Relativistic nuclear matter in generalized thermo-statistics

Transcription

1 Relativistic nuclear matter in generalized termo-statistics K. Miyazaki Abstract Te generalized Fermi-Dirac termo-statistics is developed for relativistic nuclear matter. We introduce te generalized termodynamic potential using te q-deformed exponential and logaritm. Te baryon density as a form of te q- expectation value. Ten, we use te mapping of te non-extensive termo-statistics onto te extensive one. Terefore, te Clausius extensive entropy and te pysical intensive temperature are determined troug te Gibbs termodynamic relation. Te power-law index is constrained into a region < q < 4=3 and assumed to be density dependent. Te pressure-density isoterms exibit te liquid-gas pase transition. Te calculated critical temperature agrees wit te experimental value for nite nuclei of intermediate masses. Tere is little di erence in te boiling temperatures from te standard and generalized termo-statistics, altoug te latter improves te caloric curve of nuclear liquid. It is found tat te cut-o prescription by Teweldeberan et al. for te generalized Fermi-Dirac distribution breaks te tird-law of termodynamics. Introduction Now, te termo-statistics generalized by power-law [,] as been acknowledged to be beyond te standard Boltzmann-Gibbs termo-statistics and applicable to non-equilibrium systems, small systems, gravitational systems and oters. (Te compreensive references are found in Ref. [3].) Altoug te teoretical origin [4] of te power-law is still an open problem, te generalized termo-statistics is also useful [5-7] in nuclear pysics. In te present paper we investigate warm nuclear matter in te relativistic mean- eld (RMF) model [8] witin te generalized termo-statistics. Altoug tere ave been already similar works [9,0], our investigation is essentially di erent from tem in te following respects. ) We use te RMF model of eld-dependent meson-nucleon coupling constants developed in Ref. []. Te eld-dependence is due to many-body correlation in dense nuclear medium. Te correlation migt lead to te power-law. ) Te baryon densities in Refs. [9,0] do not ave te forms of q-expectation but te normal ones. 3) We introduce te pysical intensive temperature and te conjugate extensive entropy [-4] troug te Gibbs termodynamic relation.

2 Warm nuclear matter in generalized termo-statistics of power-law Generalized Fermi-Dirac termo-statistics In te standard Fermi-Dirac termo-statistics, te termodynamic potential per volume of nuclear matter in te RMF model [8] is = m i m!! 0 i k B T d 3 k () 3 ln + exp E (k) : () k B T We ave neglected te e ect of anti-nucleon. M and E (k) = (k + M ) = are te e ective mass and te energy of nucleon in nuclear matter. Te Boltzmann constant is k B. Te spin-isospin degeneracy is = 4. Te is de ned using te cemical potential and te vector potential V of nucleon as = V: () Ten, in order to develop te generalized Fermi-Dirac termo-statistics, we extend Eq. () by replacing exp(x) and ln(x) wit q-deformed functions: Terefore, we ave were exp q (x) [ + ( q) x ] =( q) ; (3) q = m i m!! 0 i q (k) = + ln q (x) x q q : (4) d 3 k ( q (k)) () 3 q q ; (5) + ( q) E (k) q : (6) A su x q as been attaced to temperature because as sown below T q is not a pysical intensive temperature. Consequently, te baryon density is B = k BT q d 3 q () 3 ( q (k)) = were te generalized Fermi-Dirac distribution is n q (k) = ( q (k)) + ( q) E (k) q = + d 3 k () 3 (n q (k)) q ; (7) i + (q ) E (k) q : (8) Altoug te precise derivation of te generalized Fermi-Dirac distribution as been In te present paper te Fermi-Dirac termo-statistics means te termodynamics based on te quantum statistical mecanics of relativistic spin-= particle.

