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From a nonlinear string to a weakly intera ting Bose gas 1 1,2 1 1 Emilia Witkowska , Mariusz Gajda and3J2/a4n6Mostowski 2 Institute of Physi s, Polish A ademy of S ien es, al.Lotników , 02-668 Warsa5w, Poland and Fa ulty of Mathemati s and Natural S ien es, Card. Stefan Wyszy«ski University, ul. Dewajtis , 01-815 Warsaw, Poland We investigate a real s alar (cid:28)eld whose dynami s is governed by a nonlinear wave equation. We 6 show that lassi al des ription an be applied to a quantum system of many intera ting bosons 0 providedthatsome quantumingredients are in luded. Anuniversal a tion has to be introdu ed in 0 order to de(cid:28)ne parti le number. The value of this a tion should be equal to the Plan k onstant. 2 This onstrain anbeimposedbyremovinghighfrequen ymodesfromthedynami sbyintrodu ing n a ut-o(cid:27). We show that the position of the ut-o(cid:27) has to be arefully adjusted. Finally, we show a the proper hoi e of the ut-o(cid:27) ensures that all low frequen y eigenenmodes whi h are taken into J a ountare ma ros opi ally o upied. 8 1 I. INTRODUCTION lowest energy levels are ma ros opi ally o upied. One ] mightexpe tthatdis retestru tureofthematter(cid:28)eldis r e notessentialanddes riptionofthesystembya` lassi al Des riptionofintera tingmanybodyquantumsystem h wave' should be valid. This expe tation has no rigorous t itisaverydi(cid:30) ulttask. Ex eptofafewrathera ademi o justi(cid:28) ation. However an ingenious idea of Bogoliubov problemsexa tsolutionsarenota essibleandsomeap- . onsisting in substitution of the annihilation operator of t proximated methods are ne essary. Re ent attempts of a a parti le in the ondensate mode by a -number ampli- des ription of a Bose-Einstein ondensate at (cid:28)nite tem- m tude[6℄isextremelysu essfulandwidelyused. Thisap- perature [1℄-[3℄ showed that it is possible to signi(cid:28) antly - proa hleads to a mean (cid:28)eld des ription of the system in d simplifytheoreti almethods ofsolvingquantumdynam- terms of a lassi al (cid:28)elds satisfying the Gross-Pitaevskii n i s. This is due to the assumption of ma ros opi o u- equation. o pation of single parti le modes. It is worth noti ing that c Bose-Einstein ondensate is not the only system with Thus the standard theory of Bose ondensate at zero [ ma ros opi o upation of quantum states. temperature is based on the Bogoliubov method. Re- ently this idea was extended to (cid:28)nite temperatures and 1 Histori ally the (cid:28)rst and the best known system with v su h properties is ele tromagneti (cid:28)eld. As long as in- is alled the lassi al (cid:28)elds method. It is su essful in 8 des ribing equilibrium properties of Bose ondensate at tensity is large lassi al approa h based on Maxwell's 0 (cid:28)nitetemperatures,ex itationsspe trum,dissipativedy- equations is valid. The lassi al point of view was the 4 nami s of vorti es and many other (cid:28)nite temperature only one used until the beginning of the twentieth en- 1 phenomena [2℄. 0 tury. Howeverthisapproa hturnedouttobeinadequate 6 to des ribeexperimentswith small intensities and there- The lassi al(cid:28)eldsmethodisbasedonheuristi substi- 0 forethe on eptofphoton had to beintrodu ed. Onthe tution of Bose operators by -number amplitudes. This t/ other hand, physi al phenomena involving ma ros opi- substitution an be easily justi(cid:28)ed at very low tempera- a ally large (cid:28)eld amplitudes (or equivalently large num- turessin epra ti allyallparti leso upythe ondensate m berofphotons)aresu essfullydes ribedbythe lassi al mode. On the other hand it seems questionable at tem- - ele tri andmagneti (cid:28)elds[4℄. Althoughquantizationof peratures lose to the riti al temperature. Nevertheless d ele tromagneti (cid:28)eld is a well established pro edure, the the lassi al (cid:28)elds method works quite well up to the n o inversepro edure,i.e. substitutionofaquantum(cid:28)eldby ondensation point. It is known that the Bose-Einstein c a lassi al one is often