In QCD where the number of colors becomes N$_{c}$ = 3, the exponent α is reduced to 3, 9 / 2 and 15 / 2 for the orthogonal, unitary and symplectic color-singlet ensembles, respectively, and these groups are fallen under an additional unimodular-like constraint. The normalisation constant is determined by where G = 1 , 2 and 4 for the orthogonal, unitary and symplectic symmetry groups, re- spectively. It should be noted that Γ( n ) is the gamma function. The color-singlet-state canonical ensemble with a certain internal symmetry that is fallen under the additional unimodular-like condition is reduced to the following general formula, When the term ( V/β $^{3}$) is parameterised, the color-singlet-state (mixed-grand) canonical ensemble which is given in Eq.(89) becomes where The normalisation constant C$_{N}$$_{c}$$_{G}$ for short reads We can relax the definition of the normalisation constant, namely, C$_{N}$$_{c}$$_{G}$ . This modified constant C$_{N}$$_{c}$$_{G}$ will be determined from the physical constraint in order to satisfy the ther- modynamic ensemble under realistic situations. The thermodynamic ensemble is concave up with respect to the variable x = V/β 3 [1]. It is evident that the physical threshold must start from bottom point of the concave up curve. The line on the left hand side of the threshold point is unphysical and must be excluded. This behaviour furnishes an additional physical constraint that mentioned above. This provides a strong signature that analytical solution has been changed somewhere on the right hand side of the threshold point. The change in 27 C$_{N}$$_{c}$$_{G}$ = √ 2 π/N$_{c}$ N N N$_{c}$ $_{G}$G $^{α}$. (92) ˜ a (0) q ¯ qg = a (0) q ¯ $_{qg}$/ ( V β 3 ) , ˜ a (2) q ¯ qg = a (2) q ¯ $_{qg}$/ V β 3 . (91) Z$_{qqg}$ ( β, V ) = C$_{N}$$_{c}$$_{G}$ ( β $^{3}$/V ˜ a (2) qqg ) α − 2 exp ( V β 3 ˜ a (0) q ¯ qg ) . (90) Z$_{singlet}$ ( β, V ) ∼ √ 2 π/N$_{c}$ N N N$_{c}$ $_{G}$G α ( a (2) qqg ) − ( α − 1 / $^{2)}$exp ( a (0) q ¯ qg ) . (89) N N$_{c}$ G = (2 π ) N c 2 G − α N$_{c}$ ∏ n =1 Γ ( 1 + 1 2 G n ) / [ Γ ( 1 + 1 2 G )] N$_{c}$ , (88)