Conventional cluster and virial expansions are generalized to momentum dependent interparticle potentials. The model with Lorentz contracted hard core potentials is considered, e.g. as hadron gas model. A Van der Waals-type model with a temperature dependent excluded volume is derived. Lorentz contraction effects at given temperature are stronger for light particles and make their effective excluded volume smaller than that of heavy ones

Bugaev, Kyrill A.

Gorenstein, Mark I.

Greiner, Walter

Stöcker, Horst

English

Hochschulschriftenserver - Universität Frankfurt am Main

arXiv:nucl-th/0004061 v1   26 Apr 2000Van der Waals Excluded Volume Model forLorentz Contracted Rigid SpheresK.A. Bugaev1,2, M.I. Gorenstein1,2, H. St¨ ocker1 and W. Greiner11Institut f¨ ur Theoretische Physik, Universit¨ at Frankfurt, Germany2Bogolyubov Institute for Theoretical Physics, Kiev, UkraineAbstractConventional cluster and virial expansions are generalized to momentum dependent inter-particle potentials. The model with Lorentz contracted hard core potentials is considered, e.g.as hadron gas model. A Van der Waals-type model with a temperature dependent excludedvolume is derived. Lorentz contraction eﬀects at given temperature are stronger for lightparticles and make their eﬀective excluded volume smaller than that of heavy ones.Key words: Cluster integrals, hard-core, Van der Waals model, Lorentz contraction.The Van der Waals (VdW) excluded volume model has been used to describe hadronyields in relativistic nucleus–nucleus collisions (see e.g. [1, 2] and references therein). Thismodel treats the hadrons as hard-core spheres and, therefore, takes into account the hadronrepulsion at short distances. In a relativistic situation one should, however, include the Lorentzcontraction of the hard core-hadrons. This problem was discussed in the literature (see e.g.Ref. [3, 4]). In this paper the cluster and virial expansions are generalized to velocity dependentinter-particle potentials. This extension is used to construct the VdW model for Lorentzcontracted rigid spheres which may be used to simulate hadrons.The canonical partition function for the gas of N classical (Boltzmann) particles takes theformZN(V,T) =1N!  N  i=1 g dridki(2π)3 exp −ωiT  exp −UT , (1)where V and T are the system volume and temperature, g is the number of internal degreesof freedom (degeneracy factor) of the particles, ωi = (m2 + k2i)1/2 is the dispersion relation offree particles with masses m. The particle interactions described by the function U in Eq. (1)are given by the sum over pair potentials:U = 1≤i<j≤Nuij . (2)In contrast to the usual statistical mechanic treatment of the pair potentials, the uij areassumed to be both coordinate and momentum dependent uij ≡ u(ri,ki; rj,kj). This gen-eralization is necessary, if Lorentz contraction eﬀects of hard spheres are to be taken intoaccount. Introducing the Mayer functionsfij = exp −uijT − 1 , (3)Eq. (1) can be presented asZN(V,T) =1N! dx1...dxN exp −ω1 + ... + ωNT   1≤i<j≤N(1 + fij) , (4)with the short notation dxi ≡ gdridki/(2π)3. Similarly to the standard procedure one canintroduce the cluster integrals [5]b1 =1V dx1 exp −ω1T =g T 32π2 K2 mT ≡ φ(T) , (5)b2 =12!V dx1dx2 exp −ω1 + ω2T f12 , (6)b3 =13!V dx1dx2dx3 exp −ω1 + ω2 + ω3T (f12f13 (7)+ f12f23 + f13f23 + f12f23f13) ,...and present the canonical partition function in the familiar formZN(V,T) =′  {ml}N  l=1(V bl)mlml!, (8)where the summation in Eq. (8) is taken over all sets of non-negative integer numbers {ml}satisfying the conditionN  l=1lml = N . (9)Note, however, that the cluster integrals deﬁned above are diﬀerent from those used in standardstatistical mechanics [5] as here nontrivial momentum integrations are included. Condition(9) makes the calculation of ZN (8) rather complicated. This problem can be avoided in thegrand canonical ensemble: the grand canonical partition function can be calculated explicitly(z ≡ exp( /T)):Z(V,T, ) ≡∞  N=0exp  NT ZN(V,T) = exp V∞  l=1blzl . (10)In the thermodynamical limit the pressure p and particle number density n are calculated inthe grand canonical ensemble in terms of the asymptotic values of the cluster integrals:p = T limV →∞lnZV= T∞  l=1blzl , (11)n = limV →∞1V∂ lnZ∂z=∞  l=1lblzl . (12)The virial expansion represents the pressure in terms of a series of particle number densityand takes the formp = T∞  l=1alnl . (13)Substituting p (11) and n (12) into