heuristi and based on physi al ondensation is a quantum phenomenon and the riti al : intuitionratherthanformalarguments. Atthispointwe temperature depends on the Plan k onstant. v i shouldmention the Glaubertheory of oherentstates [5℄ The Plan k onstant is usually introdu ed into the- X ofele tromagneti (cid:28)eld. Ifthe (cid:28)eldisin a oherentstate ory through ommutation relations for the (cid:28)eld opera- r then ele tri and magneti (cid:28)led operators an be substi- tors. Those however are violated in the lassi al (cid:28)elds a tuted by their non-vanishing mean values in this state. approa h. Nevertheless this approa h preserves some These mean values are interpreted as lassi al (cid:28)elds. bosoni features of the system. Therefore it is justi(cid:28)ed Although oherentstatesprovidealinkbetween lassi- to ask whi h features of the quantum system are taken alandquantumtheoriesthesituationisnotthatsimple into a ount in the lassi al (cid:28)eld methods and whi h as- in ase of parti les with non-zero mass. Supersele tion sumptionsleadsto orre tpredi tionofthe ondensation rules do not allow for superposition of states with di(cid:27)er- temperature. In parti ular one may ask how does the entnumberofparti les. Therefore,one annotintrodu e Plan k onstant enter into the lassi al (cid:28)eld method. oherent states for su h (cid:28)elds. Su h a situation takes Inordertoanswerthesequestions,atleastpartially,we pla e in the ase of atomi Bose ondensates, where the hoose the following approa h. We start with a lassi al number of parti les is (cid:28)xed. system of intera ting harmoni os illators and study its In Bose ondensates at low temperatures only few dynami s. At a later stage we in lude some quantum 2 ingredients and pinpoint the moment where the Plan k (cid:28)eld of zero harge. In the following we will use the onstant appears. This approa h justi(cid:28)es the lassi al dis rete version of the model. (cid:28)elds method and shows its limitations. We will now take into a ount a nonlinearity. For The paper is organized as follows. In Se . II we in- simpli ity we assume that the nonlinear intera tion is of trodu e the model. We dis uss the numeri al te hniques short range (lo al) and the dynami al equation is: pthaartti wuelaursefoarnmdomfainintefreaa ttuiorenssobfentwumeeenrio as lislloaltuotrisonles.adAs mφ¨j =−K(2yj−φj+1−φj−1)−Λφ3j, (3) Λ tononlinearequationssimilartothosestudied byFermi, where is a real parameter. This form of intera tion is PastaandUlam[7℄. UnliketheFermi-Pasta-Ulamresults widely used iφn4various areas of physi s, in parti ular in thedynami sgivenbyourmodelleadstothermalization the so alled (cid:28)eld theory, [9℄. of the system. In Se . III we analyze the state of ther- Equation(3)isverysimilartotheonewhi happearsin mal equilibrium rea hed by the system, we study energy thefamousFermi-Pasta-Ulam(FPU)problem[7℄. There equipartition, and de(cid:28)ne the temperature of the system. are, however, two di(cid:27)eren es. First, in the FPU ase In Se . IV we analyze the system in terms of quasiparti- displa ements of the (cid:28)rst and the last os illators are set lesando upationofsingleparti lestate. Weshowthat to zero (as opposed to periodi boundary onditions as- elementaryex itations(phonons)aredistributeda ord- sumed here). Se ondly, the nonlinear term in the FPU ing to the low frequen y part of Bose statisti s. High equation is of a di(cid:27)erent form. The authors onsid- frequen y part is introdu ed by hand with the help of properly hosen ut-o(cid:27). In this way we mimi quantum (eφrejd nφojn−-l1o) ral nonlinrea=r2for ers,=fo3r example of the form − , where or . The results of FPU statisti s in the whole rangeof frequen ies. In se . V we al ulations show that the system shows `very little, if on lude by summarizing our results and showing their any, tenden y toward equipartition of energy among de- impli ations for the lassi al (cid:28)elds method. grees of freedom', [7℄. On the ontrary, as we are going to show in the following, the system des ribed by Eq.