8 GLYPH<s48> GLYPH<s46>GLYPH<s48> GLYPH<s48> GLYPH<s46>GLYPH<s53> GLYPH<s49> GLYPH<s46>GLYPH<s48> GLYPH<s49> GLYPH<s46>GLYPH<s53> GLYPH<s50> GLYPH<s46>GLYPH<s48> GLYPH<s50> GLYPH<s46>GLYPH<s53> GLYPH<s51> GLYPH<s46>GLYPH<s48> GLYPH<s51> GLYPH<s46>GLYPH<s53> GLYPH<s52> GLYPH<s46>GLYPH<s48> GLYPH<s52> GLYPH<s46>GLYPH<s53> GLYPH<s53> GLYPH<s46>GLYPH<s48> GLYPH<s48> GLYPH<s46>GLYPH<s48> GLYPH<s48> GLYPH<s48> GLYPH<s46>GLYPH<s48> GLYPH<s53> GLYPH<s48> GLYPH<s46>GLYPH<s49> GLYPH<s48> GLYPH<s48> GLYPH<s46>GLYPH<s49> GLYPH<s53> GLYPH<s48> GLYPH<s46>GLYPH<s50> GLYPH<s48> GLYPH<s48> GLYPH<s46>GLYPH<s50> GLYPH<s53> GLYPH<s48> GLYPH<s46>GLYPH<s51> GLYPH<s48> GLYPH<s48> GLYPH<s46>GLYPH<s51> GLYPH<s53> GLYPH<s32>GLYPH<s80>GLYPH<s76> GLYPH<s83>GLYPH<s77> GLYPH<s32>GLYPH<s76> GLYPH<s83>GLYPH<s77> GLYPH<s32>GLYPH<s78> GLYPH<s61>GLYPH<s54> GLYPH<s32> GLYPH<s32> GLYPH<s119> GLYPH<s61>GLYPH<s112> GLYPH<s47> GLYPH<s84> GLYPH<s47>GLYPH<s84> GLYPH<s99> GLYPH<s61>GLYPH<s48> GLYPH<s32>GLYPH<s77> GLYPH<s101> GLYPH<s86> GLYPH<s48> GLYPH<s46>GLYPH<s54> GLYPH<s48> GLYPH<s46>GLYPH<s56> GLYPH<s49> GLYPH<s46>GLYPH<s48> GLYPH<s49> GLYPH<s46>GLYPH<s50> GLYPH<s49> GLYPH<s46>GLYPH<s52> GLYPH<s49> GLYPH<s46>GLYPH<s54> GLYPH<s49> GLYPH<s46>GLYPH<s56> GLYPH<s50> GLYPH<s46>GLYPH<s48> GLYPH<s48> GLYPH<s46>GLYPH<s48> GLYPH<s48> GLYPH<s48> GLYPH<s46>GLYPH<s48> GLYPH<s53> GLYPH<s48> GLYPH<s46>GLYPH<s49> GLYPH<s48> GLYPH<s48> GLYPH<s46>GLYPH<s49> GLYPH<s53> GLYPH<s48> GLYPH<s46>GLYPH<s50> GLYPH<s48> GLYPH<s48> GLYPH<s46>GLYPH<s50> GLYPH<s53> GLYPH<s48> GLYPH<s46>GLYPH<s51> GLYPH<s48> GLYPH<s32>GLYPH<s80>GLYPH<s76> GLYPH<s83>GLYPH<s77> GLYPH<s32>GLYPH<s76> GLYPH<s83>GLYPH<s77> GLYPH<s32> GLYPH<s61>GLYPH<s49> GLYPH<s32>GLYPH<s71>GLYPH<s101> GLYPH<s86> GLYPH<s32> GLYPH<s32> GLYPH<s47>GLYPH<s115> GLYPH<s84> GLYPH<s47>GLYPH<s84> GLYPH<s99> GLYPH<s61>GLYPH<s48> GLYPH<s32>GLYPH<s77> GLYPH<s101> GLYPH<s86> FIG. 13: The bulk viscosity over entropy density ratio ζ/s as a function of the temperature for µ = 0 MeV. The solid line denotes the Polyakov linear sigma model prediction and the dashed line denotes the linear sigma model prediction. Lattice data taken from Ref.[30]. The figure is taken from Ref.[53]. FIG. 12: The equation-of-state parameter w ( T ) = p ( T ) /ε ( T ) for µ = 0 MeV. The Polyakov linear sigma model prediction (solid line) and the linear sigma model prediction (dash line) are compared with N$_{f}$ = 2 + 1 lattice QCD data for N τ = 6. Lattice data taken from Ref.[38]. The figure is taken from Ref.[53]. The set of equations can be solved for the fields as func- tions of temperature T and chemical potential µ , and the solutions of these coupled equations determine the behavior of the chiral order parameter σ$_{x}$ , σ$_{y}$ and the Polyakov loop expectation values φ , φ ∗ as a function of T and µ . Minimizing the thermodynamical potential in Eq.