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´ d $^{3}$r (2 π ) 3 d$_{i}$g$_{i}$ ( r , p ) = f$_{i}$ ( p ) . In the classical limit where g$_{i}$ ( r , p ) ≪ 1 , one reaches, 

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We can take pions which are most populated particles as an example to estimate the order of magnitude of the phase space distribution. The multiplicity rapid- ity density of π + in most central collisions (centrality 0-5%) at 200 GeV A is about 320 . Suppose most pi- ons are limited inside a momentum volume of the size (Δ p ) 3 ∼ 2 πp 2 T cosh y (Δ p$_{T}$ Δ y ) ∼ 10 GeV 3 and inside the spatial volume of about V$_{h}$ ∼ 2200 fm $^{3}$, then we get g$_{π}$ ( r , p ) ∼ 0 . 027 . This provides an upper limit, for other particles g$_{i}$ ( r , p ) ≪ 1 , therefore Eq. (6) is a quite good approximation of Eq. (5). 

In order to calculate the entropy via Eq. (5-6) from the known momentum distributions f$_{i}$ ( p ) in the QCM, we make an approximation or give an estimate for g$_{i}$ ( r , p ) by g$_{i}$ ( r , p ) ≈ [(2 π ) $^{3}$/ ( d$_{i}$V )] f$_{i}$ ( p ) , where V = V$_{q}$, V$_{h}$ are the effective volumes of quark and hadronic phases respec- tively. We can then obtain the entropy ratio R = S$_{h}$/S$_{q}$ as functions of volume ratio x = V$_{h}$/V$_{q}$ . The volumes 

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V$_{h}$ of the hadronic phase is taken to be the chemical freezeout values of the fireball which are extracted from the thermal model [19, 20] at four collisional energies, V$_{h}$ (fm $^{3}$) = 1125 (17.3 GeV), 1620 (62.4 GeV), 1900 (130 GeV), 2200 (200 GeV). The results are shown in Fig. 1. For the above set of values of V$_{h}$ , only if x > 2 . 5 − 3 . 0 , which corresponds to about 1.35-1.44 for the ratio of the fireball radii in the hadronic to quark phase, should the entropy of the hadronic phase be larger than that of the quark one. 

As a contrast we also consider an ideal case for con- stituent quarks whose phase space distribution follows Fermi-Dirac distribution, g$_{i}$ ( r , p ) = 1 / [exp( ε$_{i}$/T ) + 1] . The hadron spectra can be determined by combination of quarks. Then we use the Duhem-Gibbs relation (2) and entropy formula (5) to obtain the entropies for quark and hadronic phases. The result from the Duhem-Gibbs relation is ( E, P V, T S ) = (0.61,0.145,0.755) (quark), (0.59,0.136,0.726) (hadron) for unit volume of quark phase. We can also see that T S is almost a constant, same as in the real case. The result from the entropy for- mula shows the same behavior as in Fig. 1 except that the entropy of the hadronic phase is larger than that of the quark one happens when x ≳ 16 . 

A few comments about our approach and results are needed. Our results show that the increase of the en- tropy requires an effective and adequate expansion of the fireball during hadronization, which also implies that hadronization takes finite time to complete. Actually in hydrodynamic simulation of the evolution of the fire- ball, hadronization is indeed a dynamic process with non- trivial space-time profile, i.e. it does not take place uni- formly in the same space-time but on the freezeout hy- persurface, so different part of the fireball hadronizes in 

