# Equation of state for $\beta$-stable hot nuclear matter

Ch.C. Moustakidis and C.P. Panos
Department of Theoretical Physics, Aristotle University of Thessaloniki,
54124 Thessaloniki, Greece

###### Abstract

We provide an equation of state for hot nuclear matter in $\beta$-equilibrium
by applying a momentum-dependent effective interaction. We focus on the study
of the equation of state of high-density and high-temperature nuclear matter,
containing leptons (electrons and muons) under the chemical equilibrium
condition in which neutrinos have left the system. The conditions of charge
neutrality and equilibrium under $\beta$-decay process lead first to the
evaluation of proton and lepton fractions and afterwards of internal energy,
free energy, pressure and in total to the equation of state of hot nuclear
matter. Thermal effects on the properties and equation of state of nuclear
matter are assesed and analyzed in the framework of the proposed effective
interaction model. Special attention is dedicated to the study of the
contribution of the components of $\beta$-stable nuclear matter to the entropy
per particle, a quantity of great interest for the study of structure and
collapse of supernova.

PACS number(s): 21.65.+f, 21.30.Fe, 24.10.Pa, 26.60.+c, 26.50.+x

Keywards: Hot Nuclear Matter, Effective Interaction, Equation of State,
Nuclear Symmetry Energy, Proton Fraction, Neutron Star.

##  1 Introduction

The equation of state (EOS) of hot nuclear matter determines the structure
inside a supernova [1] and a hot neutron star [2, 3, 4] and affects the state
of matter, such as its chemical composition. In addition, the equation of
state plays important roles for the study of the supernova explosion, as well
as on determining the evolution of a neutron star at the birth stage. The
profiles of a neutron star as the density, temperature and proton fraction
during the cooling, which affect the reaction rate of neutrino process inside
the star, are determined through the equation of state.

There exist many calculations for hot nuclear matter with applications to the
properties of hot neutron stars and supernova [1, 2, 3, 4, 5, 6, 7, 8, 9, 10,
11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29,
30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40]. G. Baym et al. provided an EOS of
neutron matter [5] and Bethe et al. [6] an EOS for the gravitational collapse
of stars. Friedman and Pandharipande [7] performed variational calculations of
the equation of state of hot and cold, nuclear and neutron matter. Lattimer
and Swesty carried out calculations of the EOS for stellar collapse, using the
compressible liquid-drop model for nuclei [8]. M. Prakash et al. [2]
investigated the structure of neutron stars shortly after their birth, by
applying various nuclear models. Takatsuka et al. [9, 10] have performed
detailed calculations for supernova matter, within the framework of finite
temperature Hartree-Fock approach, with effective nucleon-nucleon interaction.
Recently Das et al. [11] have calculated the EOS of dense supernova matter
within the finite temperature Brueckner Goldstone approach with effective two-
body Sussex interaction.

The present work is based on the previous work of Prakash et al. [2]. More
specifically, in order to study the properties and the EOS of hot nuclear
matter, we apply a momentum-dependent effective interaction model (MDIM),
which is able to reproduce the results of more microscopic calculations of
dense matter at zero temperature and which can be extended to finite
temperature [2, 39, 40].

The aim of this work is to apply a momentum-dependent interaction model for
the study of hot nuclear matter EOS under $\beta$-equilibrium. The present
model has the additional property, compared to the previous ones, that the
temperature affects not only the kinetic part of the energy density, but also
influences the interaction part of the energy density as well. In that way, we
are able to study simultaneously thermal effects not only on the kinetic part
of the symmetry energy and symmetry free energy, but also on the interaction
part of the above quantities. This is important in the sense that the density
dependent behavior of the symmetry energy and symmetry free energy influence
strongly the values of the proton fraction and as a consequence the
composition of hot $\beta$-stable nuclear matter, under consideration.

Using the above method , we will show that the thermal energy (and also the
related quantities) depend sensitively on the momentum dependence of the
nuclear interaction. We concentrate our study on the properties of hot nuclear
matter in the density range $n_{0}<n<6n_{0}$ (where $n_{0}=0.16$ fm-3 is the
saturation density) and temperature range $0<T<30$ MeV, taking into account
that nuclear matter consists of neutrons, protons, electrons and muons with
their relative concentrations determined from the conditions of charge
neutrality and equilibrium under $\beta$-decay process in the absence of
neutrino trapping.

The article is organized as follows. In Sec. II the model and relative
formulas are discussed and analyzed. Results are reported and discussed in
Sec. III, whereas the summary of the work is given in Sec. IV.

##  2 The model

We start by outlining the momentum dependent interaction model, then we define
the thermodynamic quantities of nuclear matter and finally we analyze the
$\beta$-equilibrium conditions, the contribution on pressure and energy of
leptons and the total equation of state of nuclear matter.

###  2.1 Momentum dependent interaction model

The schematic potential model, employed here, is designed to reproduce the
results of the microscopic calculations of both nuclear and neutron-rich
matter at zero temperature and can be extended to finite temperature [2]. The
energy density of the asymmetric nuclear matter (ANM) is given by the relation

$\epsilon(n_{n},n_{p},T)=\epsilon_{kin}^{n}(n_{n},T)+\epsilon_{kin}^{p}(n_{p},T)+V_{int}(n_{n},n_{p},T),$
(1)

where $n_{n}$ ($n_{p}$) is the neutron (proton) density and the total baryon
density is $n=n_{n}+n_{p}$. The contributions of the kinetic parts are

$\epsilon_{kin}^{n}(n_{n},T)+\epsilon_{kin}^{p}(n_{p},T)=2\int\frac{d^{3}k}{(2\pi)^{3}}\frac{\hbar^{2}k^{2}}{2m}\left(f_{n}(n_{n},k,T)+f_{p}(n_{p},k,T)\right),$
(2)

where $f_{\tau}$, (for $\tau=n,p$) is the Fermi-Dirac distribution function
with the form

$f_{\tau}(n_{\tau},k,T)=\left[1+\exp\left(\frac{e_{\tau}(n_{\tau},k,T)-\mu_{\tau}(n_{\tau},T)}{T}\right)\right]^{-1}.$
(3)

The nucleon density $n_{\tau}$ is evaluated from the following integral

$n_{\tau}=2\int\frac{d^{3}k}{(2\pi)^{3}}f_{\tau}(n_{\tau},k,T)=2\int\frac{d^{3}k}{(2\pi)^{3}}\left[1+\exp\left(\frac{e_{\tau}(n_{\tau},k,T)-\mu_{\tau}(n_{\tau},T)}{T}\right)\right]^{-1}.$
(4)

In Eq. (3), $e_{\tau}(n_{\tau},k,T)$ is the single particle energy (SPE) and
$\mu_{\tau}(n_{\tau},T)$ stands for the chemical potential of each species.
The SPE has the form

$e_{\tau}(n_{\tau},k,T)=\frac{\hbar^{2}k^{2}}{2m}+U_{\tau}(n_{\tau},k,T),$ (5)

where the single particle potential $U_{\tau}(n_{\tau},k,T)$ is obtained by
the functional derivative of the interaction part of the energy density with
respect to the distribution function $f_{\tau}$. Including the effect of
finite-range forces between nucleons, to avoid acausal behavior at high
densities, the potential contribution is parameterized as follows [2]

$\displaystyle V_{int}(n_{n},n_{p},T)$ $\displaystyle=$
$\displaystyle\frac{1}{3}An_{0}\left[\frac{3}{2}-(\frac{1}{2}+x_{0})I^{2}\right]u^{2}+\frac{\frac{2}{3}Bn_{0}\left[\frac{3}{2}-(\frac{1}{2}+x_{3})I^{2}\right]u^{\sigma+1}}{1+\frac{2}{3}B^{\prime}\left[\frac{3}{2}-(\frac{1}{2}+x_{3})I^{2}\right]u^{\sigma-1}}$
(6) $\displaystyle+$ $\displaystyle u\sum_{i=1,2}\left[C_{i}\left({\cal
J}_{n}+{\cal J}_{p}\right)+\frac{(C_{i}-8Z_{i})}{5}I\left({\cal J}_{n}-{\cal
J}_{p}\right)\right],$

where

${\cal J}_{\tau}=\
2\int\frac{d^{3}k}{(2\pi)^{3}}g(k,\Lambda_{i})f_{\tau}(n_{\tau},k,T).$ (7)

