coefficients which appear in the exponential are calculated in the following way, and a (0) q ¯ q = (2 J + 1) V N$_{c}$ N$_{f}$ ∑ q ∫ d $^{3}$⃗ p (2 π ) 3 ln [ 1 + ( e − β ( ϵ$_{q}$ ( ⃗ p ) − µ$_{q}$ ) + e − β ( ϵ$_{q}$ ( ⃗ p )+ µ$_{q}$ ) ) + e − 2 βϵ$_{q}$ ( ⃗ p ) ] , (26) a (2) q ¯ q = (2 J + 1) V N$_{f}$ ∑ q ∫ d $^{3}$⃗ p (2 π ) 3 ( e − β ( ϵ$_{q}$ ( ⃗ p ) − µ$_{q}$ ) + e − β ( ϵ$_{q}$ ( ⃗ p )+ µ$_{q}$ ) ) [1 + ( e − β ( ϵ$_{q}$ ( ⃗ p ) − µ$_{q}$ ) + e − β ( ϵ$_{q}$ ( ⃗ p )+ µ$_{q}$ $^{)}$) + e − 2 βϵ$_{q}$ ( ⃗ p $^{)}$] − (2 J + 1) V N$_{f}$ ∑ q ∫ d $^{3}$⃗ p (2 π ) 3 ( e − β ( ϵ$_{q}$ ( ⃗ p ) − µ$_{q}$ ) − e − β ( ϵ$_{q}$ ( ⃗ p )+ µ$_{q}$ ) ) 2 [1 + ( e − β ( ϵ$_{q}$ ( ⃗ p ) − µ$_{q}$ ) + e β ( ϵ$_{q}$ ( ⃗ p )+ µ$_{q}$ $^{)}$) + e − 2 βϵ$_{q}$ ( ⃗ p $^{)}$] $_{2}$. (27) a (0) q ¯ q = (2 J + 1) ( V β 3 ) N$_{c}$ massless ∑ q =1 [ 7 π 2 360 + 1 12 ( µ$_{q}$ T ) 2 + 1 24 π 2 ( µ$_{q}$ T ) 4 ] + (massive flavors) , (28) a (2) q ¯ q = (2 J + 1) ( V β 3 ) massless ∑ q =1 1 6 ( 1 + 3 π 2 µ 2 q T 2 ) + (massive flavors) , ≈ (2 J + 1) ( V β 3 ) massless ∑ q =1 1 6 ( 1 + 3 π 2 µ 2 q T 2 ) + ( · · · ) , (29) Z$_{g}$ ( β, V ) = ∫ ∏ a ∫ [ dA a $_{0}$][ dA a $_{1}$][ dA a $_{2}$]det ( ∂$_{3}$ ) e $^{S$_{g}$}$, (30) S$_{g}$ = 1 2 ∫ β 0 dτ ∫ d $^{3}$x ( A a $_{0}$, A a $_{1}$, A a $_{2}$) D ( A a $_{0}$, A a $_{1}$, A a $_{2}$) $^{T}$. (31) 14 where the action is given by The grand canonical ensemble for the adjoint gluons can be calculated in a similar way to that one done in calculating the fundamental quark and antiquark grand canonical ensemble in the functional integral representation (see for instance Eq.(15)). For the sake of simplicity, the axial gauge is considered and the result is manifestly gauge invariant [55]. In the axial gauge, the canonical partition function for a gas of adjoint gluons reads where ( · · · ) indicates the neglected massive flavors. and The chemical potential µ$_{q}$ is decomposed to µ$_{q}$ = $_{3}$µ$_{B}$ + Sµ$_{S}$ + · · · . From the fugacity definition, it is worth to keep in mind that βµ$_{q}$ → 1 T µ$_{q}$ . In the case of massless flavors, we have