The order, shape and critical point for the quark-gluon plasma phase transition

The order, shape and critical point for the phase transition between hadronic matter and quark-gluon plasma are considered in a consistent thermodynamical approach. The hadronic phase is taken as a Van der Waals gas of all the known hadronic mass spectrum particles as well as the hadronic bubbles which correspond the Hagedorn states. The density of states for hadronic bubbles is derived by calculating the micro-canonical ensemble for a bag of quarks and gluons with a specific internal color-flavor symmetry. The mixed-grand and micro-canonical ensembles are derived for massless and massive flavors. We find that hadronic bubbles are strongly suppressed in the diluted hadronic matter and they appear just below the line of phase transition. The order of phase transition depends on the hadronic bubbles' internal color-flavor structure and volume fluctuation. On the other hand, the highly compressed hadronic matter undergoes a smooth phase transition from the gas of known hadronic mass spectrum particles to another one dominated by hadronic bubbles with a specific internal structure before the phase transition to quark-gluon plasma takes place. The phase transition is found first order for the intermediate and large chemical potentials. The phase consists bubbles with colored components but with a net colorless charge phase undergoes a higher order phase transition and the tri-critical point appears in the phase transition diagram in the hot and diluted matter. The existence of the tri-critical point depends on the modification of the bubble's internal structure specified by the phenomenological paramter $\alpha\propto\gamma(\mu_B)$ in the medium