the grand potential density becomes finite and continuous over the critical temperature and chemical potential while the order of phase transition becomes rather more difficult to be determined but not really abstruse and the system undergoes higher order deconfinement phase transition. It is evident that the value of the mass/volume power exponent α plays a decisive role in determining the order and shape of the phase transition diagram. Every choice for the exponent α leads to a different partition function and another distinctive physical behaviour. The smooth phase transition has been studied using phenomenological exponents of the hadron’s mass-volume spectral density which are set by inspired models [22–28]. The mass and volume bag spectral density can be reduced simply to either mass spectral density or volume spectral density for a model with an adequate approximation such as the MIT bag model with the assumption of the sharp boundary surface. The dependence of the deconfine- ment phase transition on the mass spectral exponent α which appears in the mass spectral density, namely, ρ = c m − α e b m for the bootstrap-like model have been studied [29]. The colorless multi-quark cluster has been considered to hint some of the puzzling results in RHIC experiments [30, 31]. The Hagedorn matter below the deconfinement phase transition is found to fit the experimental and theoretical observations [32–35]. The projection of the color-singlet state resembles Polyakov Loop approach [11–13, 36–39] where the Polyakov effective potential stems from the Vandermonde potential of a specific symmetry group and interaction. The variation of the phenomenological parameters in the Polyakov effective potential reveals the internal symmetry modification by altering the correspondent Vander- monde potential. In other words, the modification of Vandermonde potential alters the bag’s internal symmetry. The QCD phase transition is considered in the context of the random matrix model [40–42]. There is a strong indication of a new class of hadronic matter above the discrete low-lying hadronic matter and below the deconfined quark-gluon plasma and this matter is identified as hadronic matter which is dominated by Hagedorn states or the quarkyonic matter [43]. It is argued that the tri-critical point that appears in the phase transition diagram is related to the existence of the predicted quarkyonic matter [7] (or equivalently the Hagedorn matter). The role of Hagedorn states in both experimental and theoretical observations have been studied extensively [44–49]. A recent survey on the im- portance of quarkyonic matter (i.e. Hagedorn phase) in the phase transition diagram from the experimental point of view can be found in Ref. [50]. The tri-critical point in the QCD 6