Nuclear liquid-gas phase transition in generalized thermo-statistics

Transcription

1 Nuclear liquid-gas hase trasitio i geeralized thermo-statistics K. Miyazaki miyazakirorio.od.e.j Abstract A ew calculus of uclear liquid-gas hase trasitio i the stadard thermostatistics is alied to the geeralized thermo-statistics. Because the hase equilibrium is de ed i terms of itesive temerature ad cojugate extesive etroy, we have to solve higher-order simultaeous oliear equatios tha those i the stadard thermo-statistics. The features of hase trasitio are essetially the same as those i the stadard thermo-statistics. The hase trasitio is the rst order. There is o critical oit i the sectio of biodal surface. The caloric curve reroduces the exerimetal data well. We however caot d the retrograde codesatio redicted i the stadard thermo-statistics. The geeralized thermo-statistics [,2] is exected to be alicable to o-equilibrium systems ad small systems. Because they are also roduced i the multifragmetatio reactio [3,4] of heavy ios, the geeralized thermo-statistics recetly attracts cosiderable attetio [5-] i uclear hysics. O the other had, it is well kow that [2,3] the asymmetric uclear matter roduced i heavy-io reactio exhibits liquid-gas hase trasitio. It is therefore valuable to ivestigate uclear liquid-gas hase trasitio i the geeralized thermo-statistics. The asymmetric uclear matter is a biary system that has two ideedet chemical otetials of roto ad eutro. Accordig to the Gibbs coditio o hase equilibrium, both the chemical otetials i liquid ad gaseous hases are equilibrated. Withi the stadard thermo-statistics, the geometrical costructio [4-9] is usually used to ivestigate uclear liquid-gas mixed hase. To the cotrary, Refs. [20,2] have recetly develoed aother calculus, i which the two chemical otetials are calculated directly from the other itesive quatities of the system, temerature or ressure, baryo desity ad isosi asymmetry. I the reset aer we exted the works [20,2] to the geeralized thermo-statistics. Our formulatio of warm uclear matter is essetially based o Ref. [], which ivestigates isosi symmetric matter i the geeralized thermo-statistics. Its alicatio to asymmetric matter is straightforward. Usig the q-deformed exoetial ad logarithm ex q (x) [ + ( q) x ] =( q) ; ()

2 Nuclear liquid-gas hase trasitio i geeralized thermo-statistics the thermodyamic otetial er volume is give by i=; l q (x) x q q ; (2) = 2 m2 hi m2 h 3 i 2 2 m2! h! 0 i 2 2 m2 h 03 i 2 k B X Z d 3 k (2) 3 l i Eki q + ex q ; (3) k B where q is the ower-law idex, = 2 is the si degeeracy factor, k B is the Boltzma costat ad i is de ed usig the chemical otetial i ad the vector otetial V i as i = i V i : (4) Eki = (k2 + Mi 2 ) =2 is the eergy of ucleo. We have eglected the e ect of ati-ucleo. Moreover, it is oted [22,24] that is ot hysical itesive temerature but = (k B ) is oly the Lagrage multilier associated with the extremalizatio of a etroic measure. We have used the relativistic mea- eld model [23] to describe uclear matter. The e ective masses M() = m () ad the vector otetials V () = v () are determied by extremizig : M = S + m 2 h! 0 i hi hi + m 2 h 3 i h 3 i m m h! 0 i m m 2 h 03 i h 03 i m = 0; (5) V = B + m 2 h! 0 i hi hi + m 2 h 3 i h 3 i v v h! 0 i m 2 h 03 i v h 03 i v = 0; (6) M = S + m 2 hi hi + m 2 h 3 i h 3 i m m m 2 h! 0 i! m 2 h 03 i h! 0 i m h 03 i m = 0; (7) V = B + m 2 h! 0 i hi hi + m 2 h 3 i h 3 i v v h! 0 i m 2 h 03 i v 2 h 03 i v = 0; (8)

