function c-.4? (,x) b-tLrrVA (j,cc) and the(0)inequalities (8) we obtain that , which provides theproof of (a)(b) Using the explicit form of the Hamiltonian (2) one getsCO)() +-(A)ft-Then, from the inequality (j...& 0)2 (Vecp ri0—a0(A)-f_J(O) (0)and the limit E_€LflL?A_PA ) 0 we get the proof of (b).Corollary 1 The free-energy density and thepressure for the model (2) with 0 are relatedby the Legandre transform, i.e. for this model the canonical and thegrand canonical ensembles are weekly equivalent.(6)3. Condensation and the Kac DensityThe relation between thermodynamics and statistical mechanics for Bosesystems has recently been scrutinized in several aspects [9-12] Inthis section we show that in spite of the thermodynamic equivalence themodels (1) and (2) are different.Theorem 2. Let .>Oand A be a rectangular box with the equal edges(cuboid) and with the center at 0 . Then, we get for the model (2) with+oo (Dirichlet boundary condition) that(a)-t 6-tLrriK)(j&)O ; (10)a(b) (lOb)for k=,2, ,‘) and “>2. Here and are thefinite-volume canonical and grand canonical Gibbs states correspondingto the model (2); p0) and j.°k 0) are the critical parameters for thefree boson gas.Proof. (a). These two limits are the consequence of the estimate (J..U)= o) == z ft0P4 [] rt ep(-r/zv) . (11)0(b) For the cuboid A the first excited level is 2) -fold degenerate.Now, we can repeat the standard arguments about BEC to get(7)f ÷(12)+ <v)(A()),AwherefA=A(?)is the solution of equation (11) for a fixed densityfHence, taking into account (lOa) we derive from (11) that asymptotically?) o(v1)(13)L ek ovj p >where °L=O= ?=0) is equal toáeE’1(e) (14)Collecting (12)-(14) one gets (lob).Remark 1. The thermodynamic properties of the model (2) (including thevalues of the critical parameters) coincide with those for the freeboson gas. If the first single-particle excited level is degenerate,then the interaction generates the type-I BEC instead of the BEC in theground state alone. But if not (e.g. for A a rectangular box withdifferent edges) the model (2) is identical to the free boson gas.Remark 2 (A Generalized Almost-Ideal Bose Gas) From above it is clearthat up to minor technical corrections the same properties are exhibitedby the modelMTA+k , {k>0}k=o . (15)(8)Let be the distribution function (theN()finite-volume Kac density) for the random variable XA(cJ)= —v- ; itrelates the Gibbs states of different ensembles:= S0 <(1) <>A= -t <XA>A(A(2))The limit distribution K-&m. k’ (as the weak limit) playsan important role in analysing the ensemble equivalence [1-4] and BEC[5, 6, 8, 13] . The canonical and grand canonical ensembles are strongly(or statistically) equivalent if the limit distribution is degenerateKxI)=()concentrated at(jk)=p(,jA) [1].Theorem 3. For the almost-ideal boson gas (2) with>O and isotropicdilationA/lR’ of the cuboid A , the limit Kac density has the form()=? (?)(0)=0) (0)______9 (s—p ) ex F ‘ (cc—jD (o)( 16)(i) p-p°L J P[ p(o)J’?Proof. By definition, the Kac density K(xI,p)is related to the-&inofthe characteristic function of the random variable XA(u))-L<ep (L+XA)() doe €DC (17)To calculate the limit in the left-hand side of (17), we use theproduct-measure A(9)<Q-9 (+XA) (J2,frL) ==P )ep(°)H{- - ep [- (Ek(A)-)] }ftO ki - ecep - L/v)]Then, from the estimate (11) and the asymptotics (13) one gets-E-trL <ex? (XA)>A(J)eocp () =p() <0), (is)- [ -°)/]xp (Lp), ?