Modelling QCD-dynamics in and beyond mean field theory

We present model calculations featuring the two most important non-perturbative effects of QCD at temperatures below the 1 GeV-scale (spontaneous chiral symmetry breaking and confinement) which are in astonishingly good agreement with full lattice QCD calculations. The existing analysis has been extended by corrections accounting for the fermion sign problem. RPA is used to estimate pressure contributions generated by mesonic fluctuations of the two light meson modes, π and σ. Qualitatively the overall picture drawn by the PNJL model remains, confirming preceeding conclusions. Additionally, in the effort to extend the range of applicability of the PNJL model, non-local approaches are under investigation. In their construction special attention has been paid to the connection to QCD at high temperatures. We show that the non-local modifications to the PNJL model preserves the Nambu-Goldstone character of pions within the framework of this model. Furthermore the thermodynamic of this non-local PNJL model predicts a joint cross-over of chiral symmetry and deconfinement, similar to the situation in the original PNJL model