0 5 10 15 20 t (fm/c) 10 -7 10 -6 10 -5 xy (0)  xy (t) (GeV 2 /fm 6 ) 145 154 170 178 185 196 0 100 200 300 400 t (fm/c) 10 -8 10 -7 69 75 82 160 180 200 T (MeV) 2 4 6  (fm/c) T (MeV) T (MeV) 50 100 150 200 T (MeV) 0.1 1.0 10.0 /s Hadron gas + HS:  $_{B}$ = 0 Hadron gas + HS:  $_{B}$/T   0.1 Hadron gas + HS:  $_{B}$/T  UrQMD Hadron gas:  $_{B}$ = 0 KSS bound: 1/4 The shear viscosity to entropy density ratio η/s for the Hagedorn r esonance gas in equilibrium at zero baryon chemical po tential is presented in Fig. 4. Also shown in the figur e are the η/s values for hadron resonance gas without Hagedorn states in the UrQMD model [24]. Our results for η/s are comparable to the UrQMD esti- mates at te mperatures T < 150 MeV. At higher temper- atures, mas sive resonances contribute dominantly result- ing in gradu al reduction of η/s with temperature [30] and reaches a m inimum of ( η/s )$_{min}$ ≈ 0 . 34 at T$_{c}$ = 196 MeV. In contrast, the UrQMD model calculation shows almost a constant value for η/s ≈ 0 . 9 at T ≈ 125 − 166 MeV. The reduct io n of η/s with Hagedorn mass spectrum can be realized as primarily due to enhancement of massive resonances t hat leads to a decrease in the mean free path of a particle and a corresponding increase in the average binary co llis ion cross section σ . In classical transport theory, as η ∼ p/σ (where p is the average momentum of the particle), implies a reduction of shear viscosity. The ( η/s )$_{min}$ ≈ 0 . 34 at zero baryon chemical potential for the equilibrated Hagedorn resonance gas is significantly above the KSS bound of 1 / 4 π [9] while somewhat close to the upper bound of 0.24 obtained from viscous hydro- dynamic analysis [7, 8] of elliptic flow; albeit η/s is as- sumed constant throughout the hydrodynamic evolution of the system. Note however, for Hagedorn resonance gas treated within the rate equation approach [30], an upper limit of η/s estimated in the kinetic theory is found to be as small as the KSS bound near T$_{c}$ . The measured antibaryon-to-baryon ratio B/B ≈ 0 . 8 at midrapidity for central Au+Au collision at RHIC [1– 4] suggests the hadronic matter formed possess, though small, but a finite net baryon number density ρ$_{B}$ . Dur- ing the late hadronic stage of evolution essentially from chemical equilibriu m to thermal equilibrium the system could acquire non-e quilibrium chemical composition [36]. In fact, reproductio n of the measured particle yield ratios at RHIC requires a non-zero baryon chemical potential in the sequential b inary emission model [18, 19], while thermal model fits [37, 38] to the Au+Au collision data at the RHIC energ y of √$_{s}$ = 200 GeV yields a chemi- cal freeze-out temp erature of T ∼ 160 MeV and baryon chemical potential µ$_{B}$ ∼ 20 MeV. In Fig. 4 we show η/s as a function of te mperature at a finite baryon chemi- cal potential ( µ$_{B}$/ T ≈ 0 . 13) created by inducing a finite ρ$_{B}$ in the initial d istribution of Hagedorn states. At a given T , the ener gy density of the system is larger for non-zero baryon ch emical potential leading to further en- hancement of part icularly massive particle abundances. In the T − µ$_{B}$ reg ion explored, the shear viscosity η in- creases compared to that at µ$_{B}$ = 0. This increment can be understood classically − substantially large parti- cle densities thoug h increase σ$_{i}$ and thereby reduces η$_{i}$ of each species i , how ever the total contribution, η = ∑ i η$_{i}$ , from these species is greatly enhanced [22]. Since the en- tropy density of th e system also grows with T , the re- Figure 3 displays the correlation function of the shear- stress tensor as a function of time. The correlation func- tions are found to damp exponentially with time, i.e. 〈 π $^{xy}$(0) π $^{xy}$( t ) 〉 ∼ exp( − t/τ$_{π}$ ). Note at small tempera- tures (right panel) the correlations sustain for long times however show a power law tail. The relaxation times τ$_{π}$ for the shear fluxes so obtained by fitting the correlation functions are found to decrease with increasing temper- ature. The shear viscosity is estimated from Eq. (6) by integrating the correlation function over time using the exponential decay found empirically. FIG. 3: Shear-stress correlation for the hadron resonance gas with Hagedorn states as a function of time at different system temperatures. The inset shows the temperature versus relax- ation time τ π obtained from fits to the correlations using an exponential decay. FIG. 4: The shear viscosity to entropy density ratio η/s as a function of temperature for a hadron resonance gas with Hagedorn states in thermodynamic equilibrium at various baryon chemical potential µ B . The η/s for a pure hadron res- onance gas in the UrQMD model [24] is also shown (squares). 4