3 K. Miyazaki attempted in Ref. [5], Eq. (8) is widely acknowledged [9,0,6,7] to be useful. Te baryon density (7), wic as a form of q-expectation, is te same as Ref. [7] but is di erent from Refs. [9,0] were te normal expectations are used. As will be sown in Appendix we can derive te entropy in di erentiating q by T q. In te present work we owever introduce te extensive entropy S troug te Gibbs termodynamic relation [3]: q = U q T S B : (9) (Precisely, S is te entropy per volume and S= B is te entropy per baryon. Bot te quantities can be de ned only for extensive entropy but not for non-extensive one.) U q is te energy density de ned by U q = d 3 k () 3 (E (k) + V ) (n q (k)) q + m i m!! 0 i : (0) Te intensive quantity T, wic is conjugate to te extensive entropy S, is just te pysically observed temperature. To te contrary T q is only an inverse of te Lagrange multiplier [7]. In tis sense our teory is te extensive termo-statistics generalized by power-law. Here, it is noted [] tat te power-law is also involved in te non-extensive termo-statistics. If te power-law index is identical to te non-extensivity index, we can utilize te mapping [3,4,8] of te non-extensive termo-statistics onto te extensive one. Bot of te intensive temperature T and te extensive entropy S are derived from te proper non-extensive entropy S q []: S q (A B) = S q (A) + S q (B) + ( q) S q (A) S q (B) : () In practice te non-extensivity () is equivalent to te extensivity, ln + ( q) S q (A B) = ln + ( q) S q (A) + ln + ( q) S q (B) : () We can terefore de ne [,3] te Clausius extensive entropy, wic satis es te rst law of termodynamics: S B = ln C q q ; (3) were C q = + ( q) S q : (4) Te conjugate intensive temperature [-4] is given by T = C q T q : (5) 3

4 Warm nuclear matter in generalized termo-statistics of power-law Te explicit expression of S q is not necessary because it is calculated troug te Gibbs termodynamic relation (9). For a de nite value of pysical temperature T, given a trial value of S q, te virtual temperature T q is determined from Eq. (5). Once T q is given, te termodynamic potential, te energy density and te baryon density are calculated. Ten, te extensive entropy S is calculated from te Gibbs termodynamic relation (9). Finally, te non-extensive entropy S q is calculated again from Eq. (3). Te procedure is iterated until te value of S q is converged. On te oter and, for a de nite value of virtual temperature T q, te termodynamic relation becomes a nonlinear equation to determine pysical temperature T : T ln T=T q q = U q q B : (6) Te e ective mass M and te vector potential V are determined by extremizing te termodynamic potential q. Te resultant equations are formally te same as tose at T = M = S ( m ) + ( ) ( m ) m m + ( ) m v gnn m! g NN! M M = 0; V = B + ( ) v + ( ) ( m ) v m! v gnn! m g NN M M = 0; (8) were M is te free nucleon mass, M = m M and V = v M. Te scalar density is de ned as S = d 3 k M () 3 E (k) (n q (k)) q : (9) Te e ective meson-nucleon coupling constant is given by [] g NN(!) = g NN(!) = ( + ) + ( ) m v g NN(!) : (0) Te free meson-nucleon coupling constants g NN and g NN! and te renormalization parameter are determined so as to reproduce te properties of nuclear matter at saturation. Teir values are found in Table of Ref. []. 4

5 K. Miyazaki 3 Numerical analyses We ave already investigated in Ref. [9] te standard Fermi-Dirac termo-statistics of nuclear matter and found te critical temperature T = 6:47MeV. (Altoug te e ect of anti-nucleon is taken into account in Ref. [9], it is negligible below T = 0MeV.) Te value agrees well wit te empirical one [0] T = 6:6 0:86MeV. Tere is terefore no room to introduce te generalized termo-statistics. However, te above value is for in nite nuclear matter. Te critical temperatures for nite nuclei [] are lower tan it. Te present paper studies weter te properties of warm nite nuclei are reproduced in te generalized termo-statistics of nuclear matter. Using a partial integration in Eq. (5) we ave q = P + k B T 6 q k B T q k 4 3q q ; () k! were P is te pressure, P = 3 d 3 k () 3 k E (k) (n q (k)) q m i + m!! 0 i : () Te termodynamic relation P = q terefore requires < q < 4=3: (3) Tis constraint is also imposed on te rst term in Eq. (). In fact, following to te similar analysis in Ref. [0], te rst term beaves asymptotically in te limit k! : d 3 k () 3 k E (k) (n q (k)) q! k4 k q q = k 4 3q q : (4) Te asymptotic beavior of te rst term in te energy density (0) is also te same. Te pressure and te energy density are nite only if te power-law index satis es te condition (3). Moreover, te baryon density (7) beaves asymptotically as d 3 k () 3 (n q (k)) q! k3 k q q = k 3 q q ; (5) and so a constraint < q < 3= is imposed, wile te scalar density (9) beaves asymptotically as d 3 k M () 3 E (k) (n q (k)) q! k k q q = k q q ; (6) and so a constraint < q < is imposed. In general te second law of termodynamics [] also requires q <. Consequently, Eq. (3) is te condition for nite values of 5

