# Critical Properties of Symmetric Nuclear Matter in Low-Density Regime Using
Effective-Relativistic Mean Field Formalism

Vishal Parmar1, M K Sharma1, S K Patra2,3 1 School of Physics and Materials
Science, Thapar Institute of Engineering and Technology, Patiala-147004,
India.  2 Institute of Physics, Bhubaneswar 751005, India  3 Homi Bhaba
National Institute, Training School Complex, Anushakti Nagar, Mumbai 400 085,
India<EMAIL_ADDRESS>
###### Abstract

The effective field theory motivated relativistic mean-field (E-RMF) formalism
is employed to study the equation of state (EoS) for the infinite symmetric
nuclear matter at finite temperature using the recently developed forces
FSUGarnet, IOPB-I, G3, and the well known NL3 force parameter. The EoS is then
used to estimate the critical temperature $T_{c}$, pressure $P_{c}$ and
density $\rho_{c}$ of the symmetric nuclear matter for the liquid-gas phase
transition. As $T_{c}$ is not a constrained parameter in both experiments and
theoretical calculations, there is a large uncertainty around its value.
Although, the critical parameters are correlated among themselves. It is
revealed that vector self-coupling $\zeta_{0}$ of used forces play determining
role in EoS in finite temperature limit. Keeping the incompressibility in
acceptable limit i.e. 240$\pm$ 20 MeV, the lower value of $\zeta_{0}$ gives a
better result of $T_{c}$ when compared to the several experimental data. The
critical parameters however show weak correlation with the properties at
saturation density at zero temperature. The compressibility factors calculated
with these parameters are in agreement with the universal value of liquid-gas
systems. Stability conditions are examined along with binodal and spinodal
regions. Besides this, the thermodynamic properties like specific heat and
latent heat are also worked out. We have carried out detailed consistency
check of our calculations using critical exponents and standard scaling laws.
All the exponents are well within the theoretical mean-field results.

###### pacs:

21.65.+f, 26.60.+c, 26.60.-c, 25.70.Pq

##  1 Introduction

One of the vital problems in nuclear physics is to study the nuclear matter
and its dependence on density, pressure, temperature, and neutron-proton
asymmetry etc. Mahi et al. and Finn et al. suggested that the liquid-gas phase
transition of nuclear matter can be analysed by studying the variation of
yield with mass and projectile energy in a multi fragment Proton-Xenon
reaction and found that the yield follows a power-law dependence [1, 2]. This
led to several experiments which explored the above mentioned critical nuclear
properties in the subsequent years [3, 4]. The critical behaviour of nuclear
matter is an important feature in medium energy heavy ion induced reactions
and therefore numerous theoretical phase-transition predictions have been
attempted by several authors over the last two decades. The basis of these
studies is thermodynamics at equilibrium and phase diagram of nuclear matter
[5, 6, 7]. Liquid-gas phase transition also plays an important role in
describing a multi-fragmentation process where intermediate mass fragments
(IMFs) are produced in the range of 3$\leq$Z$\leq$30 [8]. It was observed that
in the lowest energy state, nucleons show liquid-like characteristics with
density $\rho_{0}\approx$ 0.15 fm-3 [9]. As the temperature increases, the
nuclear liquid evaporates and undergoes the liquid-gas phase transition [10].
Below certain maximum temperature Tc, the liquid and gas phase remain in phase
coexistence which terminates at temperature greater than $T_{c}$ .

To investigate the liquid-gas phase tranistion in nuclear matter, The
Effective-Relativistic Mean Field Theory (E-RMF) has been extended to the case
of finite temperature. Quantum chromodynamics (QCD) is the fundamental theory
of strong interaction between quarks and gluons that make up hadrons but it
can not be used to describe hadron matter as it suffers from nonperturbative
properties. Quantum hadrodynamics (QHD) on the other hand is an effective
theory of strong interaction at low energy. After the suggestion of Dürr and
Teller [11], the first systematic successful model is formed by including only
$\sigma$ and $\omega$ mesons called Walecka model [9]. This model is not
preferred because of its high incompressibility $K_{\infty}\approx 550$ MeV
for the infinite nuclear matter at saturation and its stiff equation of state
(EoS). It was improved by adding the cubic and quartic nonlinearities of the
$\sigma$ meson to account for the incompressibility and finite nuclei. The
renormalizability of these models constrained the addition of scalar-vector
and vector-vector interactions. Subsequently, a series of parameter sets were
introduced such as NL1, NL2, NL-SH etc. [12] which included $\rho$ meson to
take care of the neutron-proton asymmetry and self-couplings of $\sigma$ meson
to reduce the incompressibility. These models still give a stiff EoS at
supernormal density. However, Based on the effective-field theory (EFT)
motivated relativistic mean-field (E-RMF) approach, Furnstahl, Serot, and Tang
obtained the G1 and G2 parameter sets [13]. Here, they neglected the idea of
renormalizability and included all possible couplings up to the fourth-order
of expansion in the fields which include scalar-vector and vector-vector self-
interactions in addition to tensor couplings. It is also well established that
the effective Lagrangian with meson couplings up to fourth order is a good
approximation to predict finite nuclei and nuclear matter observables
satisfactorily [13, 14, 15]. The E-RMF Lagrangian contains all the non-
renormalizable couplings consistent with the underlying symmetries of QCD. The
ambiguity in expansion is avoided by using the concept of naturalness [16].
The contributions of each term in effective Lagrangian is determined by
counting powers in the expansion parameters. Field are expanded in terms of
nucleons mass to truncate the Lagrangian at a given level of accuracy. For the
truncation to be consistent, the coupling constants should exhibit naturalness
and none should be arbitrarily dropped out to the given order without
additional symmetry arguments [17].

The study of nuclear matter EoS at finite temperature has multifold
importance. In the laboratory frame, Nuclear Multifregmentation [18, 19],
compound nucleus [20] , and heavy-ion collision experiments can be used to
obtain critical parameters of symmetric nuclear matter. There have been
numerous attempts to estimate critical temperature using non-relativistic
approaches such as Thomas-Fermi model [21], Hartree-Fock theory [22], Skyrme
interaction and Gogny interaction [23] etc. Several calculations have also
been done in the relativistic domain using relativistic mean-field framework
[24] and the critical temperature is found in the range $14.2-16.1$ MeV. Apart
from the phenomenological models, few first-principles predictions which uses
realistic models of NN interaction for the liquid-gas phase transition have
been carried out using methods such as Finite-temperature self-consistent
Green’s function (SCGF), Brueckner-Hartree-Fock (BHF) formalism [25], chiral
effective field theory [26] etc. These microscopic calculations estimate the
$T_{c}$ in the range 14-19 MeV.

At the experimental front, the critical temperature for infinite nuclear
matter is measured by extrapolating the data from the multifragmentation
reaction on finite nuclei. In these collision experiments, the hot nuclear
matter produced has few hundred nucleons only and therefore such process is
not able to describe the infinite nuclear matter. In contrast to infinite
nuclear matter, a hot nucleus shows a considerable reduction in its critical
temperature due to effects such as surface tension, Coulomb effects, finite-
size effects etc. This reduction is of about 7 MeV leading to the critical
temperature of $\approx$ 10 MeV in the mass range A = 50-400 [27].
Furthermore, the nature of phase transition is also altered due to strong
Coulomb interaction. In this context some authors prefer to define limiting
temperature ($T_{l}$) instead of the critical temperature for the hot nuclei
[28, 29]. This is the maximum temperature above which Coulomb effects
(together with the decrease in surface tension) lead to thermal dissociation
of nuclei. This $T_{l}$ is discussed experimentally by studying the plateau in
the caloric curve for different types of reactions at different energies [30].
Therefore, extending the multifragment reaction for the case of infinite
nuclear matter has several limitations arising from finite size, Coulomb,
isospin, angular momentum, and secondary decay of excited fragments etc.
However the finite nuclei can only be excited up to some limiting temperature

Furthermore, at low density, usually the nuclear matter is not a homogeneous
gas of nuclear matter but rather exists in the form of nuclei of different
shapes known as “nuclear pasta”[31] that forms the structure of inner crust of
neutron star. The formation of these nuclei of unique shapes lowers the free
energy of the system as compared to the corresponding homogeneous phase [32].
However, here we concentrate on ideal symmetric nuclear matter undergoing
liquid-gas phase transition. In the experiments to understand the phase
transition in nuclear matter [33, 34, 35, 36, 37, 38], the critical
temperature $T_{c}$ is hardly constrained. There is large uncertainty in the
value of $T_{c}$ among these experiments. Moreover, the model dependence in
these experimental calculations arises inevitably. Therefore, the theoretical
and experimental calculation of $T_{c}$ and in general liquid-gas phase
transition can not be compared one on one. Instead, to overcome the
uncertainty, we look for the dependence of $T_{c}$ on bulk matter properties
such as incompressibility at zero temperature. The correlation of critical
parameter among themselves can also be utilised to constraint the related
quantities and consequently deducing $T_{c}$. To understand this, we use newly
developed E-RMF forces G3 [39] ,FSUGarnet [40] and IOPB-I [41] in the finite
temperature limit. These forces have comparable bulk matter properties at zero
temperature although having different numbers of adjustable parameters and
their values. Different values of couplings in these newly developed E-RMF
sets motivates us to look for the contribution of different terms such as
scalar-scalar, scalar-vector terms etc. on the critical values and EoS of SNM
at finite temperature. It is to be noted that bulk matter properties are not
unique to a particular force as a different combination of these adjustable
parameters can give the same bulk matter properties at zero temperature. These
facts are used here to study the behaviour of these force in the finite
temperature limit and consequently analysing the liquid-gas phase transition
qualitatively.

