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The first inequality indicates mechanical stability which means a system at positive isothermal compressibility re- mains stable at all densities. The second inequality stems from chemical instability which shows that energy is re- quired to change the concentration in a stable system while maintaining temperature and pressure fixed. If one of these conditions get violated, a system with two phases is energetically favorable. The two phase coexistence is governed by the Gibbs’s criteria for equal chemical po- tentials and pressures in the two phases with different densities but at the same temperature: 

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Figure 2 shows the pressure as a function of nucleon density at a fixed temperature T = 10 MeV with different values of asymmetry α in the original NL3 and FSUGold sets. Below a critical value of asymmetry α , the pressure is seen (dotted curves) to decrease with density result- ing in negative incompressibility and thereby a mechani- cally unstable system. The stable two-phase (liquid-gas) configuration at each density is obtained from Maxwell construction (solid lines). Analogues to intermediate en- ergy heavy-ion collisions [1, 2] when the hot matter in the high density (liquid) phase expands it enters the coexis- tence LGP where the pressure decreases at a fixed α ̸ = 0 for the two-component asymmetric matter. Whereas, for symmetric nuclear matter at α = 0 the pressure remains 

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constant at all densities. Finally the system leaves the co- existence region and vaporizes into the low density (gas) phase. Of particular interest here is the symmetry en- ergy effects on the isotherms. It is clearly seen that in contrast to the original NL3 with Λ$_{v}$ = 0, the softer E$_{sym}$ ( ρ ) in the original FSUGold with Λ$_{v}$ = 0 . 03 [17] en- forces the onset of pure liquid phase to a higher density resulting in a wider coexistence region for each asymme- try α . Moreover, the critical pressure P$_{c}$ above which the mixed liquid-gas phase vanishes is seen larger for this soft FSUGold set; a detailed discussion of which is presented below. 

The details of chemical evolution for the LGP transi- tion is depicted in Fig. 3 where the neutron and proton chemical potentials are shown as a function of isospin asymmetry α at a fixed T = 10 MeV and pressure P = 0 . 11 MeV/fm 3 for the NL3 (left panels) and FSUG- old (right panels) at various Λ$_{v}$ values. As usual, the bare nucleon mass has been subtracted from the chemi- cal potentials. At fixed pressure and Λ$_{v}$, the solutions of the Gibbs conditions (9) and (10) for phase equilibrium form the edges of a rectangle and can be found by geo- metrical construction as shown in Fig. 3. At each Λ$_{v}$, the two different values of α defines the high density liquid phase boundary (with small α = α$_{1}$ ( T, P )) and the low density gas phase boundary (with large α = α$_{2}$ ( T, P )). From the figure it is evident that the symmetry energy 