3 K. Miyazaki where the mea- elds are exressed [23] i terms of the e ective masses ad the vector otetials. The exlicit exressios of the derivatives of mea elds are also give i Ref. [23]. The baryo ad scalar desities are de ed by [,24] Z Bi = Z Si = d 3 k (2) 3 [ i (m i ; v i ; i ; ; q; k)] q ; (9) d 3 k Mi (2) 3 [ Eki i (m i ; v i ; i ; ; q; k)] q ; (0) where the q-deformed Fermi-Dirac distributio fuctio is [,24] 8 h >< + i (m i ; v i ; i ; ; q; k) = >: + (q ) E ki i k B i q for + (q ) E ki i k B for + (q ) E ki i k B The q-deformed Fermi itegrals are calculated usig the adative automatic itegratio with 20-oits Gaussia quadrature. > 0; 0: It is oted [22] that the hase equilibrium should be de ed i terms of itesive temerature T ad cojugate extesive etroy S. Here, they are itroduced through the Gibbs thermodyamic relatio []: () = U T S B + B ; (2) where the eergy desity U is U = 2 m2 hi m2 h 3 i 2 2 m2! h! 0 i 2 2 m2 h 03 i 2 + X Z d 3 k (2) 3 E ki [ i (m i ; v i ; i ; ; q; k)] q + X V i Bi : (3) i=; Because of Eq. (2), T ad S are determied from a sigle thermodyamic quatity. I ractice, accordig to the maig of Tsallis o-extesive thermo-statistics o the stadard Boltzma-Gibbs thermo-statistics [25-27], they are exressed by the Tsallis o-extesive etroy S q : i=; where T = C q ; (4) S B = l C q q ; (5) C q = + ( q) S q ; (6) 3

4 Nuclear liquid-gas hase trasitio i geeralized thermo-statistics B = B + B : (7) I order to guaratee C q > 0 we assume [] that the ower-law idex deeds o the baryo desity: q = 3 B = 0 B = ; (8) where 0 = 0:6fm 3 is the saturatio desity of symmetric uclear matter. The desity deedece is costraied from the coditio This is ecessary [] for the relatio P = P = 3 X Z i=; < q < 4=3: (9), where P is the ressure desity, d 3 k k 2 (2) 3 [ Eki i (m i ; v i ; i ; ; q; k)] q 2 m2 hi 2 2 m2 h 3 i m2! h! 0 i m2 h 03 i 2 : (20) Moreover, the coditio (9) guaratees [] ite values of the q-deformed Fermi itegrals i Eqs. (9), (0), (3) ad (20). For de ite values of hysical itesive temerature T, baryo desity B ad isosi asymmetry B a = B ; (2) B + B we solve 7th-rak oliear simultaeous equatios (5)-(8), (2), (7) ad (2) usig the globally coverget Newto-Rahso algorithm [28] so that the e ective masses, the vector otetials, the chemical otetials ad the o-extesive etroy are determied. The ressure desity, the eergy desity ad the extesive etroy are also determied at a time. The black curves i Figs. T = 0MeV ad a = 0:3 as fuctios of ressure. ad 2 show the solutios of chemical otetials for (We set the Boltzma costat as uit.) The black curves i Fig. 3 show the ressure-desity isotherms for several asymmetries. They exhibit tyical ature of va der Waals equatio-of-state. The solid arts o lower ressure ad desity are the braches for ure gas hase while the solid arts o higher ressure ad desity are the braches for ure liquid hase. The dashed arts corresod to the liquid-gas hase trasitio. They are ot however realized because the Gibbs coditio o hase equilibrium is ot satis ed. I fact, as the uclear matter is comressed i Figs. ad 2, we caot reach to the liquid braches from the gas braches. 4

