The averages are defined as 〈O〉 2 = 1 n 2 ∫ d Γ$_{1}$ d Γ$_{2}$ f$_{1}$ ( x, p$_{1}$ ) f$_{2}$ ( x, p$_{2}$ ) O , (10) 〈Q〉 3 = 1 n 3 ∫ d Γ$_{1}$ d Γ$_{2}$ d Γ$_{3}$ f$_{1}$ ( x, p$_{1}$ ) f$_{2}$ ( x, p$_{2}$ ) f$_{3}$ ( x, p$_{3}$ ) Q (11) with n = ∫ d Γ$_{1}$ f ( x, p$_{1}$ ) , (12) where d Γ$_{j}$ = d $^{3}$p$_{j}$ / (2 π ) $^{3}$, j = 1 , 2 , 3, and the local particle distribution function f ( x, p ) is the output from the parton cascade, f ( x, p ) = ∑ i δ $^{(3)}$[ ⃗ x − ⃗ x$_{i}$ ( t )] δ $^{(3)}$[ ⃗ p − ⃗ p$_{i}$ ]. The sum runs over all particles with individual position ⃗ x$_{i}$ and momentum ⃗ p$_{i}$ at time t . For a gg → gg or a gg → ggg process involving two incoming particles with ( x, p$_{1}$ ) and ( x, p$_{2}$ ) the relative velocity of the two particles is given by v$_{rel}$ = s/ (2 E$_{1}$E$_{2}$ ), where s = ( p$_{1}$ + p$_{2}$ ) 2 is the invariant mass. The total cross sections read σ$_{gg}$$_{→}$$_{gg}$ = 1 2! ∫ s/ 4 0 d q 2 ⊥ dσ$_{gg}$$_{→}$$_{gg}$ d q 2 ⊥ , (13) σ$_{gg}$$_{→}$$_{ggg}$ = 1 2 s ∫ d Γ ′ $_{1}$d Γ ′ $_{2}$d Γ ′ 3 |M gg → ggg | $^{2}$(2 π ) $^{4}$δ$^{(4)}$( p$_{1}$ + p$_{2}$ − p ′ 1 − p ′ 2 − p ′ $_{3}$) , (14) where d Γ ′ j = d $^{3}$p ′ $_{j}$/ (2 π ) $^{3}$/ (2 E ′ $_{j}$), j = 1 , 2 , 3. For a ggg → gg process we define I$_{ggg}$$_{→}$$_{gg}$ = 1 2! ∫ d Γ ′ $_{1}$d Γ ′ 2 |M ggg → gg | $^{2}$(2 π ) $^{4}$δ$^{(4)}$( p$_{1}$ + p$_{2}$ + p$_{3}$ − p ′ 1 − p ′ $_{2}$) , (15) which is, similar as a cross section, an integral over the final states. The factor of 1 / 2! in Eqs. (13) and (15) is due to the fact that the two outgoing gluons are identical particles. There is no such a factor (1 / 3!) in gg → ggg processes [28], because the three outgoing gluons are distinguished as the radiated gluon ( p ′ $_{3}$), the radiating gluon ( p ′ $_{2}$) and the gluon ( p ′ $_{1}$), which is only deflected and does not radiate. Thus, they are kinematically “distinguishable” particles. We note that n in Eq. (12) is the local particle density in the center of mass frame of the Au+Au collision (collision frame) and only equals the Lorentz invariant particle density if the local particle system is at rest, e.g. at the collision center. Also the local collision rates (7)-(9) are calculated in the collision frame and are smaller than those in the rest frame by a Lorentz factor. The left panel of Fig. 2 shows the time evolution of the total collision rate per particle 〈 R$_{gg}$$_{→}$$_{gg}$ + R$_{gg}$$_{→}$$_{ggg}$ + R$_{ggg}$$_{→}$$_{gg}$ 〉 . The rate is obtained as the average over all particles in 7