In Eq. (6), $I$ is the asymmetry parameter ($I=(n_{n}-n_{p})/n$) and
$u=n/n_{0}$, with $n_{0}$ denoting the equilibrium symmetric nuclear matter
density, $n_{0}=0.16$ fm-3. The asymmetry parameter $I$ is related to the
proton fraction $Y_{p}$ by the equation $I=(1-2Y_{p})$. The parameters $A$,
$B$, $\sigma$, $C_{1}$, $C_{2}$ and $B^{\prime}$, which appear in the
description of symmetric nuclear matter, are determined in order that
$E(n=n_{0})-mc^{2}=-16$ MeV, $n_{0}=0.16$ fm-3, and the incompressibility to
be $K=240$ MeV. The additional parameters $x_{0}$, $x_{3}$, $Z_{1}$, and
$Z_{2}$, which are used to determine the properties of asymmetric nuclear
matter, are treated as parameters constrained by empirical knowledge [2]. The
parameterizations used in the present model have only a modest microscopic
foundation. Nonetheless, they have the merit of being able to closely
approximate more physically motivated calculations as presented in Fig. 1.
More precisely, in Fig. 1 we compare the energy per baryon (for symmetric
nuclear matter (Fig. 1a) and pure neutron matter (Fig. 1b)) calculated by the
present schematic model (MDIM), with those of existent, state of the art
calculations by Wiringa et al. [41] and Pandharipande et al. [42].

The first two terms of the right-hand side of Eq. (6) arise from local contact
nuclear interaction which lead to power density contributions as in the
standard Skyrme equation of state. The first one concerns attractive
interaction while the second one is repulsive, and both are assumed to be
temperature independent. The third term describes the effects of finite range
interactions according to the chosen function $g(k,\Lambda_{i})$, and is the
temperature dependent part of the interaction. This interaction is attractive
and important at low momentum, but it weakens and disappears at very high
momentum. The function $g(k,\Lambda_{i})$, suitably chosen to simulate finite
range effects, has the following form [2]

$g(k,\Lambda)=\left[1+\left(\frac{k}{\Lambda_{i}}\right)^{2}\right]^{-1},$ (8)

where the finite range parameters are $\Lambda_{1}=1.5k_{F}^{0}$ and
$\Lambda_{2}=3k_{F}^{0}$ and $k_{F}^{0}$ is the Fermi momentum at the
saturation point $n_{0}$.

The main origin of the momentum dependence in Brueckner theory is the
nonlocality of the exchange interaction. Following the discussion of Bertsch
et al. [43] a single-particle potential $U(n)$ which depends only on the
baryon density is oversimplified. What is more, it is well known that nuclear
interaction has strong exchange effects which give rise to a momentum
dependence in the single-particle potential and as a consequence has an effect
on the energy density functional. The question here is how best to
parameterize the momentum dependence in modelling the potential $U(n,k)$. A
promising approach might be to adopt the relativistic mean field model, where
$U(n,k)=U_{\nu}n+\frac{U_{s}n}{\sqrt{1+k^{2}/m^{2}}}$. The above potential
exhibits a strong momentum dependence for small $k$ which diminishes to zero
at high momentum. In order to perform extensive studies in heavy ion collision
studies, Gale et al. [44] have proposed the following parametrization for the
momentum part of the single-particle

$U(n,k)\sim C\frac{n}{n_{0}}\frac{1}{1+({\bf k}-\langle{\bf
k}^{\prime}\rangle)^{2}/\Lambda^{2}}.$

This has a proper fall-off at high $k$ and Galilean invariance is assured by
measuring $k$ with respect to the average of the particles in the
neighborhood, $\langle{\bf k}^{\prime}\rangle$. For static nuclear matter we
have $\langle{\bf k}^{\prime}\rangle=0$.

The present model, which is a generalization of that proposed by Gale et al.
[44], has been successfully applied in heavy ion collisions and astrophysical
studies over the years [2, 3, 4, 27, 28, 33, 45, 46].

In order to clarify the relative contribution of the three terms of the
potential energy density mentioned above, we plot them as a function of the
baryon density in Fig. 2a. In this figure we have that

$\displaystyle V_{A}$ $\displaystyle=$
$\displaystyle\frac{1}{3}An_{0}\left[\frac{3}{2}-(\frac{1}{2}+x_{0})I^{2}\right]u^{2},$
$\displaystyle V_{B}$ $\displaystyle=$
$\displaystyle\frac{\frac{2}{3}Bn_{0}\left[\frac{3}{2}-(\frac{1}{2}+x_{3})I^{2}\right]u^{\sigma+1}}{1+\frac{2}{3}B^{\prime}\left[\frac{3}{2}-(\frac{1}{2}+x_{3})I^{2}\right]u^{\sigma-1}},$
(9) $\displaystyle V_{C}$ $\displaystyle=$ $\displaystyle
u\sum_{i=1,2}\left[C_{i}\left({\cal J}_{n}^{i}+{\cal
J}_{p}^{i}\right)+\frac{(C_{i}-8Z_{i})}{5}I\left({\cal J}^{i}_{n}-{\cal
J}_{p}^{i}\right)\right].$

As mentioned above, the first term $V_{A}$ corresponds to an attractive
interaction, whereas the second one $V_{B}$ corresponds to repulsive
interaction and dominates for high values of $n$ ($n>0.6$ fm-3). Both of these
terms are temperature independent. The third term $V_{C}$ (is plotted for
$T=0$) contains the momentum dependent part of the interaction, corresponds to
attractive interaction, and its main contribution is to compete with the
repulsive interaction of $V_{B}$ for high values of $n$ and as a consequence
avoid acausal behavior of the EOS at high densities. The term $V_{C}$ consists
of two finite range terms, one corresponding to a long-range attraction and
the other to a short-range repulsion.

Thermal effects on the momentum dependent term $V_{C}$ are displayed in Fig.
2b. The contribution of $V_{C}$ is plotted for various values of $T$. It is
therefore concluded that thermal effects are more pronounced for high values
of $T$ ($T>10$ MeV), leading to a less attractive contribution. More
precisely, we find that for small values of $n$ (i.e. $n=0.15$ fm-3 ) $V_{C}$
increases (compared to the cold case $T=0$) $3\%-20\%$ for $T=10-30$. For
higher values of $n$ the increase is even less.

An additional test for the present model is to compare the single particle
potential $U_{\tau}(n_{\tau},k,T)$ (or $U_{\tau}(n,I,k,T)$) originated from
the present version of the momentum dependent interaction with other
calculations. The single particle potential $U_{\tau}(n,I,k,T)$ (protons or
neutrons), obtained from the functional derivative of the interaction part of
the energy density (Eq. (6)) with respect to the distribution function
$f_{\tau}$, has the general form [40]

$U_{\tau}(n,I,k,T)=U_{\tau}^{A}(n,I)+U_{\tau}^{B}(n,I)+U_{\tau}^{MD}(n,I,k,T).$
(10)

It is of interest to see that the single particle potentials are separated
into two parts. The first one, $U_{\tau}^{A}(n,I)+U_{\tau}^{B}(n,I)$ is a
function only of the baryon density $n$ and the isospin asymmetry parameter
$I$. The second one, $U_{\tau}^{MD}(n,I,k,T)$ has an additional dependence on
$T$ and $k$. Actually, $U_{\tau}^{MD}(n,I,k,T)$ is mainly responsible for the
trend of the effective mass and also the effective mass splitting.
Additionally, it is connected with the effect of the temperature on the
interacting part of the energy density [40].

The single-particle potential in symmetric nuclear matter has been calculated
microscopically for several Hamiltonians by Wiringa [47]. These Hamiltonians
include nucleon-nucleon potentials fit to scattering data and three nucleon
potentials fit to binding energies of few-body nuclei and saturation
properties of nuclear matter. The potential was parameterized using the ansatz

$U(n,k)=\alpha(n)+\frac{\beta(n)}{1+(\frac{k}{\Lambda(n)})^{2}},$ (11)

where the density dependent parameters $\alpha(n)$, $\beta(n)$ and
$\Lambda(n)$, for three types Hamiltonians, are listed in Table I of Ref.
[47].

