5 20000 21000 22000 23000 24000 25000 26000 25 50 75 100 125 150 175 200 225 250 c=0, b=0.1, a=0, H=0 c=0, b=0.1, a=100, H=0 c=0, b=0.1, a=-100, H=0 c=0, b=0.1, a=-100, H=10 3 T /s FIG. 6: The shear viscosity over entropy density η/s as a func- tion of the temperature T , for cases with a second-order phase transition (solid curve), a crossover (dash-dotted curve), and with no phase transition for massive field (dashed curve) and massless field (dotted curve). The figure is taken from Ref.[52]. FIG. 7: The shear viscosity over entropy density η/s as a function of the temperature T , for the case of 1st order phase transition. 10 3 10 4 10 5 10 6 10 7 0 10 20 30 40 50 60 70 a=100,b=-0.12,c=0.000025 T /s with the negative vacuum energy density ε$_{v}$ = Ω$_{v}$ = Ω( φ ) | T $_{=0}$, and the parameter ω$_{0}$ = ω$_{0}$ ( T ) is a scale at which the perturbation theory becomes valid. From the above formula, we can see that the bulk viscosity is pro- portional to the specific heat C$_{v}$ near phase transition, thus ζ/s behaves as 1 /C 2 s near T$_{c}$ in this approximation. The bulk viscosity over entropy density ratio ζ/s as a function of T is shown in Figs.8,9,10, and 11. It is shown According to the result derived from low energy theorem, in the low frequency region, the bulk viscosity takes the form of [29, 30] The bulk viscosity is related to the correlation function of the trace of the energy-momentum tensor θ µ $_{µ}$: where s, t and u are Mandelstam variables, and λ$_{3}$ = 6 φ$_{0}$ ( b + 10 c 3 φ 2 $_{0}$+ 10 c ∫ K ¯ G ( K, ¯ φ )) and λ$_{4}$ = 12( b 2 + 5 cφ 2 $_{0}$+ 5 c K ¯ G ( K, ¯ φ )) are effective couplings. 5 c ∫ K G ( K, φ )) are effective couplings. The shear viscosity over entropy density ratio η/s in the real scalar model in shown in Fig.6 and 7 for different orders of phase transitions. There is clearly a qualitative difference in the η/s behavior between cases with and without a phase transition. It is seen that η/s shows a cusp at T$_{c}$ for the case of 2nd-order phase transition, a shallow valley near T$_{c}$ for crossover, and shows a jump at T$_{c}$ for the case of 1st-order phase transition. This behavior is qualitatively the same as that in the classic systems such as the in H$_{2}$O system as shown in Fig. 5. If there is no phase transition. η/s is always monotonically decreasing. The shear viscosity η is calculated by using the Boltz- mann equation [55]. The two-particle elastic scattering amplitude, which governs particle collisions in the Boltz- mann equation, is is the specific heat. At the critical temperature, the entropy density as well as the energy density change most quickly with temperature, thus one expect that C 2 s should have a minimum at T$_{c}$ . The equation of state p ( ε ) is an important input into hydrodynamics. The square of the speed of sound C 2 s is related to p/ε and has the form of As the standard treatment in lattice calculation, we in- troduce the normalized pressure density p which is nor- malized to vanish at T = µ = 0 and the energy density ε as p = − Ω , ε = − p + T s. (14) C 2 s = d p d ε = s T d s/ d T = s C$_{v}$ , (15) C$_{v}$ = ∂ε/∂T, (16) i T = λ$_{4}$ + λ 2 3 [ 1 s − m 2 + 1 t − m 2 + 1 u − m 2 ] , (17) where ζ = 1 9 lim ω → 0 1 ω ∫ ∞ 0 dt ∫ d $^{3}$r e iωt 〈 [ θ µ $_{µ}$( x ) , θ µ $_{µ}$(0)] 〉 . (18) ζ = 1 9 ω$_{0}$ { T 5 ∂ ∂T ( ε − 3 p ) T 4 + 16 | ε$_{v}$ | } , = 1 9 ω$_{0}$ {− 16 ε + 9 T S + T C$_{v}$ + 16 | ε$_{v}$ |} . (19)