Parton-hadron matter in- and out-off equilibrium

We study the shear and bulk viscosities of partonic and hadronic matter - as well as the electric conductivity - as functions of temperature $T$ within the Parton-Hadron-String Dynamics (PHSD) off-shell transport approach. Dynamical hadronic and partonic systems in equilibrium are studied by the PHSD simulations in a finite box with periodic boundary conditions. The ratio of the shear viscosity to entropy density $\eta(T)/s(T)$ from PHSD shows a minimum (with a value of about 0.1) close to the critical temperature $T_c$. For $T<T_c$, i.e. in the hadronic phase, the ratio $\eta/s$ rises fast with decreasing temperature due to a lower interaction rate of the hadronic system and a significantly smaller number of degrees-of-freedom. The bulk viscosity $\zeta(T)$ -- evaluated in the relaxation time approach -- is found to strongly depend on the effects of mean fields (or potentials) in the partonic phase. We find a significant rise of the ratio $\zeta(T)/s(T)$ in the vicinity of the critical temperature $T_c$, when consistently including the scalar mean-field from PHSD, which is also in agreement with that from lQCD calculations. Furthermore, we present the results for the ratio $(\eta+3\zeta/4)/s$, which is found to depend non-trivially on temperature and to generally agree with the lQCD calculations as well. Within the PHSD calculations, the strong maximum of $\zeta(T)/\eta(T)$ close to $T_c$ has to be attributed to mean-fields (or potential) effects that in PHSD are encoded in the temperature dependence of the quasiparticle masses, which is related to the infrared enhancement of the resummed (effective) coupling $g(T)$. We also find that the dimensionless ratio of the electric conductivity over temperature $\sigma_0/T$ rises above $T_c$ approximately linearly with $T$ up to $T=2.5 T_c$, but approaches a constant above $5 T_c$, as expected qualitatively from perturbative QCD (pQCD).