Furthermore, the single-particle potential has been derived by Li et al. [48].
The derivation is based upon the Bonn meson-exchange model for the nucleon-
nucleon interaction and the Dirac-Brueckner approach for nuclear matter. The
potential, named DBHF, has been parameterized as following

$U(n,k)=\alpha n+\beta n^{\gamma}+\delta\ln^{2}(\epsilon(\hbar
ck)^{2}+1)n^{\sigma}.$ (12)

The parameters $\alpha$, $\beta$, $\gamma$, $\delta$, $\epsilon$ and $\sigma$
are listed in Table I of Ref. [48].

A comparison with the results of UV14+TNI, UV14+UVII, AV14+UVII and DBHF
interactions show that (see Fig. 3) $U(n,k)$ (for $T=0$) obtained from the
present model is very reasonable for at least up to the value $k=3$ fm-1. The
agreement is not so good for high values of $k$, especially compared with the
DBHF interaction, but as has been pointed out by Li et al. [48], the
parametrization in Eq. 12 is bad for large $k$ since it continues to grow with
increasing $k$, while the exact potential becomes independent of $k$ for large
momenta. In conclusion, the present results show that the momentum dependent
interaction model, which has been applied in the present work, provides a
reliable representation of $U(n,k)$ for a wide range of density and momentum.

The energy density of asymmetric nuclear matter at density $n$ and temperature
$T$, in a good approximation, is expressed as

$\epsilon(n,T,I)=\epsilon(n,T,I=0)+\epsilon_{sym}(n,T,I),$ (13)

where

$\epsilon_{sym}(n,T,I)=nI^{2}E_{sym}^{tot}(n,T)=nI^{2}\left(E_{sym}^{kin}(n,T)+E_{sym}^{int}(n,T)\right).$
(14)

In Eq. (14) the nuclear symmetry energy $E_{sym}^{tot}(n,T)$ is separated in
two parts corresponding to the kinetic contribution $E_{sym}^{kin}(n,T)$ and
the interaction one $E_{sym}^{int}(n,T)$.

From Eqs. (13) and (14) and setting $I=1$, we find that the nuclear symmetry
energy $E_{sym}^{tot}(n,T)$ is given by

$E_{sym}^{tot}(n,T)=\frac{1}{n}\left(\epsilon(n,T,I=1)-\epsilon(n,T,I=0)\right).$
(15)

Thus, from Eq. (15) and by a suitable choice of the parameters $x_{0}$,
$x_{3}$, $Z_{1}$ and $Z_{2}$, we can obtain various forms for the density
dependence of the symmetry energy $E_{sym}^{tot}(n,T)$.

It is well known that the need to explore different forms for
$E_{sym}^{tot}(n,T)$ stems from the uncertain behavior at high density [2].
The high-density behavior of symmetry energy is the least known property of
dense matter [49, 50, 51], with different nuclear models giving contradictory
predictions. Thus, in relativistic mean field (RMF) models, the symmetry
energy increases strongly with the density of nuclear matter [52], while in
many realistic potential models of nuclear matter in the variational approach
[7, 53], the symmetry energy saturates and then bends over at higher
densities.

Recently, the density dependence of the symmetry energy in the equation of
state of isospin asymmetric nuclear matter has been studied using isoscaling
of the fragment yields and the antisymmetrized molecular dynamic calculation
[54]. It was observed that the experimental data at low densities are
consistent with the form of symmetry energy, $E_{sym}(u)\approx 31.6u^{0.69}$,
in close agreement with those predicted by the results of variational many-
body calculations. In Ref. [54] it was suggested also that the heavy ion
studies favor a dependence of the form $E_{sym}(u)\approx 31.6u^{\gamma}$,
where $\gamma=0.6-1.05$. This constrains the form of the density dependence of
the symmetry energy at higher densities, ruling out an extremely ”stiff” and
”soft” dependence [54].

Additionally, Chen et al. [55] also showed, using the isospin dependent
Boltzmann-Uehling-Uhlenbeck transport model calculations, that a stiff density
dependence of the symmetry energy parameterized as, $E_{sym}(u)\approx
31.6u^{1.05}$ explains well the isospin diffusion data [56] from NSCL-MSU
(National Superconducting Cyclotron Laboratory at Michigan State University).

In this paper, since we are interested mainly in the study of thermal effects
on the nuclear symmetry energy, we choose a specific form for it, enabling us
to reproduce accurately the results of many other theoretical studies [57,
58]. In Ref. [57] the authors carried out a systematic analysis of the nuclear
symmetry energy in the formalism of the relativistic Dirac-Brueckner-Hartree-
Fock approach, using the Bonn one-boson-exchange potential. In a very recent
work [58], the authors applied a similar method as in Ref. [57] for the
microscopic predictions of the equation of state in asymmetric nuclear matter.
In that case $E_{sym}(u)$ is obtained employing the simple parametrization
$E_{sym}(u)=Cu^{\gamma}$ with $\gamma=0.7-1.0$ and $C\approx 32$ MeV. The
authors conclude that a value of $\gamma$ close to $0.8$ gives a reasonable
description of their predictions, although the use of different functions in
different density regions may be best for an optimal fit [58]. The results of
Refs. [57, 58] are well reproduced by parameterizing the nuclear symmetry
energy according to the formula

$E_{sym}^{tot}(n,T=0)=\underbrace{13u^{2/3}}_{Kinetic}+\underbrace{17F(u)}_{Interaction}.$
(16)

For the function $F(u)$, which parametrizes the interaction part of the
symmetry energy, we apply the following form

$F(u)=u.$ (17)

The parameters $x_{0}$, $x_{3}$, $Z_{1}$ and $Z_{2}$ are chosen so that Eq.
(15), for $T=0$, reproduces the results of Eq. (16) for the function $F(u)=u$.

In one of our previous paper [59], the potential part of the symmetry energy
has been parameterized in the generalized form, $F(u)=u^{c}$, and the obtained
nuclear equations of state are applied to the systematic study of the global
properties of a neutron star (masses, radii and composition). We obtained a
linear relation between the parameter $c$ and the radius and the maximum mass
of the neutron star [59]. Additionally, we found that a linear relation
between the radius and the derivative of the symmetry energy near the
saturation density $n_{0}$ holds [59].

It is worthwhile to point out that the above parametrization of the
interacting part of the nuclear symmetry energy is used extensively for the
study of neutron star properties [2, 60], as well as for the study of the
collisions of neutron-rich heavy ions at intermediate energies [61, 62]. For a
very recent review of the applications of the proposed momentum dependent
effective interaction model and its specific parameterizations, see Ref. [4]
(and references therein).

###  2.2 Thermodynamic description of hot nuclear matter

In order to study the properties of nuclear matter at finite temperature, we
need to introduce the Helmholtz free energy $F$. The differential of the total
free energy $F_{tot}$ (the total free energy of baryons contained in volume
$V$) and total internal energy $E_{tot}$ (the total internal energy of baryons
contained in volume $V$) are given by [63, 64]

${\rm d}F_{tot}=-S_{tot}{\rm d}T-P{\rm d}V+\sum_{i}\mu_{i}{\rm d}N_{i}$ (18)
${\rm d}E_{tot}=T{\rm d}S_{tot}-P{\rm d}V+\sum_{i}\mu_{i}{\rm d}N_{i}$ (19)

where $S_{tot}$ is the total entropy of the baryons, while $\mu_{i}$ and
$N_{i}$ are the chemical potential and the number of particles of each species
respectively.

It is easy to prove that the free energy per particle $F$ is written as [63,
64]

$F(n,T,I)=E(n,T,I)-TS(n,I,T).$ (20)

In Eq. (20), $E$ is the internal energy per particle, $E=\epsilon/n$, and $S$
is the entropy per particle, $S=s/n$. From Eq. (20) is also concluded that for
$T=0$, the free energy $F$ and the internal energy $E$ coincide.

