Hadronic Equation of State and Speed of Sound in Thermal and Dense Medium

The equation of state $p(\epsilon)$ and speed of sound squared $c_s^2$ are studied in grand canonical ensemble of all hadron resonances having masses $\leq 2\,$GeV. This large ensemble is divided into strange and non-strange hadron resonances and furthermore to pionic, bosonic and femionic sectors. It is found that the pions represent the main contributors to $c_s^2$ and other thermodynamic quantities including the equation of state $p(\epsilon)$ at low temperatures. At high temperatures, the main contributions are added in by the massive hadron resonances. The speed of sound squared can be calculated from the derivative of pressure with respect to the energy density, $\partial p/\partial \epsilon$, or from the entropy-specific heat ratio, $s/c_v$. It is concluded that the physics of these two expressions is not necessarily identical. They are distinguishable below and above the critical temperature $T_c$. This behavior is observed at vanishing and finite chemical potential. At high temperatures, both expressions get very close to each other and both of them approach the asymptotic value, $1/3$. In the HRG results, which are only valid below $T_c$, the difference decreases with increasing the temperature and almost vanishes near $T_c$. It is concluded that the HRG model can very well reproduce the results of the lattice quantum chromodynamics (QCD) of $\partial p/\partial \epsilon$ and $s/c_v$, especially at finite chemical potential. In light of this, energy fluctuations and other collective phenomena associated with the specific heat might be present in the HRG model. At fixed temperatures, it is found that $c_s^2$ is not sensitive to the chemical potential.