5 K. Miyazaki So as to costruct the hysically reasoable liquid-gas mixed hase beig cosistet with the Gibbs coditio o hase equilibrium, we have to solve the followig 5th-rak simultaeous oliear equatios. The four equatios of them determie the e ective masses ad the vector otetials of roto ad eutro i gaseous hase: M (g) = (g) S + hi g m2 hi g h + m 2 3 i g m (g) h 3 i g m (g) h! 0 i g h! 0 i g m (g) m 2 h 03 i g h 03 i g m (g) = 0; (22) V (g) = (g) B + hi g m2 hi g v (g) + m 2 h 3 i g h 3 i g v (g) h! 0 i g h! 0 i g v (g) m 2 h 03 i g h 03 i g v (g) = 0; (23) M (g) = (g) S + hi g m2 hi g m (g) + m 2 h 3 i g h 3 i g m (g) h! 0 i g h! 0 i g m (g) m 2 h 03 i g h 03 i g m (g) = 0; (24) V (g) = (g) B + hi g m2 hi g v (g) + m 2 h 3 i g h 3 i g v (g) h! 0 i g h! 0 i g v (g) m 2 h 03 i g h 03 i g v (g) = 0; (25) where the su x g idicates gaseous hase. It is oted agai that the mea- elds are exressed i terms of the e ective masses ad the vector otetials. Similarly, the equatios for liquid hase are M (l) = (l) S + hi l m2 hi l + m 2 h 3 i l m (l) h 3 i l m (l) h! 0 i l h! 0 i l m (l) m 2 h 03 i l h 03 i l m (l) = 0; (26) V (l) = (l) B + hi l hi l m2 v (l) + m 2 h 3 i l h 3 i l v (l) h! 0 i l h! 0 i l v (l) m 2 h 03 i l h 03 i l v (l) = 0; (27) 5

6 Nuclear liquid-gas hase trasitio i geeralized thermo-statistics M (l) = (l) S + hi l m2 hi l m (l) + m 2 h 3 i l h 3 i l m (l) h! 0 i l h! 0 i l m (l) m 2 h 03 i l h 03 i l m (l) = 0; (28) V (l) = (l) B + hi l hi l m2 v (l) + m 2 h 3 i l h 3 i l v (l) h! 0 i l h! 0 i l v (l) m 2 h 03 i l h 03 i l v (l) = 0: (29) The other two equatios secify the desities of roto ad eutro i liquid-gas mixed hase: B = 2 ( a) B = f g (g) B + ( f g) (l) B ; (30) B = 2 ( + a) B = f g (g) B + ( f g) (l) B ; (3) where 0 f g is the ratio of gas. The baryo desities i gaseous ad liquid hases are (g) Bi = Z (l) Bi = Z d 3 k h (2) 3 i d 3 k h (2) 3 i m (g) i m (l) i q; ; v (g) i ; i ; g ; q g ; ki (32) q: ; v (l) i ; i ; l ; q l ; ki (33) It is oted that the virtual temerature is ot equilibrated. The ower-law idices i gaseous ad liquid hases are also determied self-cosistetly because they deed o each desity of the two hases: q g = 3 q l = 3 (g) B + (g) B (g) B + (g) B (l) B + (l) B (l) B + (l) B = 0 ; (34) = = 0 : (35) = Moreover, we have to assume the thermodyamic relatios i gaseous ad liquid hases: P g + U g T S g (g) B + (g) B = 0; (36) P l + U l T S l (l) B + (l) B = 0; (37) 6