The entropy density $s$ has the same functional form as that of a non
interacting gas system, given by the equation

$s_{\tau}(n,I,T)=-2\int\frac{d^{3}k}{(2\pi)^{3}}\left[f_{\tau}\ln
f_{\tau}+(1-f_{\tau})\ln(1-f_{\tau})\right].$ (21)

The total internal energy $E_{tot}$ is useful for studying isentropic
processes. In that description of a thermodynamic system, the pressure and the
chemical potential are defined as follows [63, 64]

$P=-\left(\frac{\partial E_{tot}}{\partial
V}\right)_{S,N_{i}}=n^{2}\left(\frac{\partial\epsilon/n}{\partial
n}\right)_{S,N_{i}},\qquad\qquad\qquad\mu_{i}=\left(\frac{\partial
E_{tot}}{\partial N_{i}}\right)_{S,V,N_{j\neq
i}}=\left(\frac{\partial\epsilon}{\partial n_{i}}\right)_{S,V,n_{j\neq i}}.$
(22)

Now we are going to study the properties and the EOS of nuclear matter by
considering an isothermal process. In that, the pressure and the chemical
potential are connected with the derivative of the total free energy
$F_{tot}$. More specifically, they are defined as follows

$P=-\left(\frac{\partial F_{tot}}{\partial
V}\right)_{T,N_{i}}=n^{2}\left(\frac{\partial f/n}{\partial
n}\right)_{T,N_{i}},\qquad\qquad\mu_{i}=\left(\frac{\partial F_{tot}}{\partial
N_{i}}\right)_{T,V,N_{j\neq i}}=\left(\frac{\partial f}{\partial
n_{i}}\right)_{T,V,n_{j\neq i}},\qquad\qquad$ (23)

where $f$ is the free energy density.

The pressure $P$ can be calculated also from equations [63, 64]

$VP=TS_{tot}-E_{tot}+\sum_{i}\mu_{i}N_{i},\qquad{\rm or}\qquad\qquad
P=Ts-\epsilon+\sum_{i}\mu_{i}n_{i}.$ (24)

It is also possible to calculate the entropy per particle $S(n,T)$ by
differentiating the free energy density $f$ with respect to the temperature

$S(n,T)=-\left(\frac{\partial f/n}{\partial T}\right)_{V,N_{i}}.$ (25)

The comparison of the two entropies, that is from Eqs. (21) and (25), provides
a test of the approximation used in the present work.

It is easy to show by applying Eq. (23) that (see for a proof [60] as well as
[32])

$\displaystyle\mu_{n}$ $\displaystyle=$ $\displaystyle F+u\left(\frac{\partial
F}{\partial u}\right)_{Y_{p},T}-Y_{p}\left(\frac{\partial F}{\partial
Y_{p}}\right)_{n,T},$ $\displaystyle\mu_{p}$ $\displaystyle=$
$\displaystyle\mu_{n}+\left(\frac{\partial F}{\partial Y_{p}}\right)_{n,T},$
$\displaystyle\hat{\mu}$ $\displaystyle=$
$\displaystyle\mu_{n}-\mu_{p}=-\left(\frac{\partial F}{\partial
Y_{p}}\right)_{n,T}.$ (26)

We can define the symmetry free energy per particle $F_{sym}(n,T)$ by the
following parabolic approximation (see also [32, 33])

$F(n,T,I)=F(n,T,I=0)+I^{2}F_{sym}(n,T)=F(n,T,I=0)+(1-2Y_{p})^{2}F_{sym}(n,T),$
(27)

where

$F_{sym}(n,T)=F(n,T,I=1)-F(n,T,I=0).$ (28)

It is worthwhile to notice that the above approximation is not valid from the
beginning, but one needs to check the validity of the parabolic law in the
present model before using it. As we see later, that law is well satisfied as
well as the parabolic law holding for the energy.

Now, by applying Eq. (27) in Eq. (26), we obtain the key relation

$\hat{\mu}=\mu_{n}-\mu_{p}=4(1-2Y_{p})F_{sym}(n,T).$ (29)

The above equation is similar to that obtained for cold nuclear matter by
replacing $E_{sym}(n)$ with $F_{sym}(n,T)$.

###  2.3 $\beta$-equilibrium, leptons contribution and equation of state

Stable high density nuclear matter must be in chemical equilibrium for all
types of reactions, including the weak interactions, while $\beta$ decay and
electron capture take place simultaneously

$n\longrightarrow p+e^{-}+\bar{\nu}_{e},\qquad\qquad p+e^{-}\longrightarrow
n+\nu_{e}.$ (30)

Both types of reactions change the electron per nucleon fraction, $Y_{e}$ and
thus affect the equation of state. Here, we assume that neutrinos generated in
those reactions have left the system. The absence of neutrino-trapping has a
dramatic effect on the equation of state and mainly induces a significant
change on the values of the proton fraction $Y_{p}$ [9, 10]. The absence of
neutrinos implies that

$\hat{\mu}=\mu_{n}-\mu_{p}=\mu_{e}.$ (31)

When the energy of electrons is large enough (i.e. greater than the muon
mass), it is energetically favorable for the electrons to convert to muons

$e^{-}\longrightarrow\mu^{-}+\bar{\nu}_{\mu}+\nu_{e}.$ (32)

Denoting the muon chemical potential by $\mu_{\mu}$, the chemical equilibrium
established by the above process and its inverse is given by

$\mu_{e}=\mu_{\mu}.$

Taking into account that the threshold for muons occurs for
$\mu_{\mu}=m_{\mu}c^{2}\simeq 105.7$ MeV, one may expect muons to appear
roughly at nuclear density $n=0.16$ fm-3.

Thus, in total, we consider that nuclear matter contains neutrons, protons,
electrons, and muons. They are in a $\beta$-equilibrium, where the following
relations hold

$\mu_{n}=\mu_{p}+\mu_{e},\qquad\qquad\mu_{e}=\mu_{\mu}.$ (33)

Furthermore, they obey the charge neutrality condition i.e.

$n_{p}=n_{e}+n_{\mu}.$ (34)

The leptons (electrons and muons) density is given by the expression

$n_{l}=\frac{2}{(2\pi)^{3}}\int\frac{{\rm d}{\bf
k}}{1+\exp\left[\frac{\sqrt{\hbar^{2}k^{2}c^{2}+m_{l}^{2}c^{4}}-\mu_{l}}{T}\right]}.$
(35)

One can solve self-consistently Eqs. (29),(33),(34) and (35) in order to
calculate the proton fraction $Y_{p}$, the lepton fractions $Y_{e}$ and
$Y_{\mu}$, as well as the electron chemical potential $\mu_{e}$ as a function
of the baryon density $n$, for various values of the temperature $T$.

The next step is to calculate the energy and pressure of leptons given by the
following formulae

$\epsilon_{l}(n_{l},T)=\frac{2}{(2\pi)^{3}}\int\frac{\sqrt{\hbar^{2}k^{2}c^{2}+m_{l}^{2}c^{4}}\
{\rm d}{\bf
k}}{1+\exp\left[\frac{\sqrt{\hbar^{2}k^{2}c^{2}+m_{l}^{2}c^{4}}-\mu_{l}}{T}\right]},$
(36) $P_{l}(n_{l},T)=\frac{1}{3}\frac{2(\hbar
c)^{2}}{(2\pi)^{3}}\int\frac{1}{\sqrt{\hbar^{2}k^{2}c^{2}+m_{l}^{2}c^{4}}}\frac{\
k^{2}\ {\rm d}{\bf
k}}{1+\exp\left[\frac{\sqrt{\hbar^{2}k^{2}c^{2}+m_{l}^{2}c^{4}}-\mu_{l}}{T}\right]}.$
(37)

The chemical potentials of electrons and muons are equal and according to Eqs.
(29) and (33) are

$\mu_{e}=\mu_{\mu}=\mu_{p}-\mu_{n}=4\left(1-2Y_{p}(n,T)\right)F_{sym}(n,T)=4I(n,T)F_{sym}(n,T).$
(38)

The equation of state of hot nuclear matter in $\beta$-equilibrium
(considering that it consists of neutrons, protons, electrons and muons) can
be obtained by calculating the total energy density $\epsilon_{tot}$ as well
as the total pressure $P_{tot}$. The total energy density is given by

$\epsilon_{tot}(n,T,I)=\epsilon_{b}(n,T,I)+\sum_{l=e,\mu}\epsilon_{l}(n,T,I),$
(39)

where $\epsilon_{b}(n,T,I)$ and $\epsilon_{l}(n,T,I)$ are the contributions of
baryons and leptons respectively. The total pressure is

$P_{tot}(n,T,I)=P_{b}(n,T,I)+\sum_{l=e,\mu}P_{l}(n,T,I),$ (40)

where $P_{b}(n,T,I)$ is the contribution of the baryons (see Eq. (24)) i.e.