There is a surprising similarity in behaviour near the critical point among
systems that are otherwise quite different in nature. These systems can be
partitioned based on their criticality and placed in some “universal classes”.
The systems here liquid-gas system, which belongs to one universal class
should have comparable values of critical exponents and compressibility factor
keeping with the scaling hypothesis and renormalization [42]. Therefore, a
complete statistical study including critical exponents of EoSs derived from
different force parameter is necessary to provide a complete qualitative and
quantitative understanding of phase transition properties as they all are
based on mean-field approximations.

The paper is organized as follows: In Section 2, we give a brief formalism of
the E-RMF model and its extension to finite temperature. In Section 3, results
and discussions are given. We justify the used forces in Sec. 3.1 and the EoS
at finite temperature and liquid-gas phase transition is discussed in Sec.
3.2. Stability analysis is discussed using binodals in Sec. 3.3. We have shown
the consistency of our calculations using critical exponents along with the
well-defined scaling laws or thermodynamical inequalities in Sec 3.4. We have
compared our results with the recent experimental data and various theoretical
studies. Finally, the results are summarised in section 4.

##  2  Formalism

In recent years, the effective field theory motivated relativitic mean field
approach is used extensively in many body nuclear physics problems. In this
theory, nucleons oscillate in the mean-field produced by the exchange of
mesons and photons. The isoscalar-scalar $\sigma$, isoscalar-vector $\omega$,
isovector-vector $\rho$ and isoovector-scalar $\delta$ mesons interact with
Dirac nucleons, i.e. protons and neutrons. The biggest advantage of the mean-
field theory is that mean-field approximations are thermodynamically
consistent and it obeys the relevant thermodynamic identities and the virial
theorem [43]. This provides us with the opportunity to fit our equation of
state according to known experimental values at zero temperature and then
extrapolate it to finite temperature. The basic nucleon-meson E-RMF which
involve couplings of $\sigma$, $\omega$, $\rho$ and $\delta$ mesons and the
photon with Dirac nucleon up to the fourth-order is given as [44, 45]

$\displaystyle\mathcal{E}=\psi^{\dagger}(i\alpha.\gradient+\beta[M-\Phi(r)-\tau_{3}D(r)]+W(r)+\frac{1}{2}\tau_{3}R(r)+\frac{1+\tau_{3}}{2}A(r)$
(1) $\displaystyle-\frac{i\beta\alpha}{2M}(f_{\omega}\gradient
W(r)+\frac{1}{2}f_{\rho}\tau_{3}\gradient
R(r)))\psi+\quantity(\frac{1}{2}+\frac{k_{3}\Phi(r)}{3!M}+\frac{k_{4}}{4!}\frac{\Phi^{2}(r)}{M^{2}})\frac{m^{2}_{s}}{g^{2}_{s}}\Phi(r)^{2}$
$\displaystyle-\frac{\zeta_{0}}{4!}\frac{1}{g^{2}_{\omega}}W(r)^{4}+\frac{1}{2g^{2}_{s}}\quantity\Big(1+\alpha_{1}\frac{\Phi(r)}{M})(\gradient\Phi(r))^{2}-\frac{1}{2g^{2}_{\omega}}\quantity\Big(1+\alpha_{2}\frac{\Phi(r)}{M})(\gradient
W(r))^{2}$
$\displaystyle-\frac{1}{2}\quantity\Big(1+\eta_{1}\frac{\Phi(r)}{M}+\frac{\eta_{2}}{2}\frac{\Phi^{2}(r)}{M^{2}})\frac{m^{2}_{\omega}}{g^{2}_{\omega}}W^{2}(r)-\frac{1}{2e^{2}}(\gradient
A^{2}(r))^{2}-\frac{1}{2g^{2}_{\rho}}(\gradient R(r))^{2}$
$\displaystyle-\frac{1}{2}\quantity\Big(1+\eta_{\rho}\frac{\Phi(r)}{M})\frac{m^{2}_{\rho}}{g^{2}_{\rho}}R^{2}(r)-\Lambda_{\omega}(R^{2}(r)W^{2}(r))+\frac{1}{2g^{2}_{\delta}}(\gradient
D(r))^{2}+\frac{1}{2}\frac{m^{2}_{\delta}}{g^{2}_{\delta}}(D(r))^{2}.$

Here $\Phi(r)$, W(r), R(r), D(r) and A(r) are the fields corresponding to
$\sigma$, $\omega$, $\rho$ and $\delta$ mesons and photon respectively. The
$g_{s}$, $g_{\omega}$, $g_{\rho}$, $g_{\delta}$ and $\frac{e^{2}}{4\pi}$ are
the corresponding coupling constants and $m_{s}$, $m_{\omega}$, $m_{\rho}$ and
$m_{\delta}$ are the corresponding masses. Now the field equations for
nucleons and mesons can be solved by applying variational principle within the
mean field approximations. The single particle energy of nucleon is obtained
by solving the Dirac equation constraining the quantum mechanical
normalization as $\sum_{\alpha}\psi^{\dagger}\psi_{\alpha}=1$ [46].

The effective mass of the proton and neutron changes because of its motion in
the mean potential generated by the mesons. It is given as

$\displaystyle M^{*}_{p}$ $\displaystyle=$ $\displaystyle M-\Phi(r)-D(r),$ (2)
$\displaystyle M^{*}_{n}$ $\displaystyle=$ $\displaystyle M-\Phi(r)+D(r).$

Now, the E-RMF can be extended to finite temperature. We calculate the grand
canonical thermodynamic potential $\Omega$ using statistical mechanics as [47]

$\Omega=-k_{B}\text{T}\ln Z$ (3)

and

$Z=Tr\quantity{\exp(-\beta(\hat{\mathbf{H}}-\mu\hat{\mathbf{B}}))}.$ (4)

Where Z is the grand canonical partition function, $\hat{\mathbf{H}}$ is the
Hamiltonian operator, $\hat{\mathbf{B}}$ is the baryon number operator for the
mean field and $k_{B}$ is the Boltzmann constant. The basic relations between
thermodynamic potential ($\Omega$), entropy (S), chemical potential ($\mu$)
and temperature (T) are given by

$\Omega=-pV=E-TS-\mu B$ (5)

and

$\differential{\Omega}=-S\differential{T}-p\differential{V}-B\differential{\mu}.$
(6)

Now the baryon and scalar densities can be calculated by using the ensemble
average and are given as :

$\rho_{b}=\frac{1}{V}\quantity\Big[\partialderivative{\Omega}{\mu}]=\sum_{\alpha}\frac{\gamma}{(2\pi)^{3}}\int\differential[3]k[n_{k}(T)-\bar{n}_{k}(T)]$
(7)

and

$\rho_{s}=\frac{1}{V}\quantity\Big[\partialderivative{\Omega}{M}]=\sum_{\alpha}\frac{\gamma}{(2\pi)^{3}}\int\differential[3]k\frac{m^{*}_{\alpha}}{\sqrt{k^{2}_{\alpha}+M^{*}_{\alpha}}}[n_{k}(T)+\bar{n}_{k}(T)],$
(8)

with the $n_{k}(T)$ and $\bar{n}_{k}(T)$ are the baryons and antibaryons
occupation numbers respectively, which are defined by the Fermi distribution
function at finite temperature T as

$n_{k}(T)=\frac{1}{1+\exp\quantity(\frac{(E^{*}(k)-\nu)}{T})}$ (9)

and

$\bar{n}_{k}(T)=\frac{1}{1+\exp\quantity(\frac{(E^{*}(k)+\nu)}{T})},$ (10)

with E∗ as $\sqrt{k^{2}+{M^{*}}^{2}}$. The effective chemical potential $\nu$
for proton and neutron in eqs. 9 and 10 are defined as

$\displaystyle\nu_{p}$ $\displaystyle=$
$\displaystyle\mu-W(r)+\frac{1}{2}R(r),$ (11) $\displaystyle\nu_{n}$
$\displaystyle=$ $\displaystyle\mu-W(r)-\frac{1}{2}R(r).$

The entropy density s ($S=s/\rho_{b}$) can easily be calculated and has the
same form as that for a non-interacting gas. It is given as

$s_{i}=-2\sum_{i}\int\frac{\differential^{3}k}{(2\pi)^{3}}[n_{k}\ln
n_{k}+(1-n_{k})\ln(1-n_{k})+(n_{k}\leftrightarrow\bar{n}_{k})].$ (12)

The energy and pressure expression now can be calculated form energy-momentum
tensor Tμν which is obtained from Noether’s theorem as [9]

$T_{\mu\nu}=\sum_{i}\partial_{\nu}\Phi_{i}\frac{\partial\mathcal{L}}{\partial(\partial^{\mu}\Phi_{i})}-g_{\mu\nu}\mathcal{L}.$
(13)

The energy density is calculated using the zeroth component
$\expectationvalue{T_{00}}$ and pressure of the system is obtained by the
third component $\expectationvalue{T_{ii}}$ of energy-momentum tensor, i.e.