= <X(A(p)) > p3)(= 0)Expression (16) is the result of the Fourier-transform of the right-handside of (19).Corollary 2. As for the free boson gas, the strong equivalence ofensembles for the model (2) is broken.Remark 3. The dependence of the Kac density kcip) on the parameteris strongly nonanalytic. The deviation of X(JD) from thefree boson gas Kac density(0) (0) (o)2 i? (f (j<fr0)is heavily dependent on the structure of the single-particle spectrumnear the bottom. If the region A has a shape such that the first(0)excited level is nondegenerate then K (= cIp).In this case thereis no difference between the almost-ideal and the free boson gases.(18)(0))exp [— r -(0) 0 (0)J -? >?° ( (0))(10)4. ConclusionFirst it should be stressed that Theorem 3 and Remark 3 are valid forthe model (15). The knowledge of the spectrumZ() and vectoris sufficient to determine all possible corrections to the Kac densityKA(x1p). As above, the result depends not on the absolute value of thebut on their signs only.—0,In this connection it is interesting to consider the model(2) for .<O. Then, (collapse) and one has to save thesituation by switching on a repulsive interaction as was done in [14]But the free-energy density for the model (2) with exists and hasthe form[+ (,-))] . (20)Therefore, the weak equivalence is broken in this case.For .)-j the right-hand side of (20) reaches the infinum at—c 2P.)p>pC)where ,*(?)=naxfC)and () are nontrivial roots of the equation-1,2.P +°(z=? (-))= UThe critical densityp( is defined by the relation, - ()1 * (0)<Pc ,in the other caseswhere is the solution of the equation(. ()) ( (*f)) 0(11)Thus, the properties of the model (2) are very differentfor <O(.col1apse) and for (almost-ideal boson gas). Scrutinizingthe models (2) and (15), we demonstrate how subtle BEC is: theinteraction which does not change the thermodynamics can creategeneralized BEC.AcknowledgementsWe would like to thank Robert A. Minlos for valuablediscussions that initiated this work. We are also indebted toJohn T. Lewis for numerous helpful remarks. This paper was finishedwhen one of the authors (V.A.Z.) visited the School ofTheoretical Physics, Dublin. The warm hospitality and support of theDublin Institute for Advanced Studies together with the efficient typingof the manuscript done by Catherine Mc Auley made this publicationpossible.References1. V.A. Zagrebnov, Vl.V. Papoyan, Theor. and Math.Phys. 69,1240-1253 (1986).2. J.T. Lewis, J.V. Pul, The Equivalence of Ensembles inStatistical Mechanics, Lecture Notes in Mathematics 1095,25-35 (1984).3. E. Buffet, J.V. Pu1, J.Math.Phys. 24, 1608-1616 (1983).4. Vl.V. Papoyan, V.A. Zagrebnov, Phys.Lett.113A, 8-10 (1985).5. M. Van den Berg, J.Math.Phys. 23, 1159-1161 (1982).6. M. Van den Berg, J.T. Lewis, Physica llOA, 550-564 (1982).7. D.W. Robinson, Commun.Mãth.Phys. 50, 53-59 (1976);L.J. Landau, I.F. Wilde, Commun.Math.Phys. 70, 43-51 (1979).8. M. Van den Berg, J.Stat.Phys. 31, 623-637 (1983).9. M. Van den Berg, J.T. Lewis, M. Lunn, Helv.Phys.Acta, 59,1289-1310 (1986).10. Ph.de Smedt, V.A. Zagrebnov, Phys.Rev.A35, 4763-4769 (1987).11. Ph.de Smedt, J.Stat.Phys. 45, 201-213 (1986).12. P. Mac Aonghusa, J.V. Pu1, Lett.Math.Phys. 14, 117-121(1987).13. J.T. Lewis, J.V. Pul, V.A. Zagrebnov, Helv.Phys. Acta 61,1063-1078 (1988).14, M. Van den Berg, J.T. Lewis, J.V. Pu1, Commun.Math.Phys.118, 61-85 (1988).

On Generalized Bose-Einstein Condensation in the Almost-Ideal Boson Gas

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On Generalized Bose-Einstein Condensation in the Almost-Ideal Boson Gas

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