$P_{b}(n,T,I)=T\sum_{\tau=p,n}s_{\tau}(n,T,I)+\sum_{\tau=n,p}n_{\tau}\mu_{\tau}(n,T,I)-\epsilon_{b}(n,T,I),$
(41)

while $P_{l}(n,T,I)$ is the contribution of the leptons (see Eq. (37)). From
Eqs. (39) and (40) we can construct the isothermal curves for energy and
pressure and finally derive the isothermal behavior of the equation of state
of hot nuclear matter under $\beta$-equilibrium.

##  3 Results and Discussion

The schematic potential model, which has been applied in the present work, has
been designed to reproduce the results of the more microscopic calculations of
both nuclear and neutron-rich matter up to high values of baryon density (see
Fig. 1). The behavior of the high density EOS is of great significance to the
determination of hot protoneutron stars and cold neutron stars structure. The
model has the additional advantage that with the appropriate parametrization,
is able to reproduce different forms of the density dependence of the nuclear
symmetry energy.

In view of the above discussion we calculate the equation of state of hot
asymmetric nuclear matter by applying a momentum dependent effective
interaction model describing the baryons interaction. We consider that nuclear
matter contains neutrons, protons, electrons and muons under
$\beta$-equilibrium and charge neutrality. The key quantities in our
calculations are the proton fraction $Y_{p}$ and also the asymmetry free
energy defined in Eq. (28).

In order to check the validity of the parabolic approximation (27), we plot in
Fig. 4 the difference $F(n,T,I=1)-F(n,T,I=0)$ as a function of
$(1-2Y_{p})^{2}$ at temperature $T=10$ and $T=30$ MeV for three baryon
densities, i.e., $n=0.2$ fm-3, $n=0.3$ fm-3, and $n=0.4$ fm-3. It is seen that
in a good approximation an almost linear relation holds between
$F(n,T,I=1)-F(n,T,I=0)$ and $(1-2Y_{p})^{2}$. A similar behavior of
$F_{sym}(n,T)$ is found by Xu et al. [33], applying an isospin and momentum
dependent interaction model.

It is worth to present the calculation recipe of our work. The outline of our
approach is the following: For a fixed baryon density $n$, temperature $T$,
and asymmetry parameter $I$, Eq. (4) may be solved iteratively in order to
calculate the quantity

$\eta_{\tau}(n_{\tau},T)=\frac{\mu_{\tau}(n_{\tau},T)-\tilde{U}_{\tau}(n_{\tau},T)}{T},$
(42)

where

$\tilde{U}_{\tau}(n_{\tau},T)=U_{\tau}(n_{\tau},k,T)-\tilde{U}_{\tau}(n_{\tau},k).$
(43)

Knowledge of $\eta_{\tau}(n,T)$ allows the evaluation of
$\tilde{U}_{\tau}(n_{\tau},T)$, which then may be employed to infer the
chemical potential from

$\mu_{\tau}(n_{\tau},T)=T\eta_{\tau}(n_{\tau},T)+\tilde{U}(n_{\tau},T),$ (44)

required as an input for the calculation of the Fermi-Dirac distribution
function $f_{\tau}(n_{\tau},k,T)$. The knowledge of $f_{\tau}(n_{\tau},k,T)$
permits the calculation of the bulk quantities of asymmetric nuclear matter.

$F_{sym}(n,T)$, for various values of the temperature $T$, was derived with a
least-squares fit to the numerical values according to Eq. (28) and has the
form

$\displaystyle F_{sym}(u;T=0)$ $\displaystyle=$ $\displaystyle 13u^{2/3}+17u$
$\displaystyle F_{sym}(u;T=5)$ $\displaystyle=$ $\displaystyle
3.653+28.018u-1.512u^{2}+0.185u^{3}-0.001u^{4},$ $\displaystyle
F_{sym}(u;T=10)$ $\displaystyle=$ $\displaystyle
5.995+26.157u-0.827u^{2}+0.068u^{3}-0.002u^{4},$ $\displaystyle
F_{sym}(u;T=20)$ $\displaystyle=$ $\displaystyle
13.200+21.267u+0.800u^{2}-0.193u^{3}+0.014u^{4},$ $\displaystyle
F_{sym}(u;T=30)$ $\displaystyle=$ $\displaystyle
21.087+17.626u+1.645u^{2}-0.289u^{3}+0.018u^{4}.$ (45)

where the case with $T=0$, is included as well. In that case $F_{sym}$
coincides with $E_{sym}$.

In Fig. 5 we present the behavior of the free energy, corresponding to hot
$\beta$-stable nuclear matter, as a function of the baryon density $n$, for
various values of the temperature $T$. It is obvious that the thermal effects
are more pronounced for low values of the density $n$.

In Fig. 6 we plot the calculated free energy for symmetric nuclear matter and
pure neutron matter of the proposed momentum dependent interaction model in
comparison with the values of the free energy calculated by Friedman and
Pandharipande (FP model) [7]. In the FP model the equation of state of hot and
cold nuclear and neutron matter has been calculated in the framework of a
variational calculation, where a realistic nuclear interaction containing two-
and three-nucleon body nucleon-nucleon interaction has been used. In the case
of symmetric nuclear matter the results of the two models are very similar up
to values $n=0.4-0.5$ fm-3 depending on the values of $T$. The above agreement
is expected, in the sense that a part of the parameters of the model applied
in the present work are determined from constraints provided by the empirical
properties of symmetric nuclear matter at the equilibrium density $n_{0}=0.16$
fm-3.

However, there is an obvious disagreement in the case of pure nuclear matter,
where in the two models the free energy exhibits a different trend, especially
for higher values of $n$. The above disagreement will be explained below.

In Fig. 7 we display the internal energy per particle $E(n,T)=\epsilon(n,T)/n$
given by Eq. (1) for various values of temperature. Thermal effects, as
expected, are more pronounced for low values of the baryon density $n$ and
less important for high values of $n$.

In Fig. 8 we display the internal energy of symmetric nuclear matter and pure
neutron matter, for $T=0$, calculated by the MDIM and FP models. In accordance
with the case of the free energy, there is a very good agreement in symmetric
nuclear matter, but an obvious disagreement is exhibited in pure neutron
matter. The explanation of the agreement in the first case is the same as in
the case of the free energy. The disagreement is due to the completely
different behavior of the two models of the nuclear symmetry energy, presented
in Fig. 9. In our model the parameters $x_{0}$, $x_{3}$, $Z_{1}$ and $Z_{2}$
chosen so that Eq. (15), for $T=0$, reproduce the results of Eq. (16) for the
function $F(u)=u$. Consequently, $E_{sym}(n)$ shows an increasing trend shown
in Fig. 9. In contrast, in the FP model, $E_{sym}(n)$ is a slightly increasing
function of $n$ for low $n$ and then a decreasing function of $n$ for $n>0.5$
fm-3.

In addition, we plot the nuclear symmetry energy extracted from experimental
results and presented in Ref. [54], where $E_{sym}(u)$ is parameterized
according to the relation $E_{sym}(u)\approx 31.6u^{0.69}$ as well as
experimental results extracted from Ref. [55], where $E_{sym}(u)$ is given by
$E_{sym}(u)\approx 31.6u^{1.05}$. The important point to be noted is that both
cases clearly favor a stiff density dependence of the symmetry energy at
higher densities, ruling out the very stiff and very soft predictions. These
results can thus be employed to constrain the form of the density dependence
of the symmetry energy at supranormal densities relevant for neutron star
studies [54]. In the same figure the theoretical predictions of Ref. [58] are
presented, where $E_{sym}(u)$ is parameterized by $E_{sym}(u)\approx
32u^{0.8}$.

The results of Ref. [54] are in a good agrement with the present model up to
$n=0.3$ fm-3 while the theoretical predictions of Ref. [58] are very close to
the present model up to very high values of the baryon density $n$.

However, our motivation, here, is not to perform a systematic comparison of
various models, but we would like just to present the similarities and the
deviations existing between the models. The deviations, concerning the
symmetry energy behavior of the two models (MDIM and FP model) are well
reflected on the behavior of the free energy and internal energy of pure
neutron matter as shown in Figs. 6 and 8.