$\displaystyle
E=\sum_{p,n}\frac{\gamma}{(2\pi)^{3}}\int\differential[3]{k}E^{*}_{p,n}(k)[n_{k}(T)+\bar{n}_{k}(T)]+\rho
W+\quantity\Big(\frac{1}{2}+\frac{k_{3}\Phi}{3!M}+\frac{k_{4}}{4!}\frac{\Phi^{2}}{M^{2}})\frac{m^{2}_{s}}{g^{2}_{s}}\Phi^{2}$
(14)
$\displaystyle-\frac{1}{2}\quantity\Big(1+\eta_{1}\frac{\Phi}{M}+\frac{\eta_{2}}{2}\frac{\Phi^{2}}{M^{2}})\frac{m^{2}_{\omega}}{g^{2}_{\omega}}W^{2}-\frac{\zeta_{0}}{4!}\frac{1}{g^{2}_{\omega}}W^{4}+\frac{1}{2}\rho_{3}R-\frac{1}{2}\quantity\Big(1+\eta_{\rho}\frac{\Phi}{M})\frac{m^{2}_{\rho}}{g^{2}_{\rho}}R^{2}$
$\displaystyle-\Lambda_{\omega}(R^{2}W^{2})+\frac{1}{2}\frac{m^{2}_{\delta}}{g^{2}_{\delta}}(D)^{2}.$

$\displaystyle
P=\sum_{p,n}\frac{\gamma}{3(2\pi)^{3}}\int\differential[3]{k}\frac{k^{2}}{E^{*}_{p,n}(k)}[n_{k}(T)+\bar{n}_{k}(T)]-\quantity\Big(\frac{1}{2}+\frac{k_{3}\Phi}{3!M}+\frac{k_{4}}{4!}\frac{\Phi^{2}}{M^{2}})\frac{m^{2}_{s}}{g^{2}_{s}}\Phi^{2}$
(15)
$\displaystyle+\frac{1}{2}\quantity\Big(1+\eta_{1}\frac{\Phi}{M}+\frac{\eta_{2}}{2}\frac{\Phi^{2}}{M^{2}})\frac{m^{2}_{\omega}}{g^{2}_{\omega}}W^{2}+\frac{\zeta_{0}}{4!}\frac{1}{g^{2}_{\omega}}W^{4}+\frac{1}{2}\quantity\Big(1+\eta_{\rho}\frac{\Phi}{M})\frac{m^{2}_{\rho}}{g^{2}_{\rho}}R^{2}$
$\displaystyle+\Lambda_{\omega}(R^{2}W^{2})-\frac{1}{2}\frac{m^{2}_{\delta}}{g^{2}_{\delta}}(D)^{2}.$

Here $\gamma$ is the spin-isospin degeneracy. $\gamma$ = 2 for pure neutron
matter (PNM) and $\gamma$=4 for symmetric nuclear matter (SNM).

After obtaining EoS in pressure and energy, it is straightforward to calculate
latent heat using the Clausius-Clapeyron equation

$L=T\Big{(}\frac{1}{\rho_{g}}-\frac{1}{\rho_{l}}\Big{)}\Big{(}\frac{dP}{dT}\Big{)}_{coex}.$
(16)

Here the $\frac{dP}{dT}$ is along the coexistence curve which is determined
from the Maxwell construction. It invoke the Gibbs conditions of phase
transition along the isotherms as

$\mu_{q}^{{}^{\prime}}(\rho^{{}^{\prime}},T)=\mu_{q}^{{}^{\prime\prime}}(\rho^{{}^{\prime\prime}},T),\;(q=n,p),$
(17) $P^{\prime}(\rho^{\prime},T)=P^{\prime\prime}(\rho^{\prime\prime},T).$
(18)

Where two phases are represented by prime and a double-prime. $\rho_{g}$ and
$\rho_{l}$ are the coexistence gas and liquid density respectively.
Alternatively, the latent heat can be estimated from

$L=T(s_{g}-s_{l}),\;(s=S/\rho_{b}),$ (19)

which is the amount of heat required to take one particle from the ordered
phase to disordered phase at a constant temperature, pressure and chemical
potential. The related thermodynamic variables such as free energy ($F$),
specific heat ($C_{V}$), isothermal compressibility ($K_{T}$) can be
calculated using standard thermodynamic relations.

##  3  Results and Discussions

###  3.1 Force Parameters

In the present study, the critical behaviour of the liquid-gas phase
transition is investigated using three different E-RMF sets, namely FSUGarnet
[40], IOPB-I [41] and G3 [39]. The results are also compared with NL3 [48]
force which is one of the most successful forces used in the finite system
calculations. It describes nuclear properties such as quadrupole deformation
and the charge radius exceptionally well for nuclei away from the valley of
stability besides nuclei on $\beta$-stability line [49]. The FSUGarnet is a
recent parameter set and IOPB-I is the latest in this series. These both sets
are known to reproduce the neutron-skin thickness up to a satisfactory level
along with other bulk matter properties. These give the adequate addressal of
neutron star masses in the lower (M=2.01 $\pm$ 0.04 M$\odot$ ) and upper limit
(M=2.16 $\pm$ 0.03 M$\odot$ ) [41]. These both sets have positive $k_{3}$ and
negative $k_{4}$ corresponding to cubic and quartic terms arising from the
self couplings of the $\sigma$ meson. It is to be noted that a large negative
value of $k_{4}$ leads to the divergence of a solution in the lighter mass
region of the periodic table [50]. The scalar and vector cross-couplings
$\eta_{1}$ and $\eta_{2}$ zero in these sets are zero yet they have the value
of vector self-coupling $\zeta_{0}$ within the acceptable limit. The positive
$k_{3}$, $k_{4}$ and small $\zeta_{0}$ guarantee the agreement with Dirac-
Brueckner-Hartree-Fock (DBHF) theory [17].
The G3 parameter set contains all the mesons and coupling constants in Eq 1
and therefore describe properties like skin thickness and two-neutron
separation energy exceptionally well. The main feature of G3 is that it
include the $\delta$ mesons. These are an important degree of freedom for the
infinite nuclear matter calculations. Furthermore, due to the finite
$\eta_{1}$ and $\eta_{2}$, the G3 set has $\zeta_{0}$ $\approx$ 1 and positive
$k_{3}$ and $k_{4}$. All these coupling constants have their specific role in
EoS and they, therefore, describe the characteristics of a force.

Figure 1:  (a) The pressure versus energy density for NL3, FSUGarnet, IOPB-I
and G3 parameter sets at T = 0. The lower shaded region represents the
constraint at rph=R with a 95% confidence limit. Here rph and R represents
photospheric radius and neutron star radius [51]. The upper shaded region
represents the equation of state for zero temperature dense matter with a 95 %
confidence limit from QMC + Model A [52]. (b) The pressure versus baryon
density with NL3, FSUGarnet, IOPB-I and G3 sets. The results are compared with
experimental data extracted from analyzing EoS after including pressure from
asymmetry and density dependence [53].

Any effective nuclear force is such that it can explain both finite and
infinite nuclear matter. However, it is difficult using the properties of
nuclei as constraints. Consequently, one prefers to extrapolate the
systematics of observables such as binding energy, saturation density,
compressibility, effective mass etc. The E-RMF parameter sets considered above
have a very narrow range of binding energy, saturation density and effective
nucleon mass. They also have compressibility in the range as determined from
isoscalar giant monopole resonance (ISGMR) [54] result. Currently, this value
is 240 $\pm$ 20 MeV. When compared with the experimentally derived EoS’s from
various sources at zero temperature depicted in Fig 1, all the parameter sets
are in reasonable agreement except the NL3 force due to its slightly larger
incompressibility. This disagreement can be neglected in this work as the
domain of density useful for the liquid-gas phase transition is less than 0.15
$fm^{-3}$. Moreover, NL3 set is known to work exceptionally in this density
range. Therefore, it becomes essential to investigate the implication of these
forces in the finite temperature limit realizing their comparable bulk matter
properties at zero temperature. Since the mean-field theory is
thermodynamically consistent, a true “universal” force should not only
describe nuclear matter at zero temperature but it should adequately explain
the finite temperature properties both qualitatively and quantitatively. We
aim to understand the behaviour of these forces in the context of the liquid-
gas phase transition as they have different scalar and vector self and cross-
couplings. .