(3) rea hes a state of thermal equilibrium hara terized by II. DYNAMICS OF A NONLINEAR STRING equipartition of energy. L Let us introtdu= eln/actural units: 1) unit of lenǫg=thK,L22) In this se tion we are going to introdu e the model 0 0 unit of time and 3) unit of energy . and its basi equations. We onsider a one dimensional L ρ The set of oupled nonlinear equations takes the form: delearsttio (cid:28)stnrdindgyonfalemnig tahl eqaunadtiolinnseaorfmmaostsiodnenwseitydiv.idIentohre- φ¨j =−(2φj −φj+1−φj−1)−λφ3j, (4) N 1 l =L/N 0 string into − elements of length and mass m = l0ρ where . Ea h element is repla ed by a point-like par- Λ ti le intera ting with the nearest neighbor via harmoni λ= L2 F K (5) for es. The restoring for e a ting on ea h parti le is ∆l proportionaltodispla ement 0fromitsequilibriumpo- λ F = Y(∆l /l ) Y is the nonlinear oupling onstant. We assume that , sition, − 0 0 where is the Young modulus. l0 N does not depend on . Therefore oe(cid:30) ients in Eq.(4) Thusthestring anbeviewedas parti lesmovingona N do not depend on the number of os illators . line,ea hofthem onne tedtotwoneighborsbyaspring l0 K =Y/l0 Eq. (4) leads to energy onservation: withequilibriumlength andelasti onstant . 1 1 Wedenoxtje=equjli0librjiu=m1p,o..s.it,iNonsofea hparti le(os illa- E = 2 [φ˙2j +(φj −φj−1)2+λ2φ4j]=const. (6) tor) by ( ) and their displa ements j φ X j fromequilibrium(alongtheaxisofthestring)by . The Newton equations of motion for the displa ements are: We solve the set of equations (4) numeri ally for vari- N mφ¨j =−K(2φj −φj+1−φj−1). (1) ous initial energies and vaφrjious number of oφ˙sj illators . All initial displa ements and velo ities are gener- Forthefuture oφnv=eniφen eweassumeperiodi boundary ated[ frϕo,mϕ]uniformϕprobability distribution on the inter- onditions, i.e. j j+N. Eq. (1) is used in di(cid:27)erent val − , where is a parameter. Its value has to be areas of physi s, e.g. in des ription of vibration of one- adjusted a ording to the value of initial energy. In geϕn- dimensional rystal latti e. Analyti solutions of (1) are eral, the larger initial energy the larger the value of . available in terms of plane waves, [8℄. In our numeri al simulations we adjusted the time step, Letusremark,thatinthelimitof ontinuousmedium, andthereforea ura yofnumeri alpro edure,toensure N l 0 → ∞ (i.e. 0 → ) Eq.(1) takes the form of wave energy onservation. equation: Due to assumed boundary onditions it is onvenient ∂2φ(x,t) ∂2φ(x,t) to analyze the results of our simulations in the basis of c2 =0, ∂t2 − ∂x2 (2) plane waves 1 kmax c= Y/ρ φ (t)= b (t)e−ikj, where isthevelo ityofsound. Inthelanguage j √N k (7) of lassi alp(cid:28)eldtheoriesequation(1)des ribesfrees alar k=Xkmin 3 k where the dimensionless wave ve tor takes the values k =nπ/2 n= N,...,N 1 and − − . Notethatmodeswith k > k max |k| are not present in (7). The ut-o(cid:27) of large 1.´10-8 | | is introdu ed in our model by dis retizationof spa e. Wnuemwbiellrsohfoswpathtiaatltghreid upto-ion(cid:27)tspNos)itpiolnay(soraevqeuriyvaimlenptolrytathnet ~2Èb06.´10-9 È role in our des ription. Complex amplitudes of di(cid:27)erent b k plane waves are subje t to a onstrain: 2.´10-9 ∗ b (t)=b (t), k −k (8) -150 -100 -50 0 50 100 150 φ (t) Ω j be ause the displa ement (cid:28)eld is a real fun tion. This onditionisautomati allysatis(cid:28)edinnumeri alim- plementation. 1.´10-8 III. THERMAL EQUILIBRIUM 2 ~Èb56.´10-9 The most important observation whi h follows from È numeri al solutions is that the system rea hes a state of equilibrium after some transient time. We he ked this 2.´10-9 observationby hoosingmanydi(cid:27)erentinitial onditions orrespondingtothesameinitialenergy. Theequilibrium -150 -100 -50 0 50 100 150 Ω stateis hara terizedbyrandbom(tl)y-lookingos illatibon(ts)o2f k k ea h plane wave amplitude . Its modulus | | os illates in time from zero to some maximal value. The k = 0 mode with has the largest amplitude. In general, k the largerthe waveve tor | |, the smallerthe amplitude 1.