the analytical solution behaviour indicates that the canonical ensemble transmutes from one state to another but the deconfinement phase transition is still not reached yet. Further- more, the value of the thermodynamic ensemble above the threshold of the Gross-Witten point is restricted to fall in a physical region (i.e. non-negative) as follows, The critical (threshold) point x$_{c}$ ≡ x$_{min}$ is calculated by extremizing the logarithm of the canonical ensemble. The canonical ensemble is written as follows Z$_{qqg}$ ( x ) ∼ C ( ˜ a (2) $_{qqg}$x ) − ( α − 1 / 2) exp ( ˜ a (0) $_{qqg}$x ) , ln Z$_{qqg}$ ( x ) ∼ − α − 1 / 2) ln( x ) + ˜ a (0) $_{qqg}$x + [ ln C − ( α − 1 / 2) ln ˜ a (2) qqg ] . (94) The value of the relaxation constant C = C$_{N}$$_{c}$$_{G}$ is fixed in order to regulate the solution and match some physical requirements [1]. The threshold point x$_{th}$ is determined by calculating the minimum point of the concave up curve as follows The limit of the normalisation constant C is determined by The energy threshold is found The threshold Hagedorn mass is estimated from ( V β 3 )∣ ∣ $_{threshold}$. As it will be demonstrated below when the canonical ensemble is transformed to the micro-canonical ensemble using Eq.(101), the saddle point method leads to the following dual connection In the context of standard MIT bag model, the above relation is reduced to 28 ( V β 3 ) → m 4 (˜ a (0) $_{qqg}$) 3 / $^{4}$B1 / $_{4}$, ≥ ( α − 1 2 ) ( 1 ˜ a (0) $_{qqg}$) , (99) ( V β 3 ) → 1 3˜ a (0) qqg ( 3˜ a (0) $_{qqg}$V W 3 ) 1 / $^{4}$. (98) ( V β 3 )∣ ∣ th = x$_{th}$, = ( α − 1 / 2) / ˜ a (0) $_{qqg}$. (97) ln Z ( x$_{th}$ ) ≥ ln Z$_{0}$ ≥ 0 . (96) x$_{th}$ = x$_{min}$, = ( α − 1 / 2) / ˜ a (0) $_{qqg}$. (95) ln [ Z$_{qqg}$ (threshold) ] ≥ 0 . (93)