7 K. Miyazaki where U g = 2 m2 hi 2 g + 2 m2 h 3 i 2 g 2 m2! h! 0 i 2 g 2 m2 h 03 i 2 g + X Z d 3 k h i q X (2) 3 E(g) ki i m (g) i ; v (g) i ; i ; g ; q g ; k + i=; i=; V (g) i (g) Bi ; (38) U l = 2 m2 hi 2 l + 2 m2 h 3 i 2 l 2 m2! h! 0 i 2 l 2 m2 h 03 i 2 l + X Z d 3 k h i q X (2) 3 E(l) ki i m (l) i ; v (l) i ; i ; l ; q l ; k + i=; i=; V (l) i (l) Bi : (39) The last equatio imoses the equilibrium coditio o ressures i the two hases: P g 3 X Z i=; 2 m2 hi 2 g = P l X Z 3 d 3 k k 2 h (2) 3 E (g) i ki i=; 2 m2 hi 2 l m (g) i ; v (g) i i q ; i ; g ; q g ; k 2 m2 h 3 i 2 g + 2 m2! h! 0 i 2 g + 2 m2 h 03 i 2 g d 3 k k 2 h i q (2) 3 E (l) i m (l) i ; v (l) i ; i ; l ; q l ; k ki 2 m2 h 3 i 2 l + 2 m2! h! 0 i 2 l + 2 m2 h 03 i 2 l : (40) Solvig Eqs. (22)-(3), (34)-(37) ad (40) for de ite values of temerature T, baryo desity B ad isosi asymmetry a, we have the e ective masses, the vector otetials, the o-extesive etroies ad the ower-law idices i gaseous ad liquid hases as well as the chemical otetials ad the ratio of gas (or liquid) hase. The baryo ad eergy desities i the two hases ad the equilibrated ressure are also determied at a time. The total etroy is give by f g S g + ( f g ) S l because S g ad S l are the extesive etroies. The total eergy desity is also give by U = f g U g + ( f g ) U l. The solutios of chemical otetials for T = 0MeV ad a = 0:3 are show by the red curves i Figs. ad 2. As the uclear matter is comressed betwee P = 0:049MeV=fm 3 ad 0:09MeV=fm 3, decreases while icreases. The results idicate that the ratios of eutro icrease i both of liquid ad gaseous hases. This is clearly see i Fig. 4, where the asymmetry a g = (g) B (g) B (g) B +(g) B i gaseous hase ad the asymmetry a l = (l) B (l) B (l) B +(l) B i liquid hase are show by the red ad blue curves, resectively. The ratio of liquid hase f l = :0 f g is also show by the black curve. Moreover, we see i Figs. ad 2 that =P ad =P have discotiuities at the begiig ad edig oits of the hase trasitio. It is therefore cocluded [] that the hase trasitio is the rst order. The ressure-desity isotherms i the hase trasitio are also show by the red curves i Fig. 3. As the asymmetry becomes higher, the uclear matter is comressed more weakly while a gai of ressure is larger. The liquid-gas mixed hase for a = 0:5 7

8 Nuclear liquid-gas hase trasitio i geeralized thermo-statistics is ot show because above some baryo desity we have o solutios of Eqs. (22)- (3), (34)-(37) ad (40). The roblem is revealed more geerally i the biodal surface of Fig. 5. The red ad blue curves are gas ad liquid braches. The former eds at a = 0:84 while the latter eds at a = 0:42. Cosequetly, there is o critical oit, o which the two braches coect with each other excet for equal cocetratio a = 0. The absece of critical oit is essetially due to the e ective desity-deedet ucleoucleo iteractio. I ractice, it has bee also foud [5,9,20] i the stadard thermostatistics. However, there are di ereces betwee the stadard ad geeralized thermostatistics. The regio of o gas ad liquid braches betwee a = 0:42 ad 0:84 is much wider tha those i Refs. [5,9,20]. Moreover, the gas brach i Fig. 5 does ot show the retrograde codesatio redicted i Refs. [4-20]. Next, we ivestigate the thermodyamic roerties of uclear matter for de ite values of ressure P, baryo desity B ad isosi asymmetry a. Solvig 7th-rak oliear simultaeous equatios (5)-(8), (7), (20) ad (2), the e ective masses, the vector otetials, the chemical otetials ad the virtual temerature are determied. The eergy desity is also determied at a time. The hysical temerature T is calculated from a variatio of Eq. (2): T l (T=) q = U + P B + B The, the extesive etroy is calculated from Eq. (5). The black curves i Figs. B : (4) 6 ad 7 show the solutios of chemical otetials for P = 0:03MeV=fm 3 ad a = 0:3 as fuctios of temerature T. The solid arts o lower temerature are the braches for ure liquid hase while the solid arts o higher temerature are the braches for ure gas hase. The dashed arts corresod to the liquid-gas hase trasitio. The black curve i Fig. 8 shows the caloric curve, the temerature as a fuctio of the excitatio eergy er article, which is de ed by subtractig the bidig eergy of cold asymmetric uclear matter [23] from U= B. The solid art is the ure liquid hase while the dashed art is the liquid-gas mixed hase. The triagles are the exerimetal data of Fig. 5 i Ref. [29]. The black dashed curves i Figs. 6-8 are ot realized because the Gibbs coditio o hase equilibrium is ot satis ed. I fact, as the uclear matter is heated i Figs. 6 ad 7, we caot reach to the gas braches from the liquid braches. So as to costruct the hysically reasoable liquid-gas mixed hase beig cosistet with the Gibbs coditio o hase equilibrium, for de ite values of ressure P, baryo desity B ad isosi asymmetry a, we have to solve 6th-rak simultaeous oliear equatios (22)-(3), (34), (35), the thermodyamic relatios i gaseous ad liquid hases, P + U g T S g (g) B + (g) B = 0; (42) 8