In Fig. 10 we plot the thermal energy per particle

$E_{thermal}(n,T)=E(n,T)-E(n,T=0),$

of $\beta$-stable matter as a function of the baryon density $n$ for various
values of temperature $T$. The most striking feature of $E_{thermal}(n,T)$ is
that for small values of $T$, the thermal contribution to the internal energy
is almost independent of the density $n$. For high values of $T$ the situation
is different and $E_{thermal}(n,T)$, for fixed values of $T$, is a decreasing
function of the density $n$.

$E_{thermal}(n,T)$ can be decomposed to separate contributions of the kinetic
and potential energies as follows:

$E_{thermal}(n,T)=E_{thermal}^{kin}(n,T)+E_{thermal}^{pot}(n,T).$

We find that for small values of the baryon density (i.e. $n=0.2$ fm-3) the
contribution, to $E_{thermal}(n,T)$ of the potential energy
$E_{thermal}^{pot}(n,T)$ is $20\%-10\%$ for $T=5-30$ MeV. For medium values of
$n$ (i.e. $n=0.4$ fm-3) is $43\%-20\%$ for $T=5-30$ MeV and for higher values
(i.e. $n=0.6$ fm-3) is $70\%-30\%$ for $T=5-30$ MeV. Hence, it is concluded
that the potential part of the energy (as a result of the momentum dependence
of the interaction) contributes significantly to the thermal energy, mainly
for small values of $T$ (for fixed values of $n$) and for large values of $n$
(for fixed values of $T$).

At this point, it is worthwhile to compare the results for the pressure
obtained by applying Eqs. (24) and (23). Thus, in Fig. 11 we plot $P$ of
asymmetric nuclear matter for $Y_{p}=0.1$ and $0.3$ at $T=10$ and $30$ MeV.
The full lines give the results calculated from Eq. (24), while the squares
give results obtained by differentiating $F(n,T)$ (Eq. (23)). The two
calculations for the pressure are in excellent agreement. This agreement
provides a test of the calculations performed in the present model.

It is of interest also to study the effect of the temperature on the baryon
pressure defined by equation (24). A related quantity is the thermal pressure
$P_{thermal}(n,T)$ defined as:

$P_{thermal}(n,T)=P(n,T)-P(n,T=0).$

$P_{thermal}(n,T)$ as a function of $n$, for various values of $T$ is seen in
Fig. 12. $P_{thermal}(n,T)$, in all of the cases, is an increasing function of
the baryon density.

The proton fraction affects the reaction rate of neutrino process inside that
star. If a neutron star has a large proton fraction, the cooling rate may
drastically change through the high neutrino emissivity due to the direct Urca
process. This process can occur if the proton fraction in the matter of a cold
neutron star exceeds the critical value of $0.11$-$0.15$ and would lead to the
rapid cooling of the neutron star. Thus, it is important to calculate the
proton fraction as a function of the baryon density and investigate the
temperature effects on that.

Fig. 13 displays the fractions of protons, electrons and muons as functions of
the density, for various values of $T$. The proton fraction is an increasing
function of $T$ and this effect is more pronounced for $T>10$ MeV. The proton
fraction $Y_{p}$ was derived also with a least-squares fit to the numerical
results obtained from our calculations, leading to the following relations
(for $n>0.15$ fm-3).

$\displaystyle Y_{p}(n;T=0)$ $\displaystyle=$
$\displaystyle-0.050+0.633n-0.521n^{2}+0.184n^{3},$ $\displaystyle
Y_{p}(n;T=5)$ $\displaystyle=$
$\displaystyle-0.046+0.625n-0.514n^{2}+0.179n^{3},$ $\displaystyle
Y_{p}(n;T=10)$ $\displaystyle=$
$\displaystyle-0.032+0.570n-0.436n^{2}+0.139n^{3},$ (46) $\displaystyle
Y_{p}(n;T=20)$ $\displaystyle=$ $\displaystyle
0.021+0.378n-0.163n^{2}+0.004n^{3},$ $\displaystyle Y_{p}(n;T=30)$
$\displaystyle=$ $\displaystyle
0.109-0.062n+0.908n^{2}-1.270n^{3}+0.580n^{4}.$

In Fig. 14 we plot the Fermi distribution function $f_{p,n}(n,T)$ both for
neutrons and protons for various values of $T$. We observe that the
diffuseness of $f_{p}(n,T)$ is larger than that of $f_{n}(n,T)$. We give an
explanation (see also [9]): the ratio of $T$ to the Fermi kinetic energy
$\epsilon_{Fi}$ is a measure of the thermal effect. Thus by comparing the two
ratios we have (see also Appendix)

$\left(\frac{(T/\epsilon_{Fp})}{(T/\epsilon_{Fn})}\right)={k_{F}^{n}}^{2}/{k_{F}^{p}}^{2}=Y_{n}^{3/2}/Y_{p}^{3/2}.$

But, due to $Y_{n}>Y_{p}$, we conclude that we expect the diffuseness to be
larger for the proton distribution than for the neutron one, depending of
course on the specific value of the ratio $Y_{p}/Y_{n}$. As we will see later,
this fact influences the values of the contributions of protons and neutrons
to the total entropy per particle. The entropy, according to relation (21), is
an increasing function of the diffuseness. Thus, the contribution of each
species on the total value of the entropy depends strongly on the diffuseness
of the corresponding Fermi distribution function.

We can provide a second test, concerning the accuracy of the present
calculations, by comparing the entropy per baryon calculated by applying Eqs.
(21) and (25). Thus, in Fig. 15 we plot $S$ of asymmetric nuclear matter with
$Y_{p}=0.2$ at $T=10,20,30$ MeV. The full lines give the entropy calculated
from Eq. (21), while the squares give results obtained by differentiating
$F(n,T)$ with respect to $T$ (Eq. (25)). It is obvious that there is again a
very good agreement of the results, especially for low values of $T$ and high
values of $n$.

In Fig. 16 we plot the contribution of the proton $S_{p}$, the neutron $S_{n}$
and the total entropy per baryon $S$. It is obvious that there is a strong
effect of $T$ on the values of the entropies mainly for low values of the
density. The main part of the contribution comes from neutrons, whereas the
contribution of protons is three times less. It is worthwhile to notice that,
in spite of $Y_{p}\sim(1/20-3/10)Y_{n}$, the approximate relation
$S_{p}\sim(1/4-3/7)S_{n}$ holds. This feature is understood by the previous
discussion that $f_{p}(n,T)$ is diffused more broadly than $f_{n}(n,T)$, so
the larger the diffuseness, the larger is the entropy contribution (see also
[9]).

In Fig. 17 we plot the contribution of the electronic $S_{e}$, the muonic
$S_{\mu}$ and the total (leptonic) $S_{l}$ to the entropy per baryon. The
contribution to the entropy, of $S_{e}$ depends slightly on the density, for
fixed values of $T$. Our present results are very close to those found by Onsi
et al. [19], where they employed the analytical approximate formula for the
electron entropy density $s_{e}$

$s_{e}=\frac{1}{3}\frac{\mu_{e}^{2}}{(\hbar c)^{3}}T,\qquad\qquad\mu_{e}=\hbar
c(3\pi^{2}Y_{e}n)^{1/3}.$ (47)

According to the above formula, the contribution of electrons to the entropy
per baryon has the form

$S_{e}=s_{e}/n\sim\left(\frac{Y_{e}^{2}}{n}\right)^{1/3}T.$ (48)

The quantity $\left(\frac{Y_{e}^{2}}{n}\right)^{1/3}$ is a function slightly
dependent on the density $n$, so that for a fixed value of $T$ the
contribution $S_{e}$ is almost constant. The muonic contribution to the
entropy, for fixed $T$, increases slightly as a function of the density.

In Fig. 18 we present the EOS of the $\beta$-stable hot nuclear matter by
taking into account and analyzing the contribution to the total pressure of
each component. The main contribution to the total pressure originates from
the baryons, while the contribution of the leptons is about a few percent
compared to $P_{b}$. It is worthwhile to notice that thermal effects are not
important for the calculation of $P_{e}$, but only for $P_{\mu}$, especially
for small values of $n$ ($n<0.4$ fm-3). We found that thermal effects produce
a slightly stiffer equation of state with respect to the case of cold nuclear
matter. The above EOS can be applied to the evaluation of the bulk properties
of hot neutron stars (mass and radius).

