2 tum spectra of thermal quarks which are parameterized by the blast wave model [17], | y | < 1 for the quark and hadronic phases. We can see that ( P V )$_{h}$ (for hadrons) is slightly smaller than ( P V )$_{q}$ (for quarks) because where ξ = r/R$_{max}$ with R$_{max}$ the maximum radius of the thermal source, T$_{f}$ = 165 MeV is the local tempera- ture, ρ = tanh − $^{1}$( β$_{s}$ξ $^{n}$) with β$_{s}$ the radial velocity on the surface of the firsball and n = 0 . 3 a parameter, I$_{0}$ and K$_{1}$ are the modified Bessel functions. The average radial velocity is given by 〈 β$_{r}$ 〉 = 2 β$_{s}$/ ( n + 2) . By fitting the data for transverse momentum spectra of hadrons we can fix the parameters β$_{s}$ or equivalently 〈 β$_{r}$ 〉 , see Tab. II. With these inputs the available data for the transverse momentum spectra and rapidity density of hadrons have been well described [14–16]. The reason we only consider thermal quarks is that the entropy is a bulk property which is dominated by the small p$_{T}$ region. Equipped by the above settings, we are able to calcu- late the entropy of the system before and after hadroniza- tion. Given all particle momentum distributions, one way to obtain the entropy of a system is through the Duhem- Gibbs relation [18], where S, E , P , V and T are the entropy, energy, pressure, volume, and temperature of the system respectively, and µ$_{i}$ and N$_{i}$ are the chemical potential and number for the quark/hadron species i respectively. Note that the sys- tem will undergo expansion during hadronization which is indeed a dynamic and real time process and takes place in local volume. The global volume of an expanding sys- tem is not well defined in a Lorentz invariant way. Here the volume V is only an effective parameter to character- ize the average particle occupancy in spatial dimension. The chemical potential µ$_{i}$ can be determined from the multiplicity ratio of the particle i and to its anti-particle i via N$_{i}$/N$_{i}$ = e 2 µ i /T where T is set to 170 MeV for all particles. The quantities on the right hand side of Eq. (2) are all known with, where p ≡ | p | and ε$_{i}$ = √ p 2 + m 2 i denotes the scalar mo- mentum and energy respectively. For constituent quarks we choose m$_{u}$ = m$_{d}$ = 330 MeV and m$_{s}$ = 550 MeV. In Tab. I we list the total E , P V , µN ≡ ∑ $_{i}$µ$_{i}$ N$_{i}$ and T S per unit rapidity in the central rapidity region where in the last line is due to that quarks with the same momentum direction combine. We also see that µN is very small compared to P V and E since in high energy collisions almost all quarks in central rapidity are ex- cited in vacuum. Note that the energy per unit rapidity are not exactly equal for the two phases and the differ- ence is within a few percent. The effect of the energy non-conservation is the result of the momentum conser- vation imposed during the course of a specific combina- tion, because energy and mometum conservation cannot be simultaneously fullfilled with fixed quark masses. For example two light constituent quarks with the total mass 660 MeV combine into a pion of mass 140 MeV, the en- ergy of pion is less than that of two quarks for small momenta. But the non-conservation effect is small on average. With ( P V )$_{q}$ ≈ ( P V )$_{h}$ and V$_{q}$ < V$_{h}$ we obtain P$_{q}$ > P$_{h}$ and then P$_{q}$ − B = P$_{h}$ ( B is the bag constant), which is consistent to the condition for phase transition. Finally we see that T S per unit rapidity is almost con- stant for two phases at each collisional energy upto a few percent. If we assume a sudden hadronization the tem- perature should be the same for the two phases, then we have the approimate conservation of the total entropy in hadronization. Actually the hadronization time is finite during which the system expands and cools down. There- fore the total entropy S of the hadronic phase (with lower temperature) is larger than that of the quark phase (with higher temperature). All above observations are about the directly produced hadrons right after the combina- tion. For the final state hadrons after resonance decays we also see that E , P V , µN and T S do not change much as compared to the directly produced hadrons. This ob- servation that the resonance decays cannot compensate much to the total entropy is different from previous ar- guments that resonant decays might be important. Another way of obtaining the entropy is directly through the formula, where d$_{i}$ are degeneracy factors for quark or hadron species i , the signs + / − are for bosons/fermions, and g$_{i}$ ( r , p ) are phase-space distributions satisfying S = ∑ i d$_{i}$ ˆ d $^{3}$rd$^{3}$p (2 π ) 3 {− g$_{i}$ ( r , p ) ln g$_{i}$ ( r , p ) ± [1 ± g$_{i}$ ( r , p )] ln[1 ± g$_{i}$ ( r , p )] } , (5) E = ∑ i ˆ d $^{3}$pε$_{i}$ f$_{i}$ ( p ) , P V = ∑ i ˆ d $^{3}$p p 2 3 ε$_{i}$ f$_{i}$ ( p ) , N$_{i}$ = ˆ d $^{3}$pf$_{i}$ ( p ) , (3) S = 1 T ( E + P V − ∑ i µ$_{i}$N$_{i}$ ) , (2) p 2 1 3 ε$_{1}$ + p 2 3 ε$_{2}$ − ( p$_{1}$ + p$_{2}$ ) 2 3( ε$_{1}$ + ε$_{2}$ ) = [ ( p$_{1}$ε$_{2}$ − p$_{2}$ε$_{1}$ ) 2 + 2 p$_{1}$p$_{2}$ε$_{1}$ε$_{2}$ ( 1 − p 1 · p 2 p$_{1}$p$_{2}$ )] / [3 ε$_{1}$ε$_{2}$ ( ε$_{1}$ + ε$_{2}$ )] ≈ ( p$_{1}$ε$_{2}$ − p$_{2}$ε$_{1}$ ) 2 3 ε$_{1}$ε$_{2}$ ( ε$_{1}$ + ε$_{2}$ ) ≳ 0 , (4) f$_{i}$ ( y, p $_{T}$) ∝ ˆ 1 0 dξξ √ p 2 T + m 2 $_{i}$I$_{0}$ ( p$_{T}$ sinh ρ T$_{f}$ ) × K$_{1}$ ( p$_{T}$ cosh ρ T$_{f}$ ) , (1)