The crucial phenomenon of the model is that the quarks and anti-quarks tends to be confined in the U (1) N$_{c}$ Hagedorn states (i.e. colorless states). These states can be interpreted as quark liquid bubbles. The construction of the color-singlet-state for (unimodular-like) U (1) N$_{c}$ quark-bag is different from those for the color-singlet-state bags with the unitary, orthogonal and symplectic internal symmetry groups with the unimodular-like constraint. The U (1) N$_{c}$ symmetry group maintains the minimal gluonic content in the quark-bag (i.e. the minimal interaction among the quarks and antiquarks since N$_{fun}$ = N$_{c}$ and N$_{adj}$ = N$_{c}$ ). The U (1) N$_{c}$ color-singlet-state canonical ensemble for quark and antiquark bag is constructed in the following way Z$_{qbag}$ ( β, V ) = [∫ dµ$_{U}$$_{(1)}$ ( θ ) Z$_{qqg}$ ( β, V, θ ) ] N$_{c}$ − 1 , = [ 1 2 π ∫ π − π dθ Z$_{qqg}$ ( β, V ; θ ) ] N$_{c}$ − 1 . (108) Z$_{qbag}$ ( β, V ) = ∫ ∏ i dµ$_{U}$$_{(1)}$ ( θ$_{i}$ ) 2 πδ ( N$_{c}$ ∑ i θ$_{i}$ ) N$_{c}$ ∏ i Z$_{qqg}$ ( β, V, θ$_{i}$ ) , = ∫ N$_{c}$ ∏ i dθ$_{i}$ 2 π 2 πδ ( N$_{c}$ ∑ j θ$_{j}$ ) N$_{c}$ ∏ k Z$_{qqg}$ ( β, V, θ$_{k}$ ) , (109) Z$_{qbag}$ ( β, V ) = √ 2 π N$_{c}$ V β $_{3}$˜ u (2) qqg exp ( V β $_{3}$˜ u (0) qqg ) ∫ N$_{c}$ ∏ i dθ$_{i}$ 2 π exp ( − 1 2 V β $_{3}$˜ u (2) qqg N$_{c}$ ∑ i θ 2 i ) , = 1 √$_{N}$$_{c}$ (2 π ) − N$_{c}$/ 2+1 / 2 ( V β $_{3}$˜ u (2) qqg ) − N$_{c}$/ 2+1 / 2 exp ( V β $_{3}$˜ u (0) qqg ) , (110) ˜ u (0) qqg = (2 J + 1) N$_{c}$ N$_{f}$ ∑ f =1 [ 7 π 2 360 + 1 12 ( µ$_{f}$ T ) 2 + 1 24 π 2 ( µ$_{f}$ T ) 4 ] +(2 J + 1) N$_{c}$ ( π 2 90 ) , (111) 31 where where it is restricted to the unimodular-like constraint. At extreme condition, it is reduced to Instead of considering the U (1) N$_{c}$ symmetry group, the U (1) N$_{c}$ symmetry group with the unimodular-like constraint ( ∑ N$_{c}$ i θ$_{i}$ = 0 ) is adopted in order to be consistent with unitary, orthogonal and symplectic symmetry groups those are restricted to the unimodular-like condition and have the same number of the fundamental color charges. Hence, the canonical ensemble for the U (1) N$_{c}$ color-singlet state of the quark-bag is constructed as follows