6 Warm nuclear matter in generalized termo-statistics of power-law te pressure, te energy density, te baryon and scalar densities. It is looser tan te condition < q < 5=4 in Ref. [0] because of te q-expectations in Eqs. () and (4)-(6) against te normal expectations in Ref. [0]. We ave introduced te pysical intensive temperature in Eq. (5). It may be owever negative for q > because te entropy is generally large at low density. (See Eq. (3).) So as to avoid a problem jq j sould be satis ed at low baryon densities. In te present work we assume tat te power-law index depends on te baryon density as follows: were 0 is te saturation density. q = 3 B = 0 B = 0 + ; (7) At present te pysical origin of power-law is an open problem. If it is due to te many-body correlation incorporated in te e ective coupling constant (0), te density dependence of power-law index is reasonable because te correlation is stronger at iger density. We still ave to specify te cut-o prescription [7] for te generalized Fermi-Dirac distribution (8), wic can be de ned precisely only for + (q ) E (k) > 0. Here, we follow te standard Tsallis prescription: 8 >< n q (k) = >: i + + (q ) E (k) q for + (q ) E (k) > 0 for + (q ) E (k) 0 : (8) For de nite pysical temperature T and te baryon density B we solve 4t-rank nonlinear simultaneous equations (7), (9), (7) and (8) using Newton-Rapson algoritm, so tat we ave te e ective mass M, te vector potential V, te cemical potential and te non-extensive entropy S q. Te pressure, te energy density and te extensive entropy are determined at a time. Figure sows te isoterms on pressure-density plane. (In te following calculations we set k B =.) Tey exibit typical nature of van der Waals equation-of-state. Te critical temperature is T = :09MeV and te critical point lies on P = 0:96MeV=fm 3 and B = 0:058fm 3. Te as temperature, above wic te pressure is always positive at any density, is T = 0:46MeV. Bot te temperatures are lower tan te values [9] in te standard Fermi-Dirac termo-statistics. However, te critical temperature agrees wit T = :4 0:99MeV [] for nite nuclei of 40 < A < 80. Te result suggests tat te generalized termo-statistics of nuclear matter takes into account niteness of nuclei. Te black and red curves in Fig. sow te pressure-density isoterms at T = 0MeV in te standard and generalized termo-statistics, respectively. Tere is large di erence between te dotted parts in te two curves, wic correspond to te liquid-gas pase transition. Because we are founded on te extensive entropy in Eq. 6

7 K. Miyazaki (9), te pase equilibrium condition is te same as te standard Gibbs one. Terefore, te Maxwell construction of pase equilibrium between liquid and gas pases leads to te orizontal lines, wile te dotted parts are not pysically realized. We see tat in te equilibrium pressure of liquid-gas mixed pase tere is little di erence between te standard and generalized termo-statistics. Next, at de nite pressure P = 0:0MeV=fm 3 we calculate te caloric curve by solving 4t-rank nonlinear simultaneous equations (7), (7), (8) and (), so tat we ave te e ective mass M, te vector potential V, te cemical potential and te virtual temperature T q. Te pysical temperature T is determined from Eq. (6). Ten, te extensive entropy is calculated from Eq. (3). Te result is sown by te red curve in Fig. 3 wile te black curve is te result in te standard Fermi-Dirac termo-statistics. Te orizontal lines are te boiling temperatures from te Maxwell construction of liquid-gas pase equilibrium. Te experimental data are from Ref. [3]. Altoug tere is little di erence in te two boiling temperatures, te generalized termo-statistics improves te caloric curve in te region of low excitation or nuclear liquid. We ave used te cut-o prescription (8) for te generalized Fermi-Dirac distribution, wile Ref. [0] used anoter one proposed in Ref. [7]: 8 >< n q (k) = >: i + + (q ) E (k) q i + + ( q) E (k) q for E (k) > 0 for E (k) 0 : (9) Tis is based on a requirement tat te q-deformed exponential satis es te similar relation exp q ( x) = = exp q (x) to exp ( x) = = exp (x) of te normal exponential. We ave recalculated te caloric curve using (9) but found tat te tird-law of termodynamics is broken. As a matter of fact Fig. 4 sows te Clausius entropy per baryon S= B as a function of te pysical temperature T. Te solid and dotted curves are te results using te cut-o prescription (8) and (9), respectively. Te dased curve is te result in te standard Fermi-Dirac termo-statistics. We really see lim T!0 (S= B ) = 0 for Eq. (8) but S= B = 0 at T > 0 for Eq. (9). Te problem caused by Eq. (9) migt be caracteristic of our model of te generalized termo-statistics. It is owever concluded tat te standard Tsallis prescription (8) is robust in application of te generalized Fermi-Dirac distribution. 4 Summary We ave investigated te generalized termo-statistics of nuclear matter. Our formulation starts at te q-deformed termodynamic potential expressed in terms of te q-deformed 7