9 K. Miyazaki P + U l T S l (l) B + (l) B = 0; (43) ad the equilibrium coditios o ressures i the two hases, P = 3 P = 3 X Z i=; 2 m2 hi 2 g X Z i=; 2 m2 hi 2 l d 3 k k 2 h (2) 3 E (g) i ki m (g) i ; v (g) i i q ; i ; g ; q g ; k 2 m2 h 3 i 2 g + 2 m2! h! 0 i 2 g + 2 m2 h 03 i 2 g ; (44) d 3 k k 2 h (2) 3 E (l) i ki m (l) i i q ; v (l) ; i ; l ; q l ; k i 2 m2 h 3 i 2 l + 2 m2! h! 0 i 2 l + 2 m2 h 03 i 2 l : (45) We have the e ective masses, the vector otetials, the o-extesive etroies ad the ower-law idices i gaseous ad liquid hases as well as the chemical otetials, the ratio of gas (or liquid) hase ad the iteral hysical temerature. The eergy desities i the two hases are also determied at a time. The solutios of chemical otetials for P = 0:03MeV=fm 3 ad a = 0:3 are show by the red curves i Figs. 6 ad 7. As the uclear matter is heated from T = 5:79MeV to 8:80MeV, icreases while decreases. The results idicate that the ratios of eutro decrease i both of liquid ad gaseous hases. This is clearly see i Fig. 9, where the asymmetries i the two hases are show by the red ad blue curves, resectively. The ratio of liquid f l = :0 f g is also show by the black curve. The boilig of uclear liquid uder costat ressure is show more geerally i Fig. 0, where the blue ad red curves are the boilig temerature of uclear liquid ad the codesed temerature of uclear gas, resectively. The dotted lies idicate that the uclear liquid of asymmetry a = 0:3 begis to evaorate ito highly asymmetric uclear gas of a g = 0:977, which is almost comosed of eutros. Whe the evaoratio comletes, the liquid hase is i rather symmetric state of asymmetry a l = 0:057. The caloric curve i liquid-gas hase trasitio is also show by the red curve i Fig. 8. It reroduces the exerimetal data better tha the black dashed curve. Cosequetly, we ca see that the exerimetal data rovide the clear evidece of uclear liquid-gas hase trasitio. The liquid-gas hase trasitio i asymmetric uclear matter has bee ivestigated for the rst time withi the geeralized thermo-statistics. A ew calculus of the hase trasitio i the stadard thermo-statistics is alicable to the geeralized thermo-statistics. We however have to solve higher-order simultaeous oliear equatios tha those i the stadard thermo-statistics. This is because the hase equilibrium ca be de ed oly i terms of itesive temerature ad cojugate extesive etroy. So as to determie them the Gibbs thermodyamic relatio should be imosed o the thermodyamic 9