´10-8 of os illations of the orresponding mode. We will now analyze the stationary state. It depends 2 on the energy E and the number N. Let us analyze ~Èb-56.´10-9 È ˜tbh(eωf)re=quend tyeiωsptbe (ttr)um of ea h plane wave amplitude k k . It turns out that the randomly bk(t) 2.´10-9 lookingosR illationsof haveaveryregularspe trum. Tky=pi0 ,alf5r,e5quen yspe traofplanewaveamplitudeswith -150 -100 -50 0 50 100 150 − are plotted in Fig. 1. Ω The spe trum of ea h mode is omposed of two peaks ω ω k k entered around and − . The peaks have (cid:28)nite width whi h is relatively small as ompared to the en- E = 0.1 λ=1 tral frequen y. Therefore, in the (cid:28)rst approximation, it FIG. 1: Spe trumof aNmp=lit6u4des for enker=gy0 k =, −5 , isjusti(cid:28)ed to negle tthewidthofpeaksinthefrequen y nku=m5ber of grid points and a) , bω)k −ω,k ) spe trum and assume that ea h plane wave amplitude . Note existen e of two peaks enteredat and . os illates in time with two frequen ies. The frequen y ω k is de(cid:28)ned as the weighted mean value of frequen ies ωk We he kedthatthefrequen ies ful(cid:28)llthefollowing orresponding to the positive frequen y peak: dispersion relation ω˜b (ω)2 ω = ω>0 | k | k k P ω>0|˜bk(ω)|2 (9) ωk =r4sin2 2 +ω02, (12) Pβ ω k =0 k 0 while e(cid:27)e tive amplitude is: where is the frequen y of the mode. Numeri al (cid:28)t shows that: βk = ˜bk(ω)2 kmax sωX>0| | (10) ω02 =2λα |βk|2, (13) k=Xkmin ω k In this wβay we de(cid:28)ned an kunique frequen y anbd am- α 4.5 plitude k for every mode . Thus the amplitude k has and is loseto . InFig. 2weshowthedispersionre- the following time dependen e: lation obtained from numeri al simulations (points) and b (t)=β e−iωkt+β∗ eiωkt. ompare it to the analyti formula (12). Comparison k k −k (11) shows remarkably good agreement. 4 120 0.005 100 80 Ωk ¶k0.003 60 40 20 0.001 -180 -120 -60 0 60 120 180 -180 -90 0 90 180 k k FIG. 2: DispEers=ion0.1relNatio=n.64Pointsλre=fer1to numeri al sim- FIG.3: Equipartitionofenergy. Energypermode,Eq. (16), ulation with α=,4.5 and . Curve refers to as a fun tion of wave vEe =tor0..2PoNint=s6re4fer toλn=um1eri alsimu- formula (12) with . lation of Eq. (4) with , and . ε ε = E/N Inthe ontinuouslimit4,swinh2e(rke/E2)q. (4)be omes anon- where k is the energy per mode. Obviously k . lkin2earwaveequationthe termbe omessimply Fig. (3)illustratestheequipartitionofenergy. Letusre- . This di(cid:27)eren e is due to dis retization of spa e and mind that no equipartition of energyis observedin FPU the simpli(cid:28)ed form of the dis rete version of the se ond problem. spatial derivative. Havingtheequipartitionofenergywe ande(cid:28)neatem- In the state of thermal equilibrium the dispala ment φ (t) perature of the system: j an be approximated by T˜=ε , φj(t)= √1N kmax (βke−i(kj−ωkt)+β−∗kei(kj−ωkt)). k ǫ/k k (17) k=Xkmin wheretemperatureisexpressedinunitof B ( B isthe (14) Boltzmann onstant). Normal modes of the system are plane waves. Every k ω k mode os illates with two opposite frequen ies and ω k − . Amplitudes orrespondingto these frequen ies are IV. PHONONS βk = β−k related, | | | |. Using the language of the (cid:28)eld theory we an say that the positive frequen y part or- In the previous se tion we have shown that the non- responds to parti le-like modes while the negative fre- linear hamiltonian (6) an be expressed in the diagonal quen y omponent orresponds to antiparti le-like ex i- form(15)forea hparti ularstateofthermalequilibrium. tations. Notethatit annotbedone`globally',i.e. independently The total energy of the intera ting system an be ex- β ω of the total energy of the system. This is be ause the k k pressed in terms of amplitudes and frequen ies eigenfrequen ies depend on the energy. similarly as in the linear ase: N Thediagonalhamiltonian(15)isasumofenergiesof kmax ωk E =2 ω2 β 2. independent harmoniλ os illators of frequen ies . The k| k| (15) intera tion strength enters this hamiltonian through ω k=Xkmin the eigenfrequen ies k only. Thispi tureispurely lassi al. Inwhatfollowswewill