The study of hot nuclear matter in the absence of neutrino trapping is the
first step to study the properties of hot neutron stars and supernova matter.
Next, one can study the more realistic case of neutrino-trapped matter in
$\beta$-equilibrium. In this case, the $\beta$-equilibrium conditions in
matter are altered from the case in which neutrinos have left the system and
thus the composition of matter is affected. The proton fraction increases
dramatically and influences significantly the properties of nuclear matter.
Such a work is in progress.

##  4 Summary

The evaluation of the equation of state of hot nuclear matter is an important
problem in nuclear physics and astrophysics. EOS is the basis ingredient for
the study of the supernova explosion as well as on determining the properties
of hot neutron stars. The motivation of the present work is to apply a
momentum-dependent interaction model for the study of the hot nuclear matter
EOS under $\beta$-equilibrium in order to be able to study simultaneously
thermal effects, not only on the kinetic part of the symmetry energy and
symmetry free energy, but also on the interaction part of the above quantities
as well. We calculate the proton fraction, as well as the lepton fractions, by
applying the constraints for chemical equilibrium and charge neutrality. The
free energy, the internal energy and also the pressure are calculated as
functions of baryon density and for various values of temperature. We also
concentrate on the evaluation of thermal effects on the internal energy and
baryon pressure. Special attention is dedicated to the study of the
contribution of the components of $\beta$-stable nuclear matter on the entropy
per particle, a quantity of great interest in the study of structure and
collapse of supernova. We present and analyze the contribution of each
component. Finally, we present the EOS of $\beta$-stable hot nuclear matter,
by taking into account and analyzing the contributions to the total pressure
of each component. The above EOS can be applied to the evaluation of the gross
properties of hot neutron stars i.e. mass and radius, (work in progress).

## Acknowledgments

One of the authors (Ch.C.M) would like to thank Professor Tatsuyauki Takatsuka
for valuable comments and correspondence.

## Appendix

The energy density of baryons (Eq. (1)), at $T=0$, is given by

$\displaystyle\epsilon(n,I,T=0)$ $\displaystyle=$
$\displaystyle\frac{3}{10}E_{F}^{0}n_{0}u^{5/3}\left[(1+I)^{5/3}+(1-I)^{5/3}\right]+\frac{1}{3}An_{0}\left[\frac{3}{2}-(\frac{1}{2}+x_{0})I^{2}\right]u^{2}$
$\displaystyle+$
$\displaystyle\frac{\frac{2}{3}Bn_{0}\left[\frac{3}{2}-(\frac{1}{2}+x_{3})I^{2}\right]u^{\sigma+1}}{1+\frac{2}{3}B^{\prime}\left[\frac{3}{2}-(\frac{1}{2}+x_{3})I^{2}\right]u^{\sigma-1}}$
$\displaystyle+$
$\displaystyle\frac{3}{2}n_{0}u\sum_{i=1,2}\left[C_{i}+\frac{C_{i}-8Z_{i}}{5}I\right]\left(\frac{\Lambda_{i}}{k_{F}^{0}}\right)^{3}\left(\frac{\left((1+I)u\right)^{1/3}}{\frac{\Lambda_{i}}{k_{F}^{0}}}-\tan^{-1}\frac{\left((1+I)u\right)^{1/3}}{\frac{\Lambda_{i}}{k_{F}^{0}}}\right)$
$\displaystyle+$
$\displaystyle\frac{3}{2}n_{0}u\sum_{i=1,2}\left[C_{i}-\frac{C_{i}-8Z_{i}}{5}I\right]\left(\frac{\Lambda_{i}}{k_{F}^{0}}\right)^{3}\left(\frac{\left((1-I)u\right)^{1/3}}{\frac{\Lambda_{i}}{k_{F}^{0}}}-\tan^{-1}\frac{\left((1-I)u\right)^{1/3}}{\frac{\Lambda_{i}}{k_{F}^{0}}}\right).$

The pressure of the baryons, at $T=0$, defined as

$P=n^{2}\frac{d(\epsilon/n)}{dn},$

is given by

$\displaystyle P(n,I,T=0)$ $\displaystyle=$
$\displaystyle\frac{1}{5}n_{0}E_{F}^{0}u^{5/3}\left[(1+I)^{5/3}+(1-I)^{5/3}\right]+\frac{1}{3}n_{0}u^{2}A\left[\frac{3}{2}-\left(\frac{1}{2}+x_{0}\right)I^{2}\right]$
$\displaystyle+$ $\displaystyle\frac{2}{3}B\sigma
n_{0}u^{\sigma+1}\frac{\left[\frac{3}{2}-(\frac{1}{2}+x_{3})I^{2}\right]\left(1+\frac{2}{3\sigma}B^{\prime}u^{\sigma-1}\left[\frac{3}{2}-(\frac{1}{2}+x_{3})I^{2}\right]\right)}{\left(1+\frac{2}{3}B^{\prime}\left[\frac{3}{2}-(\frac{1}{2}+x_{3})I^{2}\right]u^{\sigma-1}\right)^{2}}$
$\displaystyle+$
$\displaystyle\frac{n_{0}u^{2}}{2}\sum_{i=1,2}\left[C_{i}+\frac{C_{i}-8Z_{i}}{5}I\right]\left(\frac{\Lambda_{i}}{k_{F}^{0}}\right)^{2}\frac{(1+I)^{1/3}}{u^{2/3}}\left(1-\frac{1}{1+\frac{(1+I)^{2/3}u^{2/3}}{\left(\frac{\Lambda_{i}}{k_{F}^{0}}\right)^{2}}}\right)$
$\displaystyle+$
$\displaystyle\frac{n_{0}u^{2}}{2}\sum_{i=1,2}\left[C_{i}-\frac{C_{i}-8Z_{i}}{5}I\right]\left(\frac{\Lambda_{i}}{k_{F}^{0}}\right)^{2}\frac{(1-I)^{1/3}}{u^{2/3}}\left(1-\frac{1}{1+\frac{(1-I)^{2/3}u^{2/3}}{\left(\frac{\Lambda_{i}}{k_{F}^{0}}\right)^{2}}}\right).$

In Eq. (Appendix) $E_{F}^{0}$ is the Fermi energy of symmetric nuclear matter
corresponding to equilibrium density $n_{0}$ and is given by

$E_{F}^{0}=\frac{\left(\hbar k_{F}^{0}\right)^{2}}{2m},\qquad
k_{F}^{0}=\left(3\pi^{2}\frac{n_{0}}{2}\right)^{1/3}.$ (51)

The Fermi momenta of protons and neutrons are

$k_{F}^{p}=\left(3\pi^{2}xn\right)^{1/3}=\left(3\pi^{2}\frac{1-I}{2}n\right)^{1/3},$
$k_{F}^{n}=\left(3\pi^{2}(1-x)n\right)^{1/3}=\left(3\pi^{2}\frac{1+I}{2}n\right)^{1/3}.$

The chemical potentials of protons and neutrons, at $T=0$, are given by

$\displaystyle\mu_{n}$ $\displaystyle=$ $\displaystyle E+u\left(\frac{\partial
E}{\partial u}\right)_{Y_{p}}-Y_{p}\left(\frac{\partial E}{\partial
Y_{p}}\right)_{n},$ $\displaystyle\mu_{p}$ $\displaystyle=$
$\displaystyle\mu_{n}+\left(\frac{\partial E}{\partial Y_{p}}\right)_{n},$
(52) $\displaystyle\hat{\mu}$ $\displaystyle=$
$\displaystyle\mu_{n}-\mu_{p}=-\left(\frac{\partial E}{\partial
Y_{p}}\right)_{n},$