10 Nuclear liquid-gas hase trasitio i geeralized thermo-statistics otetial. Moreover, the ower-law idex deeds o the baryo desity so that the extesive etroy is ot egative. We have ivestigated the isothermal comressio ad the boilig uder costat ressure. The essetial features of uclear liquid-gas hase trasitio i the geeralized thermo-statistics are similar to those i the stadard thermo-statistics. The derivatives of chemical otetials have discotiuities at the eds of hase trasitio. We ca therefore coclude that the hase trasitio is the rst order. The liquid ad gas braches i the sectio of biodal surface are coected with each other oly o the equal cocetratio, but there is o critical oit. The caloric curve calculated for aroriate values of asymmetry ad ressure reroduces the exerimetal data well. We however caot d the retrograde codesatio redicted i the stadard thermo-statistics. Refereces [] C. Tsallis, Braz. J. Phys. 29 (999). [2] Eurohysics News 36 (2005) No.6 [htt:// [3] V.E. Viola et al., Phys. Re. 434 (2006) [arxiv:ucl-ex/060402]. [4] V.A. Karaukhov, Phys. Elem. Part. Atom. Nucl. 37 (2006) 32 [htt:// [5] D.B. Walto ad J. Rafelski, Phys. Rev. Lett. 84 (2000) 3 [arxiv:he-h/ ]. [6] K.K. Gudima, A.S. Parva, M. P oszajczak ad V.D. Toeev, Phys. Rev. Lett. 84 (2000) 469 [arxiv:ucl-th/ ]. [7] C.E. Aguiar ad T. Kodama, Physica A 320 (2003) 37. [8] A. Drago, A. Lavago ad P. Quarati, Physica A 344 (2004) 472 [arxiv:uclth/03208]. [9] F.I.M. Pereira, R. Silva ad J.S. Alcaiz, Phys. Rev. C 76 (2007) 0520 [arxiv:uclth/ ]. [0] T. Osada ad G. Wilk, arxiv:ucl-th/ [] K. Miyazaki, Mathematical Physics Prerit Archive (m_arc) [2] J. Richert ad P. Wager, Phys. Re. 350 (200) [arxiv:ucl-th/ ]. [3] S.D. Guta, A.Z. Mekjia ad M.B. Tsag, Advaces i Nuclear Physics, Vol. 26 (Kluwer Academic, 200) [arxiv:ucl-th/ ]. [4] H. Müller ad B. D. Serot, Phys. Rev. C 52 (995) 2072 [arxiv:ucl-th/950503]. 0

11 K. Miyazaki [5] W.L. Qia, R-K. Su ad P. Wag, Phys. Lett. B 49 (2000) 90 [arxiv:uclth/ ]. [6] V.M. Kolomietz, A.I. Sazhur, S. Shlomo ad S.A. Firi, Phys. Rev. C 64 (200) [arxiv:ucl-th/00403]. [7] P.K. Pada, G. Klei, D.P. Meezes ad C. Providêcia, Phys. Rev. C 68 (2003) 0520 [arxiv:ucl-th/ ]. [8] P. Wag, D.B. Leiweber, A.W. Thomas ad A.G. Williams, Nucl. Phys. A 748 (2005) 226 [arxiv:ucl-th/ ]. [9] J. Xu. L-W. Che, B-A. Li ad H-R. Ma, Phys. Lett. B 650 (2007) 348 [arxiv:uclth/ ]. [20] K. Miyazaki, Mathematical Physics Prerit Archive (m_arc) [2] K. Miyazaki, Mathematical Physics Prerit Archive (m_arc) [22] S. Abe, S. Martíez, F. Peii ad A. Plastio, Phys. Lett. A 28 (200) 26 [arxiv:cod-mat/0002]. [23] K. Miyazaki, Mathematical Physics Prerit Archive (m_arc) [24] A.M. Teweldeberha, A.R. Plastio ad H.G. Miller, Phys. Lett. A 343 (2005) 7. [arxiv:cod-mat/050456]. [25] E. Vives ad A. Plaes, Phys. Rev. Lett. 88 (2002) [arxiv:codmat/006428]. [26] R. Toral, Physica A 37 (2003) 209. [27] A.S. Parva, Phys. Lett. A 350 (2006) 33 [arxiv:cod-mat/ ]. [28] W.H. Press, S.A. Teukolsky, W.T. Vetterlig ad B.P. Flaery, Numerical Recies i C 2d editio, 992, Cambridge Uiversity Press, [htt:// [29] A. Ruagma et al., Phys. Rev. C 66 (2002)

