content_image/330917.jpg content_image/330905.jpg content_image/330916.jpg content_image/330915.jpg content_image/330914.jpg content_image/330913.jpg content_image/330912.jpg
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, 

<img src='content_image/330917.jpg'>

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 

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 

<img src='content_image/330905.jpg'>

<img src='content_image/330916.jpg'>

<img src='content_image/330915.jpg'>

<img src='content_image/330914.jpg'>

<img src='content_image/330913.jpg'>

<img src='content_image/330912.jpg'>

