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the UrQMD model hadron resonance gas (without HS) yields a constant value of c 2 s = 0 . 18 [24]. 

To gauge the dynamics of hadron-hadron interaction, we also investigate the thermodynamic properties in an independent statistical model for an hadron resonance gas in the grand canonical ensemble. The particle densi- ties in the statistical model is given by [31] 

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where the ± sign refers to fermions/bosons. For the i th species, g$_{i}$ is the spin-isospin degeneracy and µ$_{i}$ = µ$_{B}$B$_{i}$ + µ$_{S}$S$_{i}$ is the chemical potential with µ$_{B}$ and µ$_{S}$ the baryon and strangeness chemical potentials, respec- tively. For the observed resonances, ρ$_{i}$ ( m ) is taken as Breit-Wigner mass distribution while for HS it is replaced by ρ$_{HS}$ of Eq. (1). The results are found to be rather insensitive to the choice of upper mass limit of integra- tion for HS m$_{max}$ ≥ 40 GeV. In the limit m$_{max}$ → ∞ , as the estimated exponent α = 2 . 44 in Eq. (1), the point- particle energy density, pressure and entropy density of an ideal classical (Boltzmann) Hagedorn gas remain fi- nite at T ≤ T$_{H}$ and diverge at T > T$_{H}$ exhibiting critical behavior [27]; albeit in the excluded-volume approach, the corresponding quantities evaluated at T ≤ T$_{H}$ = T$_{c}$ would remain finite. 

In Fig. 2 (left panel) we also show the equation of state obtained in the statistical model. Compared to a pion gas (stars), inclusion of measured discrete resonances up to mass ∼ 2 GeV (triangles) results in a considerable de- crease in the system pressure. The interaction measure, 

Θ = ( ε − 3 P ) /T $^{4}$, t hen gradually increases as T → T$_{c}$ . Further inclusion of Hagedorn states in the statistical model opens up add itional degrees of freedom and pro- vides an attractive interaction that slows the pressure increase at ε > ∼ 1 . 5 G eV/fm $^{3}$. The trace anomaly Θ then increases dramatica lly near T = T$_{H}$ . The speed of sound in the statistical mo del with HS turns out to be rather small c 2 s ≈ 0 . 07 at T$_{c}$ . Comparison of this curve with se- quential binary emiss ion model result provides a measure of the dynamics of r epulsive interaction in the medium. The interaction effec t starts at temperature above the pion mass and becom es significant at high energies where massive resonances a re produced. 

The point-particle entropy density of the system can be computed using t he Gibbs formula 

where ρ$_{B}$ = ∑ i B$_{i}$ρ$_{i}$ is the net baryon density and the (initial) net strangeness density ρ$_{S}$ is set to zero in our calculation. The baryon chemical potential µ$_{B}$ is evalu- ated from the particle yield ratios at equilibrium. Fig- ure 2 (right panel) shows the excluded-volume entropy density [16] as a function of temperature at zero baryon chemical potential. In the statistical model, compared to the pion gas, larger degrees of freedom in the hadron (and Hagedorn) resonance gas at the same energy density low- ers its temperature which results in higher entropy [32]. In the sequential emission model the increase (or slope) of entropy density with temperature near T$_{c}$ is somewhat less dramatic and captures the trend associated with that of a crossover transition obtained in the lattice data [25]. 

We now extract the shear viscosity for the dynamically evolving system of an equilibrated hadron resonance gas with Hagedorn states using the Kubo relation [33]. The transport coefficients determine the dynamics of fluctu- ations of dissipative fluxes about the equilibrated state. The Kubo formalism employs the linear response theory to relate the transport coefficients as correlations of dissi- pative fluxes by considering the dissipative fluxes as per- turbations to local thermodynamic equilibrium [33–35]. The Green-Kubo relation for shear viscosity is 

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Here t refers to time after the system equilibrates which is set at t = 0, and π xy is the shear component (traceless part) of the energy momentum tensor π $^{µν}$: 

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The equivalence accounts for the point particles used in our simulation. The averaging in Eq. (6) is over the ensemble of events generated in the simulation. 

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