8 Warm nuclear matter in generalized termo-statistics of power-law exponential and logaritm. Te baryon density is directly obtained in di erentiating te termodynamic potential by cemical potential. It as a form of te q-expectation value. We ave used te mapping of te non-extensive termo-statistics onto te extensive one, so tat te Clausius extensive entropy and te conjugate intensive temperature are determined troug te Gibbs termodynamic relation. Te power-law index is constrained into a region < q < 4=3 so as to produce nite values of pressure, energy density, baryon and scalar densities. Moreover, te index is assumed to be density dependent so tat te pysical intensive temperature is not negative at low baryon densities. We ave calculated te pressure-density isoterms. Tey exibit te nature of van der Waals equation-of-state. Te calculated critical temperature is lower tan te empirical value for in nite nuclear matter but agrees wit te experimental value for nite nuclei of intermediate masses. Te generalized termo-statistics can take into account te e ect by niteness of nuclei. In te equilibrium pressure of liquid-gas mixed pase from te Maxwell construction, tere is little di erence between te standard and generalized termo-statistics. Next, we ave calculated te caloric curve. In te boiling temperature from te Maxwell construction of liquid-gas mixed pase tere is also little di erence between te standard and generalized termo-statistics. However, te latter improves te caloric curve in pure pase of nuclear liquid. Moreover, we ave investigated te two cut-o prescriptions by Tsallis and by Teweldeberan et al. for te generalized Fermi-Dirac distribution. It is found tat te latter breaks te tird-law of termodynamics. In te present paper we ave considered symmetric nuclear matter. In practice te nuclear liquid-gas pase transition is experimentally investigated in eavy-ion reactions, wic produce te asymmetric nuclear matter. It is te binary system tat as two independent cemical potentials of proton and neutron. Te pase transition in binary system cannot be described in te Maxwell construction, but we need te Gibbs construction [4] to take into account te equilibrium of two cemical potentials. In a future work we will extend te present investigation to asymmetric nuclear matter. Appendix Here, we derive te entropy from di erentiating te termodynamic potential (5) by virtual temperature. Because Eq. (6) is rewritten as E (k) = ( q (k) ) q k B T q q ; (30) we ave S q T q = k B " d 3 k ( q (k)) q () 3 q (n q (k)) q ( q (k) ) q q # : (3) 8

9 K. Miyazaki Moreover, because Eq. (8) is rewritten as q (k) = n q (k) q (k) ; (3) Eq. (3) becomes S q = k B d 3 k (nq (k)) q () 3 + ( q (k)) q n q (k) : (33) q q Finally, using q (k) = [ n q (k) ] from Eq. (3), we ave d 3 k (n q (k)) q + ( n q (k)) q S q = k B () 3 ; q d 3 k nq (k) (n q (k)) q = k B () 3 + ( n q (k)) ( n q (k)) q q q = k B d 3 k () 3 [(n q (k)) q ln q (n q (k)) + ( n q (k)) q ln q ( n q (k))]: (34) Te result is a q-deformed extension of te Boltzmann-Fermi-Dirac entropy. Te essentially same result is also derived in Ref. [7]. ; References [] C. Tsallis, Braz. J. Pys. 9 (999). [] Europysics News 36 (005) No.6 [ttp:// [3] C. Tsallis, ttp://tsallis.cat.cbpf.br/temuco3.pdf. [4] G. Wilk and. W odarczyk, Pys. Rev. Lett. 84 (000) 770 [arxiv:epp/ ]. [5] D.B. Walton and J. Rafelski, Pys. Rev. Lett. 84 (000) 3 [arxiv:ep-p/990773]. [6] K.K. Gudima, A.S. Parvan, M. P oszajczak and V.D. Toneev, Pys. Rev. Lett. 84 (000) 469 [arxiv:nucl-t/000305]. [7] C.E. Aguiar and T. Kodama, Pysica A 30 (003) 37. [8] B.D. Serot and J.D. Walecka, Advances in Nuclear Pysics, Vol. 6 (Plenum, New York, 986). [9] A. Drago, A. Lavagno and P. Quarati, Pysica A 344 (004) 47 [arxiv:nuclt/0308]. 9