###  3.2  Liquid-gas phase transition

Figure 2: The pressure versus baryon density $\rho_{b}$ at various
temperatures in MeV for the FSUGarnet, G3, IOPB-I and NL3 sets. The dot
represents the microscopic calculation by [56]

Now we discuss the implication of these force parameters at finite temperature
explicitly near nuclear matter saturation density. The equation of state is
shown in Fig 2 for the FSUGarnet, G3, IOPB-I and NL3 sets at temperature T= 0,
5, Tf, Tc, 20 MeV in the pressure vs baryon density phase diagram. Here Tf and
Tc stands for flash and critical temperature respectively. It is evident that
P-$\rho_{b}$ isotherms have a quite distinct pocket at a lower temperature.
These isotherms show typical short-range Van der Waals like property as the
nature of both Van der Waals and Nuclear force is the same. As one move
towards higher temperature, the compressibility of nuclear matter decreases
and therefore at a particular temperature, we will have a situation where
pressure is no more negative i.e. P($\rho$, Tf)=0 and consequently
$dP/d\rho=0.0$. This temperature is called the flash temperature and the
corresponding density flash density. It represents the highest temperature at
which a self-bound system can exist in hydrostatic equilibrium (P=0). Above
this temperature, the nuclear matter is unbound and starts expanding. Further
increase in temperature results in the occurrence of an inflation point which
is given by [58]

$\frac{\partial
p}{\partial\rho}\bigg{|}_{T_{c}}=\frac{\partial^{2}p}{\partial\rho^{2}}\bigg{|}_{T_{c}}=0.$
(20)

This temperature is called critical temperature $T_{c}$ after which, the
pressure is a monotonically increasing function of density. The corresponding
pressure and density are termed critical pressure $P_{c}$ and critical density
$\rho_{c}$. At the critical density, the second derivative of free energy
($F=E-TS$) as a function of baryon density is called critical
incompressibility. This is given by [55]

$K_{c}=9\rho_{b}^{2}\frac{\partial^{2}}{\partial\rho_{b}^{2}}\frac{F}{\rho_{b}}\bigg{|}_{\rho_{c}}.$
(21)

Table 1 compiles all these critical values for all the sets. There is a large
uncertainty among the several experimental values of critical temperature and
it is hardly constrained [33, 34, 35, 36, 37, 38]. This uncertainty mainly
arises because all these experiments are performed for fragmentation reaction
on finite nuclei and then extrapolated to evaluate the $T_{c}$ for the
infinite matter. These experimental calculations are model dependent. Besides,
various effects like finite size, small time scale $\approx 10^{-(22-23)}$s in
multi fragment reaction which makes it hard to study thermodynamic
equilibrium, Coulomb interaction, isospin, angular momentum etc., also add to
this uncertainty. Therefore, it is not very wise to theoretically address the
quantitative nature of the nuclear matter phase transition using any mean-
field calculation. Although one can realise based on recent experiments [36,
37, 38] that critical temperature should be greater than 15 MeV.

Table 1: The calculated critical temperature $T_{c}$ (in MeV), critical
pressure $P_{c}$ (in MeV fm-3) and critical density $\rho_{c}$ (in fm-3) with
the respective binding energy (in MeV) at the critical point along with the
flash temperature $T_{f}$ and flash density $\rho_{f}$ with FSUGarnet, IOPB-I,
G3 and NL3 parameter sets for symmetric nuclear matter. The experimental data
are given wherever available. The corresponding incompressibility, effective
mass at critical density and compressibility factor $C_{f}$ and critical
incompressibility $K_{c}$ is also shown.

| $T_{c}$ | $P_{c}$ | $\rho_{c}$ |  $\mathcal{E}/\rho_{b}$ at $T_{c}$ | Tf | $\rho_{f}$ | $K_{\infty}$ | $M^{*}_{c}/M$ | $C_{f}$ | $K_{c}$
---|---|---|---|---|---|---|---|---|---|---
| MeV | MeV/fm3 | fm-3 | MeV | MeV | fm-3 | MeV |  |  | MeV
FSUGarnet | 13.80 | 0.171 | 0.043 | -9.61 | 11.3 | 0.071 | 229.50 | 0.850 | 0.282 | -68.35
IOPB-I | 13.75 | 0.167 | 0.042 | -8.80 | 11.2 | 0.071 | 222.65 | 0.864 | 0.277 | -69.95
G3 | 15.30 | 0.218 | 0.049 | -6.68 | 12.1 | 0.075 | 243.96 | 0.879 | 0.292 | -82.91
NL3 | 14.60 | 0.202 | 0.046 | -8.38 | 11.8 | 0.070 | 271.38 | 0.846 | 0.276 | -77.85
G1 [13] | 14.30 | 0.187 | 0.046 | -8.24 | 11.5 | 0.075 | 215.00 | 0.877 | 0.285 | -72.03
G2 [13] | 14.30 | 0.184 | 0.043 | -8.04 | 11.8 | 0.080 | 215.00 | 0.879 | 0.299 | -78.77
NL1 [57] | 13.74 | 0.164 | 0.041 | 9.71 | 11.2 | 0.070 | 211.70 | 0.872 | 0.290 | -71.909
NL2 [57] | 18.63 | 0.361 | 0.056 | -3.47 | 14.3 | 0.085 | 399.20 | 0.861 | 0.345 | -111.03
NL-SH [57] | 15.96 | 0.264 | 0.052 | -6.70 | 12.7 | 0.080 | 355.36 | 0.846 | 0.315 | -90.097
FP (Microscopic) [56] | 17.5 $\pm$ 1.00 | - | - | - | - | - | 240.00 | - | - | -
Exp. 1 (1984) [33] | 12.0 $\pm$ 0.20 | - | - | - | - | - | - | - | - | -
Exp. 2 (1994) [34] | 13.1 $\pm$ 0.60 | - | - | - | - | - | - | - | - | -
Exp. 3 (2002) [35] | 16.6 $\pm$ 0.86 | - | - | - | - | - | - | - | - | -
Exp. 4 (2003) [36] | 20.0 $\pm$ 3.00 | - | - | - | - | - | - | - | - | -
Exp. 5 (2008) [37] | 19.5 $\pm$ 1.20 | - | - | - | - | - | - | - | - | -
| 16.5 $\pm$ 1.00 | - | - |  | - | - | - | - | - |
Exp. 6 (2013) [38] | 17.9 $\pm$ 0.40 | 0.31 $\pm 0.07$ | 0.06 $\pm$ 0.01 | - | - | - | - | - | 0.288 | -

Theoretically, there are two sets of RMF parameters. One, those have
incompressibility in the accepted range as predicted by ISGMR result i.e. 240
$\pm$ 20 MeV. These parameters, which include the FSUGarnet, IOPB-I and G3
forces, predict the critical temperature less than 15 MeV. One exception being
G3 which estimates $T_{c}$ = 15.3 MeV. These sets are extensively used for
Neutron star calculation where the nuclear matter is at high density. The
others set, although estimate large $T_{c}$ but have large incompressibility
and therefore are not used in nuclear astrophysical applications. Our
calculations with the set FSUGarnet, IOPB-I, G3 along with the NL3 force are
in reasonable agreement with the first set of forces.

The critical parameters obtained using mentioned E-RMF sets shows some
striking behaviour in terms of their correlations which otherwise are very
difficult to establish analytically. $T_{c}$ and $P_{c}$ have the strongest
correlation with Pearson’s coefficient 0.991 with p-value of 0.009. Here
p-value represents the significance level of correlation coefficient. This is
in consistency with classical van der Waals (VDW) gas property ; $T_{c}/P_{c}$
=8b, where b arises due to the repulsive interaction [59]. A similar trend is
observed in $T_{c}T_{f}$,$T_{c}K_{c}$,$P_{c}K_{c}$ and $\rho_{c}K_{c}$ with
p-value less then 0.05. These correlations are in consistent with the standard
analytical equation relating the $K_{c}$ and the ratio $P_{c}/\rho_{c}$ given
by [55]

$K_{c}+18\frac{P_{c}}{\rho_{c}}=0,$ (22)

Therefore, these correlation suggests the strong relationship of critical
parameter among themselves. They become important as constraining $T_{c}$ is
difficult and a direct comparison with experimental data is not suitable. If
however, extrapolating the finite nuclei experiments to infinite matter was
possible accurately and critical temperature in these experiments could be
measured precisely, one can easily get other finite temperature properties
based on established correlations.