Eq. (13) and equating the coeﬃcients of each power of z,one ﬁnds the virial coeﬃcients al in terms of the cluster integralsa1 = 1 , a2 = −b2b21, a3 =4b22b41−2b3b31, ... (14)Let us recall, ﬁrst, the derivation of the standard VdW excluded volume model. Then itis extended by adding the Lorentz contraction of the moving particles. Keeping the ﬁrst twoterms of the virial expansion (13) the following result is obtained:p(T,n) = Tn (1 + a2n) . (15)It is valid for small particle densities (i.e. n << 1/a2). The usual (momentum independent)hard core potential for spherical particles with radius ro is uij = u(|ri−rj|). Here the functionu(r) equals to 0 for r > 2ro and ∞ for r < 2ro. The second cluster integral (6) can easily becalculated in this case:b2 = − φ2(T)16π3r3o . (16)Therefore a2 = 4vo, where vo = 4πr3o/3 is the particle hard core volume. The VdW excludedvolume model is obtained as the extrapolation of Eq. (15) to large particle densities in theformp(T,n) =Tn1 − a2n. (17)For practical use the pressure is given as a function of T and   independent variables, i.e.in the grand canonical ensemble. This is done by substituting n = (∂p/∂ )T into Eq. (17),which then turns into a partial diﬀerential equation for the function p(T, ). For the VdWmodel (17) the solution of this partial diﬀerential equation can be presented in the form of atranscendental equationp(T, ) = Tφ(T)eµ/T exp −a2pT ≡ pid(T,  − a2p) . (18)Eq. (18) was ﬁrst obtained in Ref. [6] using the Laplas transform technique. With p(T, )(the solution of Eq. (18)) the particle number density, entropy density and energy density arecalculated as (ν =   − a2 p(T, ), a2 = 4vo):n(T, ) ≡ ∂p(T, )∂  T=nid(T,ν)1 + a2nid(T,ν), (19)s(T, ) ≡ ∂p(T, )∂T µ=sid(T,ν)1 + a2nid(T,ν), (20)ǫ(T, ) ≡ Ts − p +  n =ǫid(T,ν)1 + a2nid(T,ν). (21)Here the superscripts id in the thermodynamical functions (18–21) indicate those of the idealgas.The excluded volume eﬀect accounts for the blocked volume of two spheres when theytouch each other. If hard-sphere particles move with relativistic velocities it is necessary toinclude their Lorentz contraction in the rest frame of the ﬂuid. The model suggested in Ref.[4] is not satisfactory: the parameter a2 = 4vo of the VdW excluded volume model is confusedthere with the proper volume of an individual particle – the contraction eﬀect is introducedfor the proper volume of each particle. In order to get the correct result it is necessary toaccount for the excluded volume of two Lorentz contracted spheres.Let ri and rj be the coordinates of the i-th and j-th particle, respectively, and ki andkj be their momenta, ˆ rij denotes the unit vector ˆ rij = rij/|rij| (rij = |ri − rj|). Then fora given set of vectors (ˆ rij,ki,kj) for the Lorentz contracted rigid spheres of radius ro thereexists the minimum distance between their centers rij(ˆ rij;ki,kj) = min|rij|. The dependenceof the potentials uij on the coordinates ri,rj and momenta ki,kj can be given in terms of theminimal distance as followsu(ri,ki;rj,kj) =  0, |ri − rj| > rij (ˆ rij;ki,kj) ,∞, |ri − rj| ≤ rij (ˆ rij;ki,kj) .(22)The general approach to the cluster- and virial expansions described above is valid for thismomentum dependent potential, and it leads to Eqs. (17,18) witha2(T) =12φ2(T)  dk1dk2(2π)6 exp −ω1 + ω2T × (23)× dr12 Θ(r12(ˆ r12;k1,k2) − |r12|) .The new feature is the temperature dependence of the excluded volume a2 (23) which is dueto the Lorentz contraction of the rigid spheres. The pressure and particle number density arestill given by Eqs. (18,19), but with temperature dependent a2(T) (23). However, Eqs. (20,21) are now modiﬁed, e.g.ǫ(T, ) =ǫid(T,ν) − p2 da2(T)/dT1 + a2nid(T,ν). (24)In contrast to Eq. (21) the energy density (24) contains the extra term which appears also inthe entropy density. The excluded volume a2(T) (23) is always smaller than 4vo. It has beenproven rigorously that a2(T) is a monotonously decreasing function of T and, therefore, theadditional term in Eq. (24) is always positive. Let us introduce the notationa2(T) = 4vo f(T) . (25)The function f(T) depends on the T/m ratio. It can be calculated numerically and its behavioris shown in Fig. 1. The simple analytical formulaf(T) = c + (1 − c)ρs(T)φ(T)(26)withc = 1 +749π −1, ρs =g(2π)3 dkmωexp −ωT ,is found to be valid with an accuracy of a few percents for all temperatures. The asymptoticbehavior of f(T) is the following: 1 − O(T/m) at T << m and c + O(m/T) at