content_image/422615.jpg content_image/422617.jpg content_image/422618.jpg content_image/422612.jpg content_image/422613.jpg content_image/422614.jpg content_image/422598.jpg content_image/422601.jpg
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 

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 

where 

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 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 

<img src='content_image/422615.jpg'>

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. 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 bulk viscosity is related to the correlation function of the trace of the energy-momentum tensor θ µ $_{µ}$: 

<img src='content_image/422617.jpg'>

According to the result derived from low energy theorem, in the low frequency region, the bulk viscosity takes the form of [29, 30] 

<img src='content_image/422618.jpg'>

<img src='content_image/422612.jpg'>

<img src='content_image/422613.jpg'>

<img src='content_image/422614.jpg'>

<img src='content_image/422598.jpg'>

<img src='content_image/422601.jpg'>

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 

