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We study the shear and bulk viscosities of partonic and hadronic matter asfunctions of temperature T within the parton-hadron-string dynamics (PHSD)off-shell transport approach. Dynamical hadronic and partonic systems inequilibrium are studied by the PHSD simulations in a finite box with periodicboundary 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 thecritical temperature T_c, while it approaches the perturbative QCD limit athigher temperatures in line with lattice QCD (lQCD) results. For T<T_c, i.e.,in the hadronic phase, the ratio \eta/s rises fast with decreasing temperaturedue to a strong decrease of the entropy density $s$ in the hadronic phase atdecreasing T. Within statistics, we obtain practically the same results in theKubo formalism and in the relaxation time approximation. The bulk viscosity\zeta(T)---evaluated in the relaxation time approach---is found to stronglydepend on the effects of mean fields (or potentials) in the partonic phase. Wefind a significant rise of the ratio \zeta(T)/s(T) in the vicinity of thecritical temperature T_c, when consistently including the scalar mean-fieldfrom 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 isfound to depend nontrivially on temperature and to generally agree with thelQCD 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-field (or potential)effects that in PHSD are encoded in the temperature dependence of thequasiparticle masses, which is related to the infrared enhancement of theresummed (effective) coupling g(T).

Shear and bulk viscosities of strongly-interacting 'infinite' parton-hadron matter within the parton-hadron-string transport approach