in the Hilbert space, reads The gluon partition function Z$_{g}$ ( β, V ) is given by Eq.(41). The color-singlet state of the mixed-grand canonical ensemble for the chiral quark-gluon bag is projected out in the fol- lowing way where χ$_{Singlet}$ ( g ) = 1 and G refers to the group classification either unitary, orthogonal or symplectic one. The integration over the invariance Haar measure is performed in the standard way using the Gaussian integration method as that is presented in Sec. II (see also Refs. [1, 2]). Furthermore, the micro-canonical ensemble with large energy W is calculated by taking the inverse Laplace transform of the mixed-grand canonical ensemble in the following way The inverse Laplace transform is evaluated using the steepest decent method [1, 2]. The Laplace stationary point is determined by extremizing the exponential with respect to the Laplace variable β . Because of the finite value of the quark’s chiral mass, the solution of the Laplace saddle point becomes a transcendental one and can not be written in a closed form. In this case, the saddle point and the inverse Laplace transform is evaluated by iteration [1]). The first iteration is sufficient to shed the light on the chiral restoration phase transition. The standard MIT bag model furnishes the following relations between the bag’s mass and volume: where B , W , V and m are the bag’s constant, cavity energy, volume and mass, respectively. Hence, the Hagedorn’s density of states for the chiral color-singlet quark-gluon bag with a specific internal color symmetry is reduced to 40 ρ$_{H}$ ( m, σ ) ∝ Z$_{Singlet}$ ( W, V, σ ) | ( W = 3 $_{4}$m,V = m 4 $_{$_{B}$)}$. (147) m = W + BV, W = 3 $_{4}$m, V = m 4 $_{B}$, (146) Z$_{Singlet}$ ( W, V, σ ) = 1 2 π i ∫ β$_{0}$ + i ∞ β$_{0}$ − i ∞ dβ e βW Z$_{Singlet}$ ( β, V, σ ) . (145) Z$_{Singlet}$ ( β, V, σ ) = ∫ dµ$_{G}$ ( g ) χ$_{Singlet}$ ( g ) × Z$_{qqg σ}$ ( β, V, σ ) , = dµ$_{G}$ ( g ) Z$_{qqg σ}$ ( β, V, σ ) , (144) Z$_{qqg σ}$ ( β, V, σ ) = Z$_{qqσ}$ ( β, V, σ ) × Z$_{g}$ ( β, V ) . (143)