Fa tor two whi h appears in the above formula omes reformulateourresultsinthelanguageofquantummany from the fa t that ea h plane wave os illates with two bodytheory. Ourgoalistorea habetterunderstanding frequen ies of opposite sign and ea h of them gives the of foundations of the lassi al (cid:28)elds method. same ontributiontotheenergy. We he kthatdisagree- The hamiltonian (15) an be quantized with the help mentbetweenvaluesofenergygivenbyEq. (6)and(15) 5% 10% of anoni al quantization pro edure, i.e. by de(cid:28)ning po- is notlargerthan − . Therefore, the total energy βk sition and momentum in terms of amplitudes , and isasum ofenergiesofindependentmodes. The hamilto- then imposing anoni al ommutation relation between nian expressed in terms of plane wavesis diagonal. them. However, bosoni ommutation relations are vio- Oneofthemostimportantresultsofournumeri al al- latedin the lassi al(cid:28)eldsmethod(cid:21) lassi alamplitudes ulationsisthattotalenergyisevenlydistributedamong do ommute. Thus we will hoose a di(cid:27)erent approa h. allplanewavemodes. Theequilibriumstaterea heddur- First we will de(cid:28)ne dimensionless amplitudes. These ing nonlinear evolution is hara terized by equipartition amplitudes are the only dynami al obje ts appearing in of energy: ε =2ω2 β 2 =const. the lassi al (cid:28)eld method. In order to de(cid:28)ne them we k k| k| (16) havetorewriteEq.(15)usingquantitieswiththeirproper 5 dimensions: phonons(inthe` lassi allimit')thaninthethermalequi- kmax librium it should obey Bose statisti s: =m Ω2 A 2, 1 H k| k| (18) N = , k=Xkmin k e~(Ωk−µ)/kBT 1 (24) − Ω = (1/t )ω A = √2Lβ µ where k 0 k is the frequen y and k k where iΩs a hemi al potential. In the limit of low frek- k is a lassi al amplitude of ea h harmoni os illator. To quen ies , the equilibrium o upation of the mode de(cid:28)ne dimensionless amplitudes we need some universal an be approximated by: quantitywhi hhasdimensionofana tion. Su haquan- N ~(Ω µ)=k T. k k B − (25) tity does not exist in the lassi al theory. Therefore we have to~introdu e this onstant by hand. We will use ComparisonofEq. (25)andEq. (23)showsthato upa- symbol for this elementary a tion. At this moment its tion ofnormalmodesobtained in our al ulationsagrees value an be arbitrary. So far we have not mentioned withalowfrequen ylimitofBosestatisti sprovidedthat any physi al ondition whi h ould, at least in prin iple, µ=0 . determine its value. At a later stage this onstant will Equipartition of energy de(cid:28)nes the energy per mode. be identi(cid:28)ed with the Plan k onstant. k <kmax All modes with waveve tors | | are o upied by Let us de(cid:28)ne units of amplitudes: phononswhileothermodesareemptybe auseofthemo- ~ mentum ut-o(cid:27)usedintheimplementationofthemodel. A (k)= 1D 0 mΩ (19) Thisdistributionofenergyisa analogueofthePlan k r k distribution of the bla kbody radiation. The Plan k distribution says that energy density grows with frequen y and dimensionless amplitudes for ea h os illator: up to some maximal value and then falls exponentially B = L√2βk = 2mΩkLβ , to zero. This initial growth iskrelatedkt2o the phase spa e k A0(k) r ~ k (20) volu1mDe whi h in reases with as ∝ and is absent in the modelstudiedhere. Theexponentialde ayofthe B bla kbody energy density in repla ed by a sharp ut-o(cid:27) k k =k and express the Hamiltonian Eq.(18) in terms of : max at in our al ulations. kmax We will use the~similarities des ribed above to deter- H= ~Ωk|Bk|2. (21) mine the value of whi h will allow uNskto (cid:28)nd the abso- kXmin lute value of the number of phonons ~ . Following the Plan k idea we determine the value of by equating the temperatureofthesystemtotheenergy orrespondingto This form of the Hamiltonian Eq.