**The Parton-Hadron-String Dynamics (PHSD)**
is a microscopic off-shell transport approach for the description
of strongly interacting hadronic and partonic matter in and
out-of equilibrium. It is based on the solution of Kadanoffâ€“Baym equations in first-order
gradient expansion in phase space which allows to describe in a causal way the time
evolution of nonperturbative interacting systems. The PHSD consistently describes
the full evolution of a relativistic heavy-ion collision from the initial hard
scatterings and string formation through the dynamical deconfinement phase
transition to the quark-gluon plasma as well as hadronization and to the subsequent
interactions in the hadronic phase.
The PHSD has been developed by the Giessen/Frankfurt groups on the basis of the
**Hadron-String Dynamics transport approach (HSD).**

In PHSD the transition from the partonic (quarks and gluons) to hadronic degrees
of freedom is described by covariant transition rates for the fusion of
quark-antiquark pairs to mesonic resonances or three quarks (antiquarks) to baryonic
states, i.e., by a dynamical hadronization obeying flavor current-conservation,
color neutrality as well as energy-momentum conservation. The two-particle
correlations resulting from the finite width of the parton spectral functions
are taken into account dynamically in the PHSD by means of the **generalized
off-shell Cassing-Juchem transport equations** for testparticles - followed from Kadanoffâ€“Baym equations -
that go beyond the mean field or Boltzmann approximation.

The transport theoretical description of quarks and gluons in the PHSD is based
on the **Dynamical Quasi-Particle Model (DQPM)**
for partons that is constructed to reproduce lattice QCD (lQCD) results for a
quark-gluon plasma in thermodynamic equilibrium. The DQPM provides the mean-fields
for gluons/quarks and their effective 2-body interactions that are implemented in
the PHSD. Since the dynamical quarks and antiquarks become very massive close to
the phase transition, the formed resonant 'pre-hadronic' color-dipole states (q-qbar
or qqq) are of high invariant mass, too, and sequentially decay to the groundstate
meson and baryon octets. Accordingly, the dynamical hadronization process increases
the total entropy such that the second law of thermodynamics is fulfilled.

The PHSD approach has been applied to nucleus-nucleus collisions from low Super-Proton-Synchrotron (SPS) to Large-Hadron-Collider (LHC) energies in order to explore the space-time regions of 'partonic matter'. It provides a consistent description of the bulk properties of heavy-ion collisions —rapidity spectra, transverse mass distributions, azimuthal asymmetries (v1,v2,v3,v4) of various particle species— from low SPS to top RHIC energies and was successfully used also for the analysis of dilepton production from hadronic and partonic sources at SPS, RHIC and LHC energies.

We have also studied equlibrium properties of the QGP by performing PHSD calculations in a finite box with periodic boundary conditions for fixed temperature T. In particular we find that the ratio of the shear viscosity to entropy density from PHSD shows a minimum (with a value of about 0.1) close to the critical temperature Tc=160 MeV, while it approaches the perturbative QCD (pQCD) limit at higher temperatures in line with lattice QCD results. This indicates that the QGP shows the properties of a strongly interacting liquid (sQGP) rather than —as expected initially— a weakly interacting gas of partons.