Hot neutron star in generalized thermo-statistics

The hot neutron star (NS) is investigated for the …rst time in the generalized thermo-statistics so as to take into account the breaking of the standard BoltzmannGibbs thermo-statistics under the long-ranged gravitational potential. It is found that at sub-saturation density in hot NS matter the gravitation leads to a sti¤er equation of state than the standard thermo-statistics. The hot NSs in the generalized thermo-statistics are therefore much more massive and larger than those in the standard thermo-statistics. The study of neutron star (NS) is an important subject in nuclear physics and astrophysics. The equation of state (EOS) of NS matter is developed [1-3] in a microscopic nuclear model of highly dense and asymmetric nuclear matter. Then, the EOS is applied to Einstein equation or Tolman-Oppenheimer-Volkov equation [4] for non-magnetized and non-rotating NS.We distinguish the local microscopic physics from the global macroscopic physics. There is no correlation between the two physics because the gravitation can be neglected as compared with nuclear force. The complete separation of the local and global pictures is really reasonable for cold NS, while it does not necessarily succeed in hot NS encountered during the evolution [5,6] of proto-neutron star. Although the gravitation does not play an explicit role in nuclear EOS, it may have an implicit e¤ect on the EOS through the breaking of the standard Boltzmann-Gibbs thermo-statistics under the long-ranged potential. In fact, it is recently shown in the analysis [7] of the temperature ‡uctuations of cosmic microwave background that the generalized thermo-statistics [8-10] is really satisfactory under gravitational potential. The present paper therefore calculates for the …rst time the EOS of hot NS in the generalized thermo-statistics in contrast to the preceding papers [11-16] that assumed a priori the validity of the standard thermo-statistics under gravitational potential. Here, we easily expect that the e¤ect of gravitation on the EOS is taken into account implicitly through the phenomenological power-law index although the theoretical derivation of the index is a subject of future investigations. In this work we make use of the relativistic mean-…eld (RMF) model of nuclear matter developed in Ref. [16]. The RMF model is reasonable for NS matter because the general 1 Hot neutron star in generalized thermo-statistics relativity is based on the validity of special relativity at local space-time, where the microscopic nuclear model is developed. We extend the thermodynamic potential in Ref. [16] using the q-deformed exponential and logarithm: expq(x) [ 1 + (1 q)x ] 1=(1 q) ; (1) lnq(x) x q 1 1 q : (2) Consequently, the thermodynamic potential q in the generalized thermo-statistics is q = 1 2 m h i 2 + 1 2 m h 3i 2 + 1 2 m h i 2 1 2 m! h!0i 2 1 2 m h 03i 2 1 2 m h 0i 2 2 kBT X B=p;n; ; +; 0; ; 0; Z dk (2 ) lnq 1 + expq B E kB kBT + lnq 1 + expq B E kB kBT 2 kBT X l=e ; Z dk (2 ) lnq 1 + expq l ekl kBT + lnq 1 + expq l ekl kBT ; (3) where kB is the Boltzmann constant and B is given by the chemical potential B and the vector potential VB of baryon as B = B VB: (4) M B = m BMB and E kB = (k 2 +M B ) 1=2 are the e¤ective mass and the energy of baryon while l and ekl = (k 2 +ml ) 1=2 are the chemical potential and the energy of lepton. The scalar mean-…elds h i, h 3i and h i in Eq. (3) are expressed [16] in terms of three independent e¤ective masses of p, n and while the vector mean-…elds h!0i, h 03i and h 0i are expressed [16] in terms of three independent vector potentials of p, n and . Then, M p , M n, M , Vp, Vn and V are determined from extremizing the thermodynamic potential q in terms of them. The results are p + X Y 6= g Y Y ! g nn + I3Y g Y Y g nn! g !=g g Y Y g nn g pp! g nn + g nn! g pp Y g nn m 2 ! h!0i+ g nn!m h 03i g !=g g nn m 2 h 0i g pp! g nn + g nn! g pp = 0; (5) 2