2 p $_{i}$( t $^{+}$) = p $_{i}$( t $^{−}$) − 2 δt M$_{α}$ ( r $_{i}$, t ) E$_{i}$ ( t ) ⃗ ∇ $_{r}$M$_{α}$ ( r $_{i}$, t ) + coll. r $_{i}$( t $^{+}$) = r $_{i}$( t $^{−}$) + 2 δt p $_{i}$( t ) E$_{i}$ ( t ) (4) p $^{µ}$∂$_{µ}$ f $^{±}$( x, p ) + M ( x ) ∂$_{µ}$M ( x ) ∂ µ $_{p}$f $^{±}$( x, p ) = C ( x, p ) (2) M$_{α}$ − m 8 gN$_{c}$ = M$_{α}$ [ ∫$_{Λ}$ d $^{3}$p (2 π ) 3 1 E − ω Δ V$_{α}$ ( A$_{α}$ ∑ i =1 1 E$_{i}$ − ˜ A$_{α}$ ∑ i =1 1 ˜ $_{E$_{i}$}$)] (5) M ( x ) − m 4 g N$_{c}$ = M ( x ) ∫ d $^{3}$p (2 π ) 3 1 − f $^{−}$( x, p ) − f $^{+}$( x, p ) E$_{p}$ ( x ) (3) to the N$_{f}$ = 2 , N$_{c}$ = 3 for calculations. As well know the theory is non-renormalizable, hence a cut-off Λ has to be introduced as a free parameter. The numerical re- sults shown in the following are derived using the Buballa parametrization: Λ = 588 MeV, g Λ 2 = 2 . 88, m = 5 . 6 MeV [21] that among the variety of parameterizations entails a behavior of ϵ, P, c 2 s closer to the lQCD results. A transport theory for the NJL model has been derived in the closed-time-path formalism combined with the ef- fective action method [22]. The main steps of deriva- tion are to perform a Wigner transformation of the Dirac equation of motion and of the related gap-equation asso- ciated to the L $_{NJL}$, Eq.(1). Then one exploits the semi- classical approximation widely used for applications in heavy-ion collisions [23, 24] evaluating the expectation value of the four-point fermion interaction in the Hartree approximation (i.e. at mean field level). Finally only the scalar and vector components of the Wigner func- tion are retained thanks to the spin saturated nature of the systems we are interested in. One finally obtains the Boltzmann-Vlasov transport equations for the (anti- ) quark phase-space distribution function f $^{±}$: where C ( x, p ) is the Boltzmann-like collision integral, main ingredient of the several cascade codes already de- veloped [11, 25, 26]. We notice that respect to the al- ready implemented cascades the NJL dynamics intro- duces a new term associated to the mass generation. Also Eq.(2) is formally the same as the widely used rel- ativistic transport approaches for hadronic matter like RBUU,uRQMD, RLV [24, 27, 28], but with a vanishing vector field. However a key difference is that particles do not have a fixed mass and a self-consistent deriva- tion couples Eq.(2) to the mass gap equation of the NJL model that extended to the case of non-equilibrium can be written as: and determines the local mass M ( x ) at the space-time point x in terms of the distribution functions f $^{±}$( x, p ). Eqs.(2) and (3) form a closed system of equations constituting the Boltzmann-Vlasov equation associated to the NJL Lagrangian that allows to obtain self- consistently the local effective mass M ( x ) affecting the time evolution of the distribution function f $^{±}$( x, p ). A seminal work on the transport equation associated to the NJL dynamics was done in Refs.[29, 30], but without a collision term, not at finite η/s and never applied to the physical conditions of ultra-relativistic heavy-ion colli- sions. For the numerical solutions of Eqs.(2) and (3) we use a three dimensional lattice that discretize the space as described in Ref.[11, 26]. The standard test par- ticle methods that sample the distribution function f by mean of an ensemble of points in the phase-space is employed. The normalization condition is given by: ∫ d Γ f ± = ω A ( ˜ A ) = N$_{q}$ ( N$_{q}$ ) with Γ the phase space, A$_{α}$ ( ˜ A$_{α}$ ) the number of test particles (antiparticles) which are inside the considered cell and ω the proper normal- ization factor that relates the test particles to the real particle number. In such a way it is possible to get a solution of the transport equations propagating the momenta of the test (anti-)particles by mean of the relativistic Hamilton’s equation. For the numerical implementation they can be written in the discretized form as: with t ± = t ± δt and δt the numerical mesh time. The term coll. on the right hand side of Eq.(4) indicates the effects of the collision integral as described in Ref.[11, 26]. By mean of a reiterating procedure on time steps one gets the solutions of the transport equation coupling Eqs.(4) with the gap equation, Eq.(3) that discretized on lattice and for point-like test particles becomes: where Δ V$_{α}$ = τA$_{T}$ tanh η$_{α}$ is the volume of each cell of the space lattice given by A$_{T}$ = 0 . 5 fm 2 the area in the transverse direction and η$_{α}$ the space-time rapidity of the center of the cell. The integral is instead the vac- uum contribution to the gap-equation which is a diver- gent quantity and it is regularized by a cutoff, Λ and has a simple analytical expression. The space-time dependence of the mass M$_{α}$ ( r, t ) = m − 2 g 〈 ¯ ψψ 〉 influences the momenta of the particles be- cause the finite gradient of the condensate generates a force which changes the momentum of a particle pro- portionally to ⃗ ∇ r 〈 ¯ ψψ 〉 , see Eq.(4). The last is negative because the phase transition occurs earlier in the sur- face of the expanding QGP fireball. Therefore the phase transition which take place locally results in a negative contribution to the particle momenta that makes the sys- tem more sticky respect to a free massless gas. Shear viscosity to entropy density ratio - The effect of the NJL mean-field can be evaluated looking at the so- called interaction measure normalized by T $^{4}$, ϵ − 3 P T 4 , that gives the deviation from the free gas relation between the energy density ϵ and the pressure P that is also a measure of the breakdown of conformal symmetry. In lattice QCD it is known to be quite large with a peak at T$_{c}$ and a non negligible value up to T ∼ 2 − 3 T$_{c}$ . In Fig.1 (left) its behavior is shown for three different