Figure 3: The contribution arising from the different meson interaction on
total pressure at T=10 MeV is shown for the different parameter sets. (a)
Total Pressure, contribution due to (b) kinetic part (c) scalar-self
interaction (d) vector-self interaction and (e) scalar-vector cross
interaction

On the other hand, weak correlation of critical parameters with that of
saturation properties at zero temperature is also interesting. Naively, one
could argue that the binding energy at saturation should be related to
critical temperature. Instead, there is no such correlation between the
binding energy $e_{0}$ and $T_{c}$ or $T_{f}$. Similarly these models also do
not satisfy the empirical relations [60] between saturation density $\rho_{0}$
and $\rho_{c}$ or $\rho_{f}$. We observe that the $T_{c}$ has a spread of 1.55
MeV while the saturation energy and saturation density lie within 0.27 MeV and
0.005 $fm^{-3}$ respectively. It indicates that the properties at saturation
have no dictation over the position of critical points. Therefore, any
relation, relating properties at saturation and finite temperature can not be
generalised. It should also be noted that each of the above force i.e.
FSUGarnet, IOPB-I, G3 and NL3 have different adjustable parameters. One can
not make any prediction directly by these adjustable parameters. In other
words, none of the scalar and vector couplings (self and cross)
($k_{3},k_{4},\zeta_{0}$) individually has a direct correlation with the
critical parameters. However, accept the NL3 force, a greater value of
$\zeta_{0}$ gives smaller $\rho_{c}$ and force with greater incompressibility
estimates the larger $T_{c}$ . A close investigation of the contribution
arising from scalar and vector channel suggests very strongly that these
forces only differ in contribution from vector self-coupling term and
therefore, $\zeta_{0}$ becomes an important quantity in finite temperature
EoS. This is shown in Figure 3 where contribution from different interaction
in total pressure is shown at T=10 MeV. The only major difference is in graph
(d) which represents the vector-self interaction term in the pressure. The
effect of small value of $\zeta_{0}$ in G3 and absence of it in NL3 is
visible. The NL3 consequently have a large negative $k_{4}$ which is a
possible reason for it to not show the common characteristic of other three
forces. A soft contribution of vector-self interaction in total pressure
therefore seems to be the reason for larger $T_{c}$ in G3 set. The
compressibility factor $C_{f}$, which signify the deviation from ideal gas is
given by [61]

$C_{f}=\frac{P_{c}V_{c}}{T_{c}}=\frac{P_{c}}{\rho_{c}T_{c}},$ (23)

can be calculated for the respective force parameters. The universal class
corresponding to the liquid-gas system have the compressibility factor
$C_{f}\approx$ 0.292 [62] following the law of corresponding states. Table 1
also shows the value of $C_{f}$ for each force parameter. All the forces
except NL2 and NL-SH, which have high incompressibility have $C_{f}$ close to
0.292. This further validates the importance of an EoS well within the
acceptable incompressibility range [61].
The recent parameters FSUGarnet, IOPB-I and G3 are known to work remarkably at
zero temperature high-density regime. This work also aims to find their
validity at the low dense finite temperature limit. For this, EoS for each
force in Fig. 2 is also compared with the two-nucleon and three-nucleon
interactions called $\nu_{14}$ and TNI [56] at T= 0, 5 and 20 MeV. This
microscopic variational calculation is known to work better at low density but
is not quite suitable for infinite nuclear matter [63]. Whereas, here we are
looking at the low-density behaviour of these forces at finite temperature.
The EoS for each of the force at the mentioned temperature traces the
microscopic calculation at density $<$ 0.06 MeV but then deviate at higher
density. This deviation gets better at a higher temperature and the parameter
set G3 is nearest to the microscopic variational calculation at T=20 MeV. This
is simply because the fitting procedure of any force does not take into
account the critical temperature of the liquid-gas phase transition. The
critical temperature, therefore, can be an important tool in making a force
universally acceptable at both low and high-density regime at zero and finite
temperature. It can be used along with the other bulk matter properties at
zero temperature as a constraint for EoS. The parameter set G3 can be a good
candidate for that.

Figure 4: Variation of free energy, pressure and rigidity (1/KT) with density
for critical isotherm for the FSUGarnet, G3, IOPB-I and NL3 sets.

The temperature also impacts the incompressibility at saturation
($K_{\infty}$) or rigidity ($K_{T}^{-1}=\rho_{b}\partial P/\partial\rho_{b}$)
of nuclear matter. At large temperature, the nucleons are more prone to be
free due to extra thermal energy and hence incompressibility decreases. Figure
4 shows the variation of Free energy, pressure and rigidity (1/KT) with
density for critical isotherm for different parameter sets. For critical
isotherm, the free energy is a smooth monotonically increasing function with
G3 showing the least slope. Below the critical isotherm, the free energy has a
well-defined pocket which disappears at T=Tc. The respective rigidity (1/KT)
is shown for each parameter set. It first increases and then starts decreasing
and at the critical density, becomes zero. At critical density, one observes
the inflation point in P-$\rho$ isotherm, which makes rigidity equals zero at
the critical point.

Figure 5: Chemical potential as a function of pressure for various values of
temperatures.

The order of the transition is depicted in Fig 5 in the chemical potential vs
pressure graph. The chemical potential plays an important role in determining
the values of Fermi function which consequently determine the baryon density.
For each parameter set, as one moves from zero pressure to point ‘b’, chemical
potential increases sharply. The potential then decreases till point ‘a’ and
pressure becomes negative. The path ‘ba’ marks the instability region. After
point ‘a’, the potential increases monotonically with a different slope as
compared to the path 0 to ‘b’. The area of path ‘obao’ is largest for the set
G3 and NL3 for each temperature. Because the system wants to attain minimum
potential and free energy, a real system will not follow the path ‘obao’ and
will take the path with the least potential. It will move from zero pressure
to point ‘o’ and then will again increase. The path ‘obao’ keeps decreasing
with an increase in temperature and vanishes as one reaches the critical
temperature. At critical isotherm, the slop changes discontinuously validating
the argument of nuclear matter undergoing a first-order liquid-gas phase
transition at low density. The set G3 and NL3 estimate the largest instability
area for a given isotherm, suggesting the dominance of strong attractive
$\Phi$ field over repulsive vector $W$ field at low density for these sets,
which play important role in determining the instability boundary. This is
directly the consequence of the value of $\zeta_{0},k_{3},k_{4}$ in these
forces. The low value of $\zeta_{0}$ in G3 and NL3 sets account for the less
repulsive force as compared to the high value of $\zeta_{0}$ in FSUGarnet and
IOPB-I forces. Therefore the instability area is inversely proportional to the
value of $\zeta_{0}$ in finite temperature limit.
The isobaric entropy as a function of temperature is shown in Figure 6 for the
FSUGarnet, G3, IOPB-I and NL3 sets. The values of pressure are 0.05, 0.1,
0.15, 0.2, 0.25 and 0.3 MeV fm-3 from left to right. There is a sharp
discontinuity at the transition temperature marking the presence of two phases
in the system. As one goes beyond the critical temperature (refer Table 1),
the curve becomes smooth and continuous showing that system now only exists in
the gaseous phase. One can see a sharp difference in isobar for 0.2 MeV fm-3
for all the sets. For IOPB-I and FSUGarnet, this pressure is above $P_{c}$
showing continuous curve and for G3 and NL3, this is very close to $P_{c}$
thus still showing a discontinuity. Furthermore, the relative separation
between isobars decreases as we increase the pressure.

The isobaric entropy is a function of chemical potential as is shown in Eq. 12
and consequently dictated by the respective EoS. Again the vector self-
coupling $\zeta_{0}$ plays an important role. It can, therefore, be
apprehended that, if a force parameter is designed to have the $\zeta_{0}$
$\approx$ 1.0 keeping the incompressibility within the limit, it can address
the qualitative as well as quantitative nature of phase transition. This is
visible in these parameter sets. They only differ much in the value of
$\zeta_{0}$ with almost the same bulk matter properties. The force G3 stands
out due to its low $\zeta_{0}$ and pressence cross-couplingss
($\eta_{1},\eta_{2}$) of scalar $\sigma$ and vector $\omega$ meson.

Figure 6: The entropy at constant pressure for 0.05, 0.1, 0.15, 0.2, 0.25 and
0.3 MeV fm-3 as a function of temperature for various force parameters.

###  3.3  Stability Analysis

Let us consider the symmetric nuclear matter interacting only through strong
interaction as Coulomb interaction is zero hypothetically due to absence of
any source current [64]. It, therefore, reduces to a one-component system and
Gibbs phase rule then allows only one degree of freedom i.e. only one pressure
for each temperature. The phase transition of nuclear matter liquid into
gaseous phase start occurring when the stability condition [58]

$F(T,\rho)<(1-\lambda)F(T,\rho^{\prime})+\lambda F(T,\rho^{\prime\prime}),$
(24)

with

$\rho=(1-\lambda)\rho^{\prime}+\lambda\rho^{\prime\prime},\hskip
8.5359pt(0<\lambda<1),$ (25)

where F is Helmholtz free energy per volume and $\lambda$ is volume fraction,
is violated.

The two phases are denoted by a prime and a double prime. This implies that a
system should follow the inequalities

$\displaystyle C_{V}>0\;or\;\Big{[}\frac{\partial S}{\partial
T}\Big{]}_{\rho}>0,$ (26) $\displaystyle K_{T}>0\;or\Big{[}\frac{\partial
P}{\partial\rho}\Big{]}_{T}>0,$ (27)

which corresponds to the dynamical and mechanical stability, to prevent phase
change. If anyone of these is violated, a system with more than one phase is
energetically favourable. In our calculations, the dynamical stability
condition is never violated. In the limit $T\rightarrow 0$, the specific heat
vanishes as $dS/dT\rightarrow 0$. At higher temperature, the specific heat
asymptotically approaches the noninteracting limit 3/2 for low densities.