(31) with respective to σ$_{x}$ , σ$_{y}$ , φ and φ $^{∗}$, we obtain a set of equations of motion Here, N$_{q}$ = 2, N$_{s}$ = 1, and E$_{q}$ = √ ⃗ p 2 + m 2 q is the va- lence quark and antiquark energy for u and d quarks, for strange quark s , it is E$_{s}$ = √ ⃗ p 2 + m 2 $_{s}$, and m$_{q}$ , m$_{s}$ is the constituent quark mass for u , d and s . The purely mesonic potential is Fig.12 shows the pressure density over energy density p/ε , which is represented in terms of equation-of-state (EOS) parameter, at zero density and finite density, re- spectively. We observe that the pressure density over en- ergy density increases with temperature and saturates at high temperature. Both the linear sigma model and the Polyakov linear sigma model give very similar results at high temperature, the pressure density over energy den- sity p/ε saturates at a value smaller than 1 / 3. Another common feature of the p/ε in the linear sigma model and the Polyakov linear sigma model is that there is a bump appearing at low temperature region, which is also observed in the lattice result. Around the critical tem- perature T$_{c}$ , the pressure density over energy density p/ε shows a downward cusp. However, the minimum value of the p/ε around T$_{c}$ is 0 . 2 in the linear sigma model, which is much larger than the result from the Polyakov linear sigma model and the lattice QCD data. For the Polyakov linear sigma model, the minimum of p/ε around T$_{c}$ is 0 . 075, which is consistent with the lattice QCD data [38]. In Ref. [34], we have investigated the equation of state and bulk viscosity in the real scalar model and O (4) model in the case of 2nd order phase transition, crossover and 1st order phase transition, and we have found that the thermodynamic properties and transport properties in these simple models near the critical temperature T$_{c}$ at strong coupling are similar to those of the complex QCD system. In a more realistic QCD effective model, i.e, the Polyakov linear sigma model [53], we have systematically investigated the thermodynamic properties and bulk vis- cosity and found these properties match with lattice data very well in the case of zero chemical potential. We fur- ther evaluate the chiral phase transitions of u, d and s quarks and deconfinement phase transition at finite tem- perature and finite density, and show the T − µ phase structure of the Polyakov linear sigma model in Fig.14. In Fig.13, we plot the bulk viscosity over entropy den- sity ratio ζ/s as a function of the temperature for zero chemical potential. It is shown that, at zero chemical po- tential µ = 0, the bulk viscosity over entropy density ζ/s decreases monotonically with the increase of the temper- ature in both the Polyakov linear sigma model and lin- ear sigma model, and at high temperature, ζ/s reaches its conformal value 0. In [30], the bulk viscosity over entropy density of the three flavor system is extracted from lattice result, which is shown by the square. It is observed that ζ/s in PLSM near phase transition is in very good agreement with the lattice result in [30], i.e, it rises sharply near phase transition. U ( σ$_{x}$, σ$_{y}$ ) = m 2 2 ( σ 2 x + σ 2 $_{y}$) − h$_{x}$σ$_{x}$ − h$_{y}$σ$_{y}$ − c $_{2}$√$_{2}$ σ 2 $_{x}$σ$_{y}$ + λ$_{1}$ 2 σ 2 $_{x}$σ 2 y + 1 8 (2 λ$_{1}$ + λ$_{2}$ ) σ 4 x + 1 4 ( λ$_{1}$ + λ$_{2}$ ) σ 4 $_{y}$. (33) ∂ Ω ∂σ$_{x}$ = 0 , ∂ Ω ∂σ$_{y}$ = 0 , ∂ Ω ∂φ = 0 , ∂ Ω ∂φ ∗ = 0 . (34)