(21) is a familiar one. the position of the maximum in the energy distribution. The energy of ea h plane wave mode is equal to the Ωk In our ase this `maximum' orresponds to the ut-o(cid:27): energy of elementary quantum of a given frequen y times some positive real number: ~Ωkmax =kBT. (26) N = B 2, k k | | (22) Eq. (26) plays a ru ial role in establishing a link between lassi al (cid:28)elds approa h and the quantum theory whi h an be alled a number of phonons. We want to of Bose system. It gives the value of the Plan k on- stresson emorethatnumberofphononsde(cid:28)nedaboveis stant up to a numeri al fa tor of the order of one. This notaninteger. Thereforeit anhaveaphysi almeaning fa t establishes a limitation of a ura y of the method, of the number of quasiparti les only if its value is large N in parti ular the a ura y of the number of parti les, or k as ompared to one. Moreover, the value of depend~s equivalently the value of the temperature of the system. still on the (arbitrary) value of the elementary a tion . By omparing Eq.(26) and Eq.(23) we see that number Be ausethereisnolimitationonthemaximalamplitude N of parti les o upying the mode with the largest energy k ofaharmoni os illator,thevalue of anbe arbitrar- is equal to one: ily large. Our lassi al (cid:28)eld annot therefore orrespond N =1. to parti les obeying fermioni statisti s. However it an kmax (27) orrespond to highly ex ited Bose (cid:28)eld. k Equipartition of energy Eqs. (16), (17), dis ussed in This means that o upation of all modes with smaller N k N >1 k max k the previous se tion, an be expressed in terms of : than isma ros opi , ,whi hisinagreement ~Ω N =k T, witha ommonunderstandingofthe lassi allimitofthe k k B (23) quantum (cid:28)eld. Indeed in the lassi al (cid:28)elds method the k k T = ǫT˜ number of modes used in numeri al implementation has B B where T˜ is the Boltzmann onstant and to be arefully adjusted. As suggested in [10℄ it should where is known from our numeri al simulations. On besu hthatthe o upationofthe highest mode isequal N k the other hands if orresponds to the number of to 1. 6 1 V. CONCLUDING REMARKS 0.5 In this paper we started from the des ription of n a purely lassi al system. Our numeri al simulations €€€€€€€0€€€€€ 0.05 N proved that nonlinear intera tions drive the system to- Ph wardsanequilibrium. Energyintheequilibriumisevenly k k max distributed among all modes of | | ≤ . Modes or- k > k max 0.005 responding to | | are not ex ited and do not ontributetothetotalenergy. Theyareabsentinourap- 0 0.02 0.04 0.06 0.08 0.1 proa hbe ause of the ut-o(cid:27) whi h is an essentialingre- T dientofthemethod. Inorderto `quantize'the dynami s we introdu ed the Plan k onstant. Then we ould de- k = 0 (cid:28)ne dimensionless amplitudes of ea h os illator and the FIG. 4: Relative number of phonons in mode as a number of ex itation quanta of ea h mode (cid:21) phonons. fun tion ofNtPemh p=era8t5ure for di(cid:27)erent totaNl Pnuhm=be1r7o5f quasi- Ho~wever, the number of phonons depends on the value parti les: (gray ir les) and (bla k λ ir= le1s). Points refer to numeri aαl s=im4u.l5ations of Eq.(4) with of whi h, at this stage of the appro~a h, an be arbi- . Lines representa (cid:28)t with . trary. In order to assign a value to we followed the Plan k's idea. We used the fa t that Plan k's distribution of energy leads to its equipartition for small values k k of and exponential drop for large . Energy distribu- ω In what follows we give an illustrative example of ap- tion rea hesamaximum~ωat freq=uekn yT kmax loseto this pli ation of the de(cid:28)ned abovepro edureof`quantization satisfying the relation kmax B . In our approa ~h of a s alar (cid:28)eld' studied in this paper. We solve the dy- the same ondition wasused to deNtermine the value of Ph nami al equations for various total energy of the system and thus the number of phonons . N andvariousnumberofgridpoints . Inour al ulations We believe that our studies shed more light on the N N =16 weusedrathersmallvaluesof rangingfrom up lassi al (cid:28)elds method used for des ription of a weakly N = 1024 to . We he k if the system rea hed the state intera tingBose-Einstein ondensate. Thewayofreason- of thermal equilibrium by determining a frequen y spe - ing goes the opposite way in `derivation' of this