In Fig. 2, one can observe that for each isotherm, there are three values of
densities for each small positive pressure for $T<T_{c}$. At very low density,
the thermal and Fermi degeneracy pressure makes the total pressure positive.
As density increases slightly, the attractive force arising due to attractive
scalar $\Phi$ field, increases, which try to reduce the volume of the system
and pressure goes from positive to negative. This corresponds to the part
where $dP/d\rho<0$. It violates the mechanical stability inequality in Eq. 27
and system transform into a two-phase system. Instability is only a sufficient
condition for phase separation to take place, therefore, it may even occur in
the one phase-stable system. This is the case of the metastable state [65].
Once the two-phase system is favoured over the one-phase system, the solution
of Eqns. 17 and 18 gives us the coexistence densities i.e. $\rho_{g}$ for
gaseous phase and $\rho_{l}$ for liquid phase for each isotherm. This density
pair for each temperature and pressure then determine the coexistence phase
boundary or binodal. Also, the locus of points where the second derivative of
free energy is zero for each isotherm marks the boundary of spinodal. The
point where the binodal and spinodal curve meet defines the critical
parameters given in Table 1. This is shown in Figure 7 where the binodal or
coexistence boundary and spinodal or instability boundary is plotted for
FSUGarnet, G3, IOPB-I and NL3 sets. The middle red shaded region marks the
region of instability and outer blue region signify the metastable region. The
outer blue solid line is the coexistence boundary which gives us the
($\rho_{g},\rho_{l}$) couple.

Figure 7: The binodal and spinodal curve for FSUGarnet, G3, IOPB-I and NL3
sets .

The liquid at saturation density is in equilibrium with the zero density
gaseous phase at zero temperature. As one increases the temperature, the
liquid coexistence density decreases and gaseous coexistence density increases
until the critical temperature is encountered. Beyond critical temperature,
the liquid-gas coexistence vanishes and the system only remain in the gaseous
phase. At small temperatures i.e. below 5-6 MeV, there is no solution to Eqns.
17 and 18 and system only remain in one stable liquid phase. For more insight
and closer comparison among all the force parameters, the binodal are plotted
in Figure 8. Here, critical density and temperature are used for scaling in
the right panel. As discussed, the least value of $\zeta_{0}$ gives us the
maximum binodal area for G3. The NL3 shows a similar behaviour. on the other
hand, FSUgarnet and IOPB-I with comparable $\zeta_{0}$ show similar behaviour
if we look at the right panel of the Figure 8. The comparatively large
instability region in G3 is the consequence of soft scalar-self coupling as
also evident in graph (c) of Figure 3.

Figure 8: The binodal curve for FSUgarnet, G3, IOPB-I and NL3 sets along with
dimensionless density and temperature

One of the unique features of the first-order phase transition is that the
first-order derivatives of Gibbs free energy are discontinuous. It gives rise
to the release of heat at a constant temperature which is called the latent
heat of vaporization. The latent heat can be determined using the Clausius
Clapeyron equation given in Eq 16. This equation requires the derivative of
pressure with temperature along the binodal or coexistence curve. This is
called vapour pressure. Figure 9 shows the variation of vapour pressure along
with the temperature for the NL3, FSUGarnet, IOPB-I, and G3 sets. The outer
graph is in the reduced dimensionless form where the respective critical
parameter is used to making them dimensionless. The inner graph uses absolute
values of the nuclear matter parameters. There is not much deflection in
different force parameter sets. At higher temperatures, the curve has a linear
dependence which can be expressed as $\pi=a\tau$ where $\pi$ and $\tau$ are
reduced pressure and temperature respectively. This characteristic helps to
determine the behaviour of the curve in the neighbourhood of critical
parameters. At lower temperature, pressure rises very slowly and there is a
sudden increase after 0.5$T_{c}$ for all forces. Below this temperature system
only remains in one stable liquid phase and therefore system encounter very
less pressure. At higher temperature, the thermal excitation and existence of
a two-phase system drive the pressure towards higher values.

Figure 9: Vapour pressure as a function of temperature with absolute and
reduced dimensionless values. Figure 10: Latent heat of vaporization for
symmetric nuclear matter $L=T(s_{g}-s_{l})$ for the FSUGarnet, IOPB-I, NL3 and
G3 sets

Figure 10 shows the variation of latent heat of vaporization with temperature
for different parameter sets using the relation $L=T(s_{g}-s_{l})$ by virtue
of Clausius Clapeyron equation. This method is numerically more stable as
compared to the calculation of the derivative of the vapour pressure shown in
Figure 9. We have checked the values of latent heat with both the methods and
they estimate a similar trend. The figure shows the characteristic behaviour
of latent heat with temperature. In the zero-temperature limit, we can write
the Clausius Clapeyron equation as [66]

$\small\lim_{T\to 0}L(T)=\lim_{T\to
0}T\Big{(}\frac{1}{\rho_{g}}-\frac{1}{\rho_{l}}\Big{)}\frac{dP_{vapour}}{dT}=-\mu_{g}=-\mu_{l}=e_{0},$
(28)

because the liquid-gas coexistence density approaches zero for vanishing
temperature. Therefore, in this limit, latent heat is simply the heat required
to remove a nucleon from the saturated liquid with energy density e0. As one
increases the temperature, the latent heat rises linearly up to a maximum
value which is close to $\approx$ 30 MeV. This linearity at low temperature
indicates the absence of interactions for low-density gas. This trend is
similar to all the forces considered and there is very less deflection among
different parameters in both left panel where absolute values are plotted and
in the right panel where reduced parameters are depicted to minimise the mean-
field dependence i.e. effect of different vector and scalar self and cross-
couplings of the various models. The latent heat reaches the peak value in
between 8-10 MeV or (0.5-0.7)Tc in all cases and then fall sharply to zero at
the critical temperature. The maximum $L/e_{0}$ in right panel has a very
narrow range of 1.8-1.9. The Latent heat, therefore, is a thermally correlated
parameter. Furthermore, the maximum latent heat LH is the least correlated
parameter at finite temperature. A similar trend is seen if we use flash
temperature as the scaling parameter. Therefore, flash temperature along with
the critical temperature is suitable to constraint the EoS for finite
temperature. At low temperature, the latent heat has linear relationship with
temperature and as the system reaches near the critical temperature, all the
force parameters give the same slope in the right panel of the graph which is
a significant validation of the well defined thermodynamical theories [66].

###  3.4 Scaling Laws

The critical points related to the phase transition are of special interest as
they are used to subsequent characterisation. The behaviour of a given system
in the neighbourhood of these points forms a fundamental problem in view of
the phase transition phenomenon. Near the critical point, various physical
quantities encounter the singularity. It becomes essential to express these
singularities in terms of power laws. These power laws are characterised by
critical exponents and in turn, determine the qualitative nature of a given
system. The exponent should also follow well known scaling laws which make
only two exponents independent at a given time. The nature of exponents is
such that they do not differ much from one system to other in one universality
due to their dependence on very less number of parameters [67]. The E-RMF
calculations are needed to numerically satisfy these laws to check their
independence from the underlying interactions. Therefore, these are used here
to qualitatively check the consistency of our analysis. The exponents $\beta$,
$\lambda$, $\delta$, $\alpha$ and $\tau$ are defined as [68]

$\displaystyle\rho_{l}-\rho_{g}$ $\displaystyle\sim$
$\displaystyle\Big{(}\frac{T_{c}-T}{T_{c}}\Big{)}^{\beta},$ (29)
$\displaystyle K_{T}$ $\displaystyle\sim$
$\displaystyle\Big{|}\frac{T-T_{c}}{T_{c}}\Big{|}^{-\lambda},$ (30)
$\displaystyle|P-P_{c}|_{T_{c}}$ $\displaystyle\sim$
$\displaystyle\Big{|}\frac{\rho-\rho_{c}}{\rho_{c}}\Big{|}^{\delta},$ (31)
$\displaystyle C_{V}$ $\displaystyle\sim$ $\displaystyle(T_{c}-T)^{-\alpha},$
(32)

and finally, from Fisher’s analysis [69],

$\displaystyle C_{f}=\frac{\zeta(\tau)}{\zeta(\tau-1)}.$ (33)

A simple power law fitting was done in the form $y=aX^{b}$ and corresponding
values of these exponents along with the mean field reaults and other liquid-
gas systems are shown in Table 2. All the critical exponents are close to
their respective values derived from mean field results.