10 Warm nuclear matter in generalized termo-statistics of power-law [0] F.I.M. Pereira, R. Silva and J.S. Alcaniz, Pys. Rev. C 76 (007) 050 [arxiv:nuclt/ ]. [] K. Miyazaki, Matematical Pysics Preprint Arcive (mp_arc) [] S. Abe, S. Martínez, F. Pennini and A. Plastino, Pys. Lett. A 8 (00) 6 [arxiv:cond-mat/000]. [3] E. Vives and A. Planes, Pys. Rev. Lett. 88 (00) 0060 [arxiv:condmat/00648]. [4] R. Toral, Pysica A 37 (003) 09. [5] H.H. Aragão-Rêgo, D.J. Soares, L.S. Lucene, L.R. da Silva, E.K. Lenzi and Kwok Sau Fa, Pysica A 37 (003) 99. [6] U. T{rnakl{, F. Büyükk{l{ç and D. Demiran, Pys. Lett. A 45 (998) 6 [arxiv:cond-mat/98096]. [7] A.M. Teweldeberan, A.R. Plastino and H.G. Miller, Pys. Lett. A 343 (005) 7. [arxiv:cond-mat/050456]. [8] A.S. Parvan, Pys. Lett. A 350 (006) 33 [arxiv:cond-mat/050639]. [9] K. Miyazaki, Matematical Pysics Preprint Arcive (mp_arc) [0] J.B. Natowitz et al., Pys. Rev. Lett. 89 (00) 70 [arxiv:nucl-ex/00405]. [] J.B. Natowitz et al., arxiv:nucl-ex/ [] S. Abe and A.K. Rajagopal, Pys. Rev. Lett. 9 (003) 060 [arxiv:condmat/ ]. [3] J.B. Natowitz et al., Pys. Rev. C 65 (00) [arxiv:nucl-ex/00606]. [4] K. Miyazaki, Matematical Pysics Preprint Arcive (mp_arc)

11 K. Miyazaki 0.5 T=6 Pressure (MeV/fm 3 ) T=.09 T=0 T=0.46 T=8 T= Baryon density (fm 3 ) Figure : Te isoterms of pressure-density plane for symmetric nuclear matter in te generalized termo-statistics.

12 Warm nuclear matter in generalized termo-statistics of power-law generalized Pressure (MeV/fm 3 ) 0 0. standard Baryon density (fm 3 ) Figure : Te black and red curves are te pressure-density isoterms at T = 0MeV in te standard and generalized termo-statistics, respectively. Te orizontal lines are te equilibrium pressures of liquid-gas mixed pases from te Maxwell construction.

13 K. Miyazaki 0 8 T (MeV) 6 4 standard generalized E * /A (MeV) Figure 3: Te black and red curves are te caloric curves of nuclear matter at P = 0:0MeV=fm 3 in te standard and generalized termo-statistics, respectively. Te orizontal lines are te boiling temperatures from te Maxwell construction of liquid-gas pase equilibrium. Te experimental data are from Ref. [3]. 3

14 Warm nuclear matter in generalized termo-statistics of power-law.5 Tsallis Teweldeberan Fermi Dirac Entropy / A T (MeV) Figure 4: Te Clausius entropy per baryon versus pysical temperature under constant pressure P = 0:0MeV=fm 3. Te solid and dotted curves are te results using te cut-o prescription (8) and (9), respectively. Te dased curve is te result in te standard Fermi-Dirac termo-statistics. 4