where

$\displaystyle\left(\frac{\partial E}{\partial u}\right)_{Y_{p}}$
$\displaystyle=$
$\displaystyle\frac{1}{5}E_{F}^{0}u^{-1/3}\left[(1+I)^{5/3}+(1-I)^{5/3}\right]+\frac{1}{3}A\left[\frac{3}{2}-(\frac{1}{2}+x_{0})I^{2}\right]$
(53) $\displaystyle+$
$\displaystyle\frac{\frac{2}{3}Bu^{\sigma-1}\sigma\left[\frac{3}{2}-(\frac{1}{2}+x_{3})I^{2}\right]\left(1+\frac{2}{3\sigma}B^{\prime}u^{\sigma-1}\left[\frac{3}{2}-(\frac{1}{2}+x_{3})I^{2}\right]\right)}{\left(1+\frac{2}{3}B^{\prime}u^{\sigma-1}\left[\frac{3}{2}-(\frac{1}{2}+x_{3})I^{2}\right]\right)^{2}}$
$\displaystyle+$
$\displaystyle\frac{1}{2}\sum_{i=1,2}\left[C_{i}+\frac{C_{i}-8Z_{i}}{5}I\right]\left(\frac{\Lambda_{i}}{k_{F}^{0}}\right)^{2}\frac{(1+I)^{1/3}}{u^{2/3}}\left(1-\frac{1}{1+\frac{(1+I)^{2/3}u^{2/3}}{\left(\frac{\Lambda_{i}}{k_{F}^{0}}\right)^{2}}}\right)$
$\displaystyle+$
$\displaystyle\frac{1}{2}\sum_{i=1,2}\left[C_{i}-\frac{C_{i}-8Z_{i}}{5}I\right]\left(\frac{\Lambda_{i}}{k_{F}^{0}}\right)^{2}\frac{(1-I)^{1/3}}{u^{2/3}}\left(1-\frac{1}{1+\frac{(1-I)^{2/3}u^{2/3}}{\left(\frac{\Lambda_{i}}{k_{F}^{0}}\right)^{2}}}\right),$
$\displaystyle\left(\frac{\partial E}{\partial Y_{p}}\right)_{n}$
$\displaystyle=$
$\displaystyle\frac{1}{2}E_{F}^{0}u^{2/3}\left[(1+I)^{2/3}-(1-I)^{2/3}\right]-\frac{1}{3}Au(\frac{1}{2}+x_{0})I$
(54) $\displaystyle-$
$\displaystyle\frac{\frac{2}{3}Bu^{\sigma}(\frac{1}{2}+x_{3})I}{\left(1+\frac{2}{3}B^{\prime}u^{\sigma-1}\left[\frac{3}{2}-(\frac{1}{2}+x_{3})I^{2}\right]\right)^{2}}$
$\displaystyle+$
$\displaystyle\frac{3}{2}\sum_{i=1,2}\left[\frac{C_{i}-8Z_{i}}{5}I\right]\left(\frac{\Lambda_{i}}{k_{F}^{0}}\right)^{3}\left(\frac{\left((1+I)u\right)^{1/3}}{\frac{\Lambda_{i}}{k_{F}^{0}}}-\tan^{-1}\frac{\left((1+I)u\right)^{1/3}}{\frac{\Lambda_{i}}{k_{F}^{0}}}\right)$
$\displaystyle-$
$\displaystyle\frac{3}{2}\sum_{i=1,2}\left[\frac{C_{i}-8Z_{i}}{5}I\right]\left(\frac{\Lambda_{i}}{k_{F}^{0}}\right)^{3}\left(\frac{\left((1-I)u\right)^{1/3}}{\frac{\Lambda_{i}}{k_{F}^{0}}}-\tan^{-1}\frac{\left((1-I)u\right)^{1/3}}{\frac{\Lambda_{i}}{k_{F}^{0}}}\right)$
$\displaystyle+$
$\displaystyle\frac{1}{2}\sum_{i=1,2}\left[C_{i}+\frac{C_{i}-8Z_{i}}{5}I\right]\left(\frac{\Lambda_{i}}{k_{F}^{0}}\right)^{2}\frac{u^{1/3}}{(1+I)^{2/3}}\left(1-\frac{1}{1+\frac{(1+I)^{2/3}u^{2/3}}{\left(\frac{\Lambda_{i}}{k_{F}^{0}}\right)^{2}}}\right)$
$\displaystyle-$
$\displaystyle\frac{1}{2}\sum_{i=1,2}\left[C_{i}-\frac{C_{i}-8Z_{i}}{5}I\right]\left(\frac{\Lambda_{i}}{k_{F}^{0}}\right)^{2}\frac{u^{1/3}}{(1-I)^{2/3}}\left(1-\frac{1}{1+\frac{(1-I)^{2/3}u^{2/3}}{\left(\frac{\Lambda_{i}}{k_{F}^{0}}\right)^{2}}}\right).$

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Figure 1:  Energy per baryon of symmetric (a) and pure neutron matter (b) of
the present model (MDIM) in comparison with those originated from realistic
calculations. More details for the models UV14+TNI, UV14+UVII and AV14+UVII in
Ref. [41] and for the models A18+UIX and A18+du+UIX∗ in Ref. [42].

Figure 2: a) The contribution of the various terms $V_{A}$, $V_{B}$, $V_{C}$
and the total potential energy density $V_{int}$ as a function of the baryon
density b) The momentum dependent term $V_{C}$ as a function of the baryon
density at temperature $T=0$, $T=10$ and $T=30$ MeV.

Figure 3: A comparison of the single-particle potential of symmetric nuclear
matter from the present model (MDIM) with the microscopic calculations of
Wirigna [47] and Li et al. [48], for densities $n=0.1$ fm-3, $n=0.3$ fm-3 and
$n=0.5$ fm-3.

Figure 4: The difference $F(n,T,Y_{p})-F(n,T,Y_{p}=1/2)$ as a function of
$(1-2Y_{p})^{2}$ at temperatures a) $T=10$ and b) $T=30$ MeV, for three baryon
densities.  Figure 5: The Helmholtz free energy $F(n,T,I)$ of $\beta$-stable
matter versus the baryon density $n$, for various values of $T$ (in MeV).

Figure 6: (a) The free energy of symmetric nuclear matter (for $T=5$ MeV and
$T=20$ MeV) and (b) for pure neutron matter (for $T=3$ MeV and $T=20$ MeV) of
the proposed model (MDIM) in comparison with the free energy calculated by
Friedman and Pandharipande model (FP) [7].  Figure 7: The internal energy
$E(n,T)$ of $\beta$-stable matter as a function of the baryon density $n$ for
various values of $T$.  Figure 8: The internal energy (for $T=0$ MeV) for
symmetric nuclear matter (SNM) and pure neutron matter (PNM) calculated with
the MDIM in comparison with the FP model.  Figure 9: The nuclear symmetry
energy calculated with the MDIM in comparison with the FP model as well as the
results of Refs. [54], [58] and [55].  Figure 10: The thermal energy
$E_{thermal}(n,T)=E(n,T)-E(n,T=0)$ of $\beta$-stable matter versus the baryon
density $n$, for various values of $T$.

Figure 11: The pressure $P$ of asymmetric nuclear matter for $Y_{p}=0.1$ and
$0.3$ at a) $T=10$ and b) $30$ MeV. The full lines give the results calculated
from Eq. (24), while the squares represent results obtained by differentiating
$F(n,T)$ (Eq. (23)).  Figure 12: The thermal pressure
$P_{thermal}(n,T)=P(n,T)-P(n,T=0)$ of $\beta$-stable matter versus the baryon
density $n$, for various values of $T$.  Figure 13: The fractions of protons
$Y_{p}$ electrons $Y_{e}$ and muons $Y_{\mu}$ of $\beta$-stable matter as
functions of the baryon density $n$, for various values of $T$.

Figure 14: The Fermi-Dirac distribution function $f_{\tau}(n,T)$ for protons
and neutrons ($\tau=p,n$ respectively), for $n=0.2$ fm-3, $n=0.4$ fm-3 and
$n=0.6$ fm-3 and various values of $T$.  Figure 15: The entropy per particle
$S$ of asymmetric nuclear matter with $Y_{p}=0.2$ at $T=10,20,30$ MeV. The
full lines give the entropy calculated from Eq. (21), while the squares give
results obtained by differentiating $F(n,T)$ (Eq. (25)).  Figure 16:
Contributions to the total entropy per particle of protons ($S_{p}$) (up
triangles) neutrons ($S_{n}$) (upside down triangles) and the total entropy
($S_{b}$) (squares). Figure 17: Contributions to the total entropy per
particle of electrons ($S_{e}$) (up triangles) muons ($S_{\mu}$) (upside down
triangles) and the total $S_{l}$ (squares). Figure 18: The pressures of
baryons $P_{b}$ leptons $P_{l}$ (electrons+muons) and the total pressure $P$
versus the baryon density, $n$ for various values of $T$.