Table 2: Calculated critical exponent for NL3, IOPB-I, FSUGarnet and G3 parameter sets.  Critical | NL3 | IOPB-I | FSUGarnet | G3 | Mean | Liquid-Gas
---|---|---|---|---|---|---
Exponent |  |  |  |  | Field | System [70]
$\beta$ | 0.37 | 0.40 | 0.39 | 0.40 | 0.5 | 0.32-0.35
$\lambda$ | 0.97 | 1.12 | 1.02 | 1.01 | 1 | 1.2-1.3
$\delta$ | 3.65 | 3.62 | 3.50 | 3.38 | 3 | 4.6-5.0
$\alpha$ | 0.00 | 0.00 | 0.00 | 0.00 | 0 | 0.0-0.2
$\tau$ | 2.209 | 2.210 | 2.167 | 2.224 | - | 2.1-2.25

The consistency check of the critical exponents can be done using standard
scaling laws or thermodynamic inequalities [42]. These are given by
Rushbrooke, Griffiths and Widom inequalities and can be respectively written
as [68]

$\displaystyle\alpha+2\beta+\gamma\geq 2,$ (34)
$\displaystyle\alpha+\beta(1+\delta)\geq 2,$ (35)
$\displaystyle\beta(\delta-1)-\gamma\leq 0,$ (36)

and from Fisher’s droplet model as ,

$\displaystyle\frac{\beta}{\gamma}-\frac{\tau-2}{3-\tau}=0.$ (37)

Table 3 compiles all the calculated results of these scaling laws for each
parameter set. The critical exponents derived from each parameter satisfy
these scaling laws up to a good approximation. We, can therefore strongly
conclude that E-RMF parameter sets with incompressibility comparable to
accepted experimental value are very consistent in the finite temperature
limit satisfying the statistical inequalities. They can consequently be used
in the calculation of hot EoS which is of primary importance in processes like
binary neutron star merger where temperature plays a very important role.

Table 3: Result of scaling laws for NL3, IOPB-I, FSUGarnet and G3 parameter sets. Force Parameter | Rushbrooke | Griffiths | Widom | Fisher
---|---|---|---|---
NL3 | 1.72 | 1.74 | 0.02 | 0.13
IOPB-I | 1.92 | 1.84 | -0.07 | 0.09
FSUGarnet | 1.79 | 1.75 | -0.04 | 0.10
G3 | 1.80 | 1.75 | -0.05 | 0.10

##  4  Summary and Conclusions

In summary, we have discussed that the nuclear matter dictated by strong
interactions which have an attractive (scalar) and a repulsive (vector ) part
undergo a first-order liquid-gas phase transition at low densities alike Van
der Waals equation of state. We have used three E-RMF force FSUGarnet, IOPB-I
and G3 in the finite temperature limit realizing there narrow range of bulk
matter properties at zero temperature. They are subsequently compared with one
of the most used set NL3. They all have different fitting procedures and
therefore different coupling constants. This fact is utilized to understand
the role of these couplings in the liquid-gas phase transition of nuclear
matter. It is shown that these forces can describe this phase transition in an
excellent manner qualitatively using instability analysis and scaling laws. In
our calculations, the critical temperature for the symmetric nuclear matter
lies between (13.75-15.3) MeV. The corresponding range of pressure $P_{c}$ and
density $\rho_{c}$ are between (0.167-0.218) MeV fm-3 and (0.042-0.049)fm-3,
respectively. The estimated critical parameters are in agreement with other
RMF forces which have incompressibility in the range 240 $\pm$ 20 MeV.
The critical parameters i.e. $T_{c}$, $P_{c}$ and $\rho_{c}$ are clearly model
dependent. However, there is a direct correlation among parameter such as
$\rho_{c}\rho_{f}$, $T_{c}T_{f}$, $P_{c}T_{c}$, etc. The critical temperature
is not constrained in theoretical as well as in experiments. Consequently,
there is large uncertainty in the value of critical temperature among
experiments as well as theoretical calculations with same bulk matter
properties. Both the experimental and theoretical calculations of critical
temperature are model dependent. The value of critical temperature $T_{c}$,
therefore, can be used for constraining the EoS.
The vector-self coupling is a determining factor in these E-RMF forces. Force
with a lower value of $\zeta_{0}$ estimate large coexistence area and large
$T_{c}$. Similarly, $T_{c}$ is positive correlated with incompressibility at
saturation except in NL3 set. A large negative value of $k_{4}$ makes the NL3
behave differently in some cases as compared to other forces. The parameter
sets used in the present work estimate the compressibility factor (Cf) close
to the universal value of liquid-gas systems. Whereas, the parameters having
higher incompressibility deviate much from this value. We have also checked
the consistency of our calculation using critical exponents and scaling laws
and emphasize that E-RMF parameters sets with acceptable incompressibility are
consistent and reliable at finite temperature limit. All the critical
exponents are very close to the mean-field results and experimental liquid-gas
systems.
We have also studied the stability conditions in terms of binodal and spinodal
curves using IOPB-I, FSUGarnet, G3 and NL3 force parameters. Subsequently,
they are used to determine thermodynamic properties like specific and latent
heat. The specific heat has linear dependence for a large number of particles.
For a lesser number of particles, it asymptotically approaches the classical
limit. Latent heat, on the other hand, is found to be a thermally correlated
variable and does not depend on the effective field.
The E-RMF sets explain the phase transition in the low-density regime
exceptionally well. A little discrepancy in the experimental and theoretical
values might be attributed to two main factors: (1) symmetric nuclear matter
being an ideal system which is too difficult to simulate in experiments and
(2) finite size effect and very short time scale of multi fragment reactions
which make it difficult to study thermodynamic equilibrium. Moreover, there is
also a need to look for higher-order vector interaction terms which might
influence the EoS at finite temperature keeping other properties intact.

##  5 References

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11institutetext: Centro de Pensamiento Antonio Sanchez de Cozar, Vereda Los
Egidos, San Gil, Santander
11email<EMAIL_ADDRESS>22institutetext: Universidad Nacional de
Colombia, Bogota, Colombia
22email<EMAIL_ADDRESS>
# NL Is Strictly Contained in P

Santiago Flum 11 J. Andres Montoya 22

###### Abstract

We prove that NL is strictly contained in P. We get this separation as a
corollary of the following result: the set of context-free languages is not
contained in NL. The reader should recall that CFL is contained in
DTIME($n^{3}$), see [7].

To the memory of Neil Jones

##  1 Mealy Machines and Pebble Automata

A Mealy machine is a deterministic real-time finite-state machine
$\mathcal{M}$ that is provided with two tapes, a one-way input tape together
with a one-way output tape [4].

###### Definition 1

A Mealy machine is a tuple

$\mathcal{M}=\left(Q,q_{0},\Sigma,\Gamma,\delta,\lambda\right)$

such that:

  1. 1.

$Q$ is a finite set of states.

  2. 2.

$q_{0}\in Q$ is the initial state.

  3. 3.

$\Sigma$ is the input alphabet.

  4. 4.

$\Gamma$ is the output alphabet.

  5. 5.

$\delta:Q\times\Sigma\rightarrow Q$ is the transition function.

  6. 6.

$\lambda:Q\times\Sigma\rightarrow\Gamma^{\ast}$ is the output function.

Let $w\in\Sigma^{n}$ be an input of $\mathcal{M}$. We define the output of
$\mathcal{M}$, on input $w,$ as follows:

  * •

$\widehat{\lambda}\left(q,a\right)=\lambda\left(q,a\right),$ for all $q\in Q$
and for all $a\in\Sigma.$

  * •

$\widehat{\lambda}\left(q,w\right)=\lambda\left(q,w\left[1\right]\right)\widehat{\lambda}\left(\delta\left(q,w\left[1\right]\right),w\left[2,...,n\right]\right).$

  * •

$O_{\mathcal{M}}\left(w\right)=\widehat{\lambda}\left(q_{0},w\right).$

Mealy machines can be used to compute syntactic morphisms and more general
reductions between formal languages. We consider one of those reductions.

###### Definition 2

Let $L\subset\Sigma^{\ast},T\subset\Gamma^{\ast}$be two languages. We say that
$L$ is Mealy-reducible to $T$ if and only if there exists a Mealy machine

$\mathcal{M}=\left(Q,q_{0},\Sigma,\Gamma,\delta,\lambda\right)$

such that for all $w\in\Sigma^{\ast}$ the condition

$w\in L\text{ if and only if }O_{\mathcal{M}}\left(w\right)\in T$

holds. We use the symbol $L\preceq_{M}T$ to indicate that $L$ is Mealy
reducible to $T.$

Most grammar classes are closed under Mealy reductions.

The one-way principle. Suppose $L\preceq_{M}T$ and suppose that $T$ is
accepted by a one-way machine $\mathcal{M}$ that has some specific structure
and uses some specific amount of resources. Language $L$ can be accepted by a
one-way machine $\mathcal{N}$ that has the same structure and uses the same
amount of resources as $\mathcal{M}$.

The soundness of this principle is based on the following fact:

Suppose $L\preceq_{M}T$ and suppose that $T$ is accepted by a one-way machine
$\mathcal{M}$. Let $\mathcal{N}$ be a Mealy machine that reduces $L$ in $T.$
We can use machine $\mathcal{M}$ to accept $L.$ To this end, and given $w,$ we
answer the membership query for $w$ by processing
$O_{\mathcal{N}}\left(w\right)$ with machine $\mathcal{M}$. The above
principle allows us to assert that:

  1. 1.

