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dependence of Λ$_{v}$ in NL3 and FSUGold [17, 25] leads to different phase boundaries α$_{1}$ ( T, P ) and α$_{2}$ ( T, P ) and hence should predict different thermodynamic properties for the LGP transition. 

As the pressure increases the system encounters a critical pressure P$_{c}$ beyond which the matter is stable but below which the second inequality (8) gets violated and the system becomes chemically unstable. The crit- ical pressure P$_{c}$ is determined by the inflection point ( ∂µ/∂α )$_{T,P}$ c = ( ∂ $^{2}$µ/∂α$^{2}$)$_{T,P}$ c = 0. The disappearance of chemical instability at P$_{c}$ results in the neutron (pro- ton) chemical potential to decrease (increase) with de- creasing asymmetry α . Figure 3 also shows the chemical potential isobars at the critical pressure (dashed lines). The rectangle from Gibbs condition then collapses into a line vertical at α ≡ α$_{c}$ . Correspondingly, ( P$_{c}$, α$_{c}$ ) defines the critical point at a given temperature that refers to the upper boundary of instability with respect to pres- sure variation. Note at T = 10 MeV, the critical val- ues ( P$_{c}$, α$_{c}$ ) at Λ$_{v}$ = 0 . 0 , 0 . 02 , 0 . 03 are respectively at (0.210, 0.652), (0.276, 0.741), (0.331, 0.797) for the NL3 set and at (0.209, 0.638), (0.266, 0.725), (0.303, 0.789) for the FSUGold set. Interestingly, we also find at a fi- nite temperature the stiffness of symmetry energy has a significant influence on the phase-separation boundaries of LGP transition [20]. In general, a softer symmetry en- ergy (larger Λ$_{v}$) gives systematically larger critical pres- sure and an enhanced asymmetry in the system. More- over at a finite α , the relatively softer symmetry energy E$_{sym}$ ( ρ, T = 0) in the NL3 compared to FSUGold [17] translates to a larger critical pressures and asymmetry for the LGP transition. 

All the pairs of solutions of Gibbs conditions, α$_{1}$ ( T, P ) and α$_{2}$ ( T, P ), form the phase-separation boundary or the binodal surface. In Fig. 4 we show the section of the binodal surface under isothermal compression of asym- metric nuclear matter at T = 10 MeV in the NL3 (top panel) and FSUGold (bottom panel). As expected the point of equal concentration (EC) corresponding to sym- metric nuclear matter is independent of Λ$_{v}$. The criti- cal point (CP) and EC divide the binodal section into two branches. One branch is the high-density (liquid) phase that is less asymmetric while the other branch corresponds to the more asymmetric low-density (gas) phase. Thus the matter on the left (right) of the binodal surface represents stable liquid (gas) phase. It is clearly seen here that the critical point ( P$_{c}$, α$_{c}$ ) depends on the density dependence of the symmetry energy associated with different Λ$_{v}$ values. 

We also indicate on the binodal surface the maximal isospin asymmetry (MA), α$_{MA}$ , of the system. Thus more neutron-rich matter on the right side of the sur- face when compressed/expanded at fixed α will never encounter a coexistence phase. Note here the maximal asymmetry is also quite sensitive to Λ$_{v}$ i.e. on E$_{sym}$ ( ρ, T ). Such effects found in the present study should have strong influence on the experimentally observed isospin distilla- tion phenomena [31] where the gas phase is more neutron- 

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rich (large n/p ratio) compared to the more asymmetric liquid phase. However for pressures P ≥ 0 . 10 MeV/fm 3 the magnitude of isospin distillation is more sensitive to the symmetry energy used. 

A new feature for LGP transition in asymmetric sys- tem, refereed to as retrograde condensation [21], arises when a nucleon gas prepared at an asymmetry α$_{c}$ < α < α$_{MA}$ is compressed at fixed total α . The matter re- mains mechanically stable but chemically unstable. Thus a coexisting liquid phase emerges which finally vanishes when the system leaves the binodal surface as a pure gas. As the extent Δ α = α$_{MA}$ − α$_{c}$ is found to decrease for softer symmetry energy with higher Λ$_{v}$, the possibility of such unique-phase condensation phenomena also be- comes minimal. 

The present study clearly suggests that for liquid-gas phase transition in hot asymmetric nuclear matter, the critical values of pressure and isospin asymmetry, the maximal asymmetry and the area and shape of the bin- odal surface are quite sensitive to the density dependence of symmetry energy with a stiffer symmetry energy leads to consistently smaller values of these thermodynamic variables. 

The existence of critical isospin asymmetry parameter α$_{c}$ at a given temperature indicates that for α > α$_{c}$ the system will not change completely into the liquid phase. Conversely, this suggests that at a fixed α there exists a critical temperature T$_{c}$ beyond which the system can only be in the gas phase at all pressures. In Fig. 5 we present T$_{c}$ as a function of α in the FSUGold set for different 