12 Nuclear liquid-gas hase trasitio i geeralized thermo-statistics µ / Asymmetry = 0.3 T = 0 MeV Pressure (MeV/fm 3 ) Figure : The roto chemical otetial as a fuctio of ressure for T = 0MeV ad a = 0:3. The black dashed curve does ot satisfy the Gibbs coditio o hase equilibrium while the hysically recise liquid-gas hase trasitio is exressed by the red curve. 2

13 K. Miyazaki µ / Asymmetry = 0.3 T = 0 MeV Pressure (MeV/fm 3 ) Figure 2: The same as Fig. but for eutro chemical otetial. 3

14 Nuclear liquid-gas hase trasitio i geeralized thermo-statistics a=0.5 a=0.4 a=0.4 a=0.3 Pressure (MeV/fm 3 ) a=0.2 a=0. a=0.3 0 T=0MeV a= a= Baryo desity (fm 3 ) Figure 3: The ressure-desity isotherms at T = 0MeV for the asymmetries from a = 0: to a = 0:5. The black dashed curves do ot satisfy the Gibbs coditio o hase equilibrium while the hysically recise liquid-gas hase trasitios are exressed by the red curves. 4

15 K. Miyazaki 0.8 Ratio of liquid Asymmetry of gas Asymmetry of liquid Pressure (MeV/fm 3 ) Figure 4: The black curve is the ratio of liquid f l = f g i the liquid-gas hase trasitio for T = 0MeV ad a = 0:3. The red ad blue curves are the asymmetries i gaseous ad liquid hases, a g = (g) B (g) B +(g) B (g) B ad a l = (l) B (l) B +(l) B (l) B. 5

16 Nuclear liquid-gas hase trasitio i geeralized thermo-statistics T = 0MeV Pressure (MeV/fm 3 ) liquid gas Asymmetry Figure 5: The sectio of biodal surface at T = 0MeV. 6

17 K. Miyazaki Asymmetry = 0.3 P = 0.03 MeV/fm 3 µ / T (MeV) Figure 6: The roto chemical otetial as a fuctio of hysical temerature for P = 0:03MeV=fm 3 ad a = 0:3. The black dashed curve does ot satisfy the Gibbs coditio o hase equilibrium while the hysically recise liquid-gas hase trasitio is exressed by the red curve. 7

18 Nuclear liquid-gas hase trasitio i geeralized thermo-statistics µ / Asymmetry = 0.3 P = 0.03 MeV/fm T (MeV) Figure 7: The same as Fig. 6 but for eutro chemical otetial. 8

19 K. Miyazaki 0 8 T (MeV) Asymmetry = 0.3 Pressure = 0.03 MeV/fm E * /A (MeV) Figure 8: The caloric curves for P = 0:03MeV=fm 3 ad a = 0:3. The black solid curve reresets the ure liquid hase. The black dashed curve does ot satisfy the Gibbs coditio o hase equilibrium while the hysically recise liquid-gas mixed hase is exressed by the red curve. 9

20 Nuclear liquid-gas hase trasitio i geeralized thermo-statistics Ratio of liquid Asymmetry of gas Asymmetry of liquid T (MeV) Figure 9: The black curve is the ratio of liquid f l = f g i the liquid-gas hase trasitio for P = 0:03MeV=fm 3 ad a = 0:3. The red ad blue curves are the asymmetries i gaseous ad liquid hases, a g = (g) B (g) B +(g) B (g) B ad a l = (l) B (l) B +(l) B (l) B. 20

21 K. Miyazaki 9 8 gas 7 Temerature (MeV) liquid P = 0.03 MeV/fm Asymmetry Figure 0: The boilig (red) ad codesed (blue) curves of uclear liquid ad gas uder costat ressure P = 0:03MeV=fm 3. 2