T >> m.If one assumes that all types of hadrons have at rest the same hard core radius then theLorentz contraction eﬀect leads to diﬀerent VdW excluded volumes for moving particles withdiﬀerent masses: for light particles (e.g. pions) the excluded volume (at given T) is smallerthan that for heavy ones. Fig. 1 shows that at T ∼ = 150 MeV the value of a2 in the nucleongas (m ∼ = 939 MeV) decreases by 10% in comparison to its nonrelativistic value 4vo, whereasfor pions (m ∼ = 140 MeV) a2 shrinks at the same T by almost a factor 2. This is simplybecause light particles are more relativistic than heavy ones at given temperature typical forhigh energy nuclear collisions, T = 120 ÷ 170 MeV.As an example, Fig. 2 shows the particle number density of the pion gas (  = 0,g = 3)with ro = 0.5 fm. The particle number density is calculated according to Eq. (19) for threediﬀerent models: the ideal pion gas (a2 = 0), the VdW model with constant excluded volume(a2 = 4vo) and the VdW model with Lorentz contraction (a2(T) is given by Eq. (23)). It canbe seen from Fig. 2 that at low T the pion density is small and excluded volume corrections areunimportant. Therefore, all three models are similar. The situation changes with increasingT: the suppression due to the excluded volume eﬀects are large and diﬀerent for a2 = 4vo anda2(T) (23). The ratios of particle number densities and energy densities of the pion gas fortwo versions of the VdW model (a2 = 4vo and a2(T) (23)) are shown in Fig. 3 as functionsof the temperature. From Fig. 3 one can observe the deviations between these two models.These deviations increase with temperature. They are larger for the energy density due to theadditional positive term in Eq. (24).In conclusion, the traditional cluster and virial expansions can be consistently generalizedto momentum dependent pair potentials. Hard-core potentials with Lorentz contraction eﬀectslead to a VdW model with a temperature dependent excluded volume a2(T) (23). For lightparticles the eﬀect of Lorentz contraction is, evidently, stronger than for heavy ones. Notethat smaller values of the pion hard core radius rπ were introduced in Refs. [7, 8] within thestandard VdW excluded volume model to ﬁt hadron yield data better. The smaller value ofthe pion excluded volume appears as a consequence of stronger Lorentz contraction for lightparticles.AcknowledgmentsThe authors would like to thank R. Pisarski for valuable comments and L. McLerran,D.H. Rischke and R. Venugopalan for discussions. K.A.B. gratefully acknowledges the warmhospitality of the BNL Nuclear Theory Group and the BNL-RIKEN Center, where parts ofthis work were done. M.I.G. acknowledges ﬁnancial support of DFG, Germany. K.A.B. isgrateful to the Alexander von Humboldt Foundation for the ﬁnancial support.References[1] G.D. Yen and M.I. Gorenstein, Phys. Rev. C 59 (1999) 2788.[2] P. Braun-Munzinger, I.Heppe and J. Stachel, Phys. Lett. B 465 (1999) 15.[3] J.I. Kapusta, and K.A. Olive, Nucl. Phys. A 408 (1983) 478.[4] Q.R. Zhang, Z. Phys. A 353 (1995) 345.[5] J.E. Mayer and M. Goeppert-Mayer, ”Statistical Mechanics” (1977).[6] D.H. Rischke, M.I. Gorenstein, H. St¨ ocker and W. Greiner, Z. Phys. C51 (1991) 485.[7] R.A. Ritchie, M.I. Gorenstein and H.G. Miller, Z. Phys. C 75, 2002 (1997).[8] G.D. Yen, M.I. Gorenstein, W. Greiner and S.N. Yang, Phys. Rev. C 56, 2210 (1997).0 0.5 1 1.5 2  T / m00.511.5 f(T)Fig. 1. f(T) as the function of the temperature-to-mass ratio. For heavy particles (e.g.,nucleons m >> T) the volume reduction is just a few per cents, whereas for pions (m ≈ T) itis about 50%.50 100 150 200  T (MeV)0.00.10.2 ρ (fm−3)Fig. 2. The particle number density for three models of the pion gas (  = 0,g = 3): thesolid line corresponds to the VdW model of the Lorentz contracted spheres (ro = 0.5 fm), thedashed one corresponds to the ideal gas of point-like particles, and the dotted one correspondsto the VdW model without Lorentz contraction for the spheres of a constant radius 0.5 fm.0 50 100 150 200  T (MeV)11.251.51.75Fig. 3. The dashed line shows the ratio of the particle number densities of the pion gas(g = 3,  = 0,ro = 0.5 fm): the VdW model with Lorentz contraction divided by the VdWmodel without Lorentz contraction. The solid line shows a similar ratio for the energy densities.

Van der Waals excluded volume model for Lorentz contracted rigid spheres

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