Equation of state and sound velocity of a hadronic gas with a hard-core interaction

Thermodynamic properties of hot and dense hadronic systems with a hard-sphere interaction are calculated in the Boltzmann approximation. Two parametrizations of pressure as a function of density are considered: the first one, used in the excluded-volume model and the second one, suggested earlier by Carnahan and Starling. The results are given for one-component systems containing only nucleons or pions, as well as for chemically equilibrated mixtures of different hadronic species. It is shown that the Carnahan–Starling approach can be used in a much broader range of hadronic densities as compared to the excluded-volume model. In this case the superluminal sound velocities appear only at very high densities, in the region where the deconfinement effects should be already important.

Figure 1

Compressibility factor $Z$ and function $\psi$ [see Eq. (21)] for different values of packing fraction $\eta$ calculated within the EVM and CSA.

Figure 2

Sound velocity of nucleon gas as a function of density for two values of temperature $T=100$ (thick lines) and 200 (thin lines) MeV. The solid and dashed curves are calculated by using compressibility functions proposed in the CSA and EVM. The dash-dotted lines are obtained in the limit of point-like nucleons $\left(R=0\right)$.

Figure 3

Sound velocity of nucleonic matter as a function of density for different values of parameter $R$. The dashed and solid curves are calculated, respectively, in the EVM and CSA. The dash-dotted line corresponds to point-like nucleons.

Figure 4

Chemical potential of nucleon gas as a function of (a) density and (b) pressure calculated in the CSA (solid lines) and EVM (dashed lines) at temperatures of 100 and 200 MeV. The dash-dotted lines correspond to point-like nucleons.

Figure 5

Equilibrium density of pions as a function of temperature for different values of parameter $R$. The dashed and solid curves are obtained, respectively, in the EVM and CSA. The dash-dotted line corresponds to ideal gas of point-like pions.

Figure 6

Sound velocity of pion gas as a function of temperature for different values of parameter $R$. The dashed and solid curves are calculated, respectively, in the EVM and CSA. The dash-dotted line corresponds to point-like pions.

Figure 7

(a) The average energy per baryon of $N+\mathrm{\Delta }$ matter $E/B$ and partial contributions of nucleons and $\mathrm{\Delta }\mathrm{s}$ as functions of temperature (all energies are given in GeV). (b) The dashed, solid, and dash-dotted curves show, respectively, relative contributions of resonances to energy, baryon charge, and heat capacity. The dotted line shows temperature dependence of the total heat capacity per baryon minus 3/2.

Figure 8

Sound velocity of $N+\mathrm{\Delta }$ matter as a function of temperature for several values of baryon density $n_B$. Thick and thin curves give the results of CSA and EVM, respectively.

Figure 9

Sound velocities of baryon matter as functions of temperature for different values of baryon density $n_B$. Thin and thick lines correspond, respectively, to the nucleon matter and to the $N+\mathrm{\Delta }$ mixture. All calculations are made in the CSA.

Figure 10

Equilibrium pion density in $\pi +N+\mathrm{\Delta }$ mixture as a function of baryon density for $T=200\text{MeV}$. The solid and short-dashed curves are calculated within CSA and EVM assuming equal sizes of hadrons. The long-dashed line is obtained in the limit of point-like pions. The dashed-dotted curve corresponds to the ideal gas.

Figure 11

Sound velocity of $\pi +N+\mathrm{\Delta }$ matter as a function of baryon density for (a) $T=150\text{MeV}$ and (b) 200 MeV. Thick and thin curves give, respectively, the results of CSA and EVM. The solid lines correspond to the case of equal sizes of baryons and pions. The dashed lines show the results in the limit of point-like pions. The dash-dotted curves correspond to the ideal gas.

Figure 12

Sound velocity of $\pi +N+\mathrm{\Delta }$ matter as a function of temperature for different values of baryon density $n_B$. Thick and thin lines are calculated within the CSA assuming, respectively, the pion radii $R_{\pi }=0.39\text{fm}$ and $R_{\pi }=0$.

Figure 13

Total density of hadronic matter as well as its contributions from baryons, antibaryons, and mesons as functions of the net baryon density $n_B$ for (a) $T=150\text{MeV}$ and (b) $200\text{MeV}$. All calculations are made in the CSA assuming the hadronic radius $R=0.39\text{fm}$.

Figure 14

Sound velocity of hadronic matter as a function of net baryon density for (a) $T=150\text{MeV}$ and (b) $200\text{MeV}$. The solid and dashed lines correspond, respectively to the (anti)baryon-meson and $N+\mathrm{\Delta }+\pi$ matter. The calculations are made in the CSA assuming the radius $R=0.39\text{fm}$ for all hadrons. The dash-dotted lines show the results in the limit of ideal gas.

Figure 15

Pressure of hadronic gas as a function of temperature for different values of baryon chemical potential ${\mu }_B$. The EVM calculations, assuming the radius $R=0.39\text{fm}$ for all hadrons, are shown by thick lines. Thin curves show the results in the Boltzmann approximation.

Figure 16

Same as Fig. 15 but for baryon density $n_B$.