method. ω β trum k andamplitudes k atvariousstagesoftimeevo- One starts with quantum many body theory of a Bose lution. When the stationary state is rea hed we ontrol gas with shortgr/aVnge interga tions. Two parti le intera - equipartition of energy and determine the temperture. tion energy is wheVre is proportional to the s-wave Finally, o upation of di(cid:27)erent eigenmodes is obtained s attering length and is the volume. The number of from the following relation: ma ros opi ally o upied modes is being assumed ak pri- max ori by hoosing a value of the ut-o(cid:27) momentum . Nk = β|βk|2 2. (28) sAunbnsithitiulatteidonboyp elraastsoir salofamthpelsietusdeeles tαekdamnoddeHseaisreentbheerng | kmax| operator equations transform into equations for lassi al α k amplitudes : Evidently Nkmax = 1, what is equivaleNntPtho=the Nonkdi- i~ddtαk = ~22mk2αk+ gVNp α∗k1αk2αk1−k2+k, (29) tion (26). If total number of phonons is kX1,k2 di(cid:27)erent than assumed we have to repeat theP al ulations hangingthe number of gNrid points. In generalthe where Np is the total number of parti les. This quan- smallerthe energythe smaller hasto be usedin order NPh tity orresponds to the number of phonons i the to keep the same number of phonons. This way we are Np dis ussed lassi al model. Note that does not enter able to get o upations of all normalmodes as fun tions dynami alequationsalone(cid:21)it ismultiplied bytheinter- of temperature for a (cid:28)xed total number of phonons. g k = 0 a tion strength . k In Fig.(4) we present o upation of mode Relative o upation of di(cid:27)erent modes is losely re- as a fun tλion=of1temperature for nonlinear intera tion lated to the lassi al amplitudes: strength and two di(cid:27)erent total number of N phonons. The solid line represents a (cid:28)t to the numer- αk 2 = k =nk. | | N (30) i al results. The resultsshow that relativeo upationof p k =0 the spatially uniform mode grows quite rapidly at temperatures lose to zero. There is no phase transition Let us observe that the Plan k onstant appears in the in the studied system be ause, (cid:28)rst of all, our system equations right from the beginning. But its value an α k is one-dimensional, and also elementary ex itations are be arbitrary (cid:21) the lassi al amplitudes are a dire t B k massless. The main goal of these al ulations was to il- analogue of amplitudes of our model. lustratehowthe lassi al(cid:28)eldmethod worksin pra ti e. Dynami s of lassi al (cid:28)elds leads to the equipartition 7 of energy: The approa hwe used gives a new interpretation of this k T reasoning. ~ω n = B =const. k k N (31) Moreover, our studies show limits of the `predi tive p power'ofthe lassi al(cid:28)eldsmethod. Be ausethePlan k ~ωknk onstant is determined with limited a ura y all predi - Only the value of is known from numeri al al- tionsabout thetotal number ofparti lesortemperature ulations, therefore (31) allows for determination of the kBT/Np T Np ofthesystemarea urateuptoanumeri alfa torofthe ratio of but not the values of , separately. Np g order of one. It seems therefore that attempts of more Notethatboth and arenotuniquelydeterminedbe- gNp a uratedeterminationofthetemperature[11℄ annotbe ause only produ t is an initial ontrol parameter. free of the ambiguity related to the position of the ut- A ordingto the pres~ent studies this problem an be re- o(cid:27). Inadditionthe lassi al(cid:28)eldsmethod annotbeused solved if the value of is determined by the requirement fordes riptionofverysubtlee(cid:27)e tssu hasashift ofthe thattheenergydistributionof lassi al(cid:28)eldsmimi s the riti altemperatureofBose-Einstein ondensationdueto quantum distribution (cid:21) the position of maximum ought intera tions. Foratypi alsystemthisshift isverysmall. to be approximately given by Some e(cid:27)orts [3℄ of obtaining the shift of riti al tem- ~ω =k T. kmax B (32) peratureusing lassi al(cid:28)eldsmethodmustbeinevitably biased by the approximate hara terof Eq.(32). 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