Suppose that $T$ is regular, we have that $L$ is also regular.

  2. 2.

Suppose that $T$ is a context-free language, we have that $L$ is also a
context-free language.

  3. 3.

Suppose that $T$ is a deterministic context-free language, we have that $L$ is
also deterministic.

###### Definition 3

Let $\mathcal{C}$ be a set of languages and let $L$ be a language in
$\mathcal{C}$. We say that $L$ is $\mathcal{C}$-hardest (or
$\mathcal{C}$-complete) if and only if any language in $\mathcal{C}$ is Mealy-
reducible to $L.$

Greibach studied the notions of Mealy-reducibility and completeness [3], [2].
If $L\preceq_{M}T$ holds, she would say that $L$ is an inverse gsm image of
$T.$ And, if $T$ is $\mathcal{C}$-hardest, Greibach would say that
$\mathcal{C}$ is generated by $T$ and that $\mathcal{C}$ is a principal AFL.
She proved two important results:

  1. 1.

The set CFL has a hardest language under syntactic morphisms [2], that is:
there exists a context-free language $L_{0}$ such that for all $L$ in CFL
there exists a syntactic morphism $h$ for which

$\text{either }L=h^{-1}\left(L_{0}\right)\text{ or
}L=h^{-1}\left(L_{0}\cup\left\\{\varepsilon\right\\}\right)$

Notice that syntactic morphisms are a special type of Mealy reductions.

  2. 2.

There does not exist a language that is DCFL-hardest [3]. To prove this
Greibach used a hierarchy argument based on the concept of jump pushdown store
acceptors [3].

###  1.1 Pebble Automata

Pebble automata are two-way automata provided with a bounded amount of
pebbles.

###### Definition 4

Let $k\geq 0,$ a nondeterministic $k$-pebble automaton is a tuple

$\mathcal{M}=\left(Q,q_{0},\Sigma,H,A,\delta\right)$

such that:

  1. 1.

$Q$ is a finite set of states.

  2. 2.

$q_{0}\in Q$ is the initial state.

  3. 3.

$H\subset Q$ is the set of halting states.

  4. 4.

$A\subset H$ is the set of accepting states.

  5. 5.

$\Sigma$ is the input alphabet.

  6. 6.

$\delta:Q\times\Sigma\times\mathcal{P}\left(\left\\{0,...,k\right\\}^{2}\right)\rightarrow\mathcal{P}\left(Q\times\mathcal{P}\left(\left\\{0,...,k\right\\}^{2}\right)\times\left\\{-1,0,1\right\\}\right)$
is the transition function

The $k$-pebble automaton $\mathcal{M}$ is a two-way nondeterministic finite
state automaton provided with $k$ pebbles which can be distinguished from each
other. Moreover, we have:

  * •

$\mathcal{M}$ can place any one of its pebbles on the tape.

  * •

$\mathcal{M}$ can sense the pebbles that lie on the position of the input
head.

  * •

$\mathcal{M}$ can pick specific pebbles from this cell when necessary.

Let $\left(q,a,A,B\right)$ be a tuple such that $A,B$ are disjoint subsets of
$\left\\{1,...,k\right\\}.$ Let us suppose that $q$ represents the inner state
of $\mathcal{M}$. Let us suppose that $a$ is the character being scanned by
$\mathcal{M}$. And, finally, let us suppose that $A$ is the set of pebbles
that are placed on the actual cell, while $B$ is the set of available pebbles.
Suppose that $\left(p,C,D,\varepsilon\right)\in\delta\left(q,a,i,j\right)$ and
suppose that $A\cup B=C\cup D.$ Automaton $\mathcal{M}$ can
nondeterministically choose to change its inner state from $q$ to $p,$ leave
the pebbles in $C$ on the actual cell, and move its head in the direction
indicated by $\varepsilon$.

We use the symbol $\mathcal{REG}_{k}$ to denote the set of languages that can
be accepted by nondeterministic automata provided with $k$ pebbles. Notice
that a binary tape of size $\log\left(n\right)$ can be simulated using two
pebbles, while a pebble can be tracked using one binary tape of logarithmic
size. It is not hard to check that the class
NSPACE$\left(\log\left(n\right)\right)$ is equal to
$\mathop{\displaystyle\bigcup}\limits_{k\geq 1}\mathcal{REG}_{k},$ see [6] and
[5].

###### Remark 1

We use the symbol NL to denote the complexity class
NSPACE$\left(\log\left(n\right)\right)$

It can be proved that $\mathcal{REG}_{1}$ is equal to the set of regular
languages [1]. The hierarchy

$\mathcal{REG}_{1}\subseteq\mathcal{REG}_{2}\subseteq\mathcal{REG}_{3}\subseteq\cdots$

is a strict hierarchy [5]. We call this hierarchy the pebble hierarchy. The
levels of the pebble hierarchy are not invariant under Mealy reductions.
However, we can prove something that suffices for our purposes.

###### Theorem 1.1

Suppose $L\preceq_{M}T$ and suppose that $T\in\mathcal{REG}_{k}$ we have that
$L\in\mathcal{REG}_{k+1}.$

###### Proof

Let us prove that $L$ can be accepted with $k+1$ pebbles. Let $\mathcal{M}$ be
a $k$-pebble automaton that accepts $T,$ and let $\mathcal{N}$ be the Mealy
machine that reduces $L$ in $T.$ We construct a $k+1$-pebble automaton
$\mathcal{S}$ that receives as input $w\in\Sigma^{\ast}$ and simulates the
computation of $\mathcal{M}$ on input $O_{\mathcal{N}}\left(w\right).$ The
simulation works as follows:

Suppose that $\mathcal{M}$ is in the state $q$ with its head located on the
right end of $O_{\mathcal{N}}\left(w\left[1\cdots i\right]\right).$ Suppose
that the pebbles on this cell are the pebbles in
$A\subset\left\\{1,...,k\right\\}.$ Suppose that
$B\subset\left\\{1,...,k\right\\}$ is the set of available pebbles. Suppose
also that the simulation has run well until this point. This means that:

  1. 1.

The head of $\mathcal{S}$ is located on cell $i.$

  2. 2.

The set of pebbles placed on the last cell of
$O_{\mathcal{N}}\left(w\left[1\cdots i\right]\right)$ and the set of pebbles
placed on cell $i$ are equal. If $B$ is the set of available pebbles for
$\mathcal{M}$, then $B\cup\left\\{k+1\right\\}$ is the set of available
pebbles for $\mathcal{S}$. Moreover, if a pebble
$P\in\left\\{1,...,k\right\\}$ is located on the last cell of
$O_{\mathcal{N}}\left(w\left[1\cdots l\right]\right),$ the corresponding
pebble $P,$ used by automaton $\mathcal{S}$, is located on cell $i.$

  3. 3.

$\mathcal{S}$ keeps in its inner memory the state
$\widehat{\delta_{\mathcal{N}}}\left(q_{0},w\left[1,...,i\right]\right).$

Notice that $\mathcal{S}$ can use its inner memory to compute the string

$\lambda\left(\widehat{\delta_{\mathcal{N}}}\left(q_{0},w\left[1,...,i\right]\right),w\left[i\right]\right)$

and to evaluate the transition function of $\mathcal{M}$. This function is
nondeterministic. Thus, suppose that $\mathcal{M}$ chooses a transition that
forces it to move the input head one position to the right. In this case
automaton $\mathcal{S}$ moves its workhead to the right and updates its inner
memory. Now suppose that $\mathcal{M}$ chooses a transition that forces
$\mathcal{M}$ to move its workhead one position to the left. Automaton
$\mathcal{S}$ has to use pebble $k+1$ in this case. This is so because
$\mathcal{S}$ has to reconstruct the value of

$\lambda\left(\widehat{\delta_{\mathcal{N}}}\left(q_{0},w\left[1,...,i-1\right]\right),w\left[i-1\right]\right)$

Let us see how the simulation proceeds in this case:

  * •

Automaton $\mathcal{S}$ places pebble $k+1$ on cell $i-1.$

  * •

$\mathcal{S}$ moves its workhead until the left end of the tape.

  * •

$\mathcal{S}$ begins to move rightward searching for the pebble $k+1$ while
simultaneously computing the transduction $\lambda.$

  * •

$\mathcal{S}$ computes

$\lambda\left(\widehat{\delta_{\mathcal{N}}}\left(q_{0},w\left[1,...,i-1\right]\right),w\left[i-1\right]\right),$

and updates its inner memory.

It is not hard to check that everything works well this way. The theorem is
proved.

##  2 The Main Result

We are ready to prove the main results of this work.

###### Theorem 2.1

CFL is not contained in NL.

###### Proof

Suppose that CFL is contained in NL. Let $L_{0}$ be Greibach’s hardest
context-free language [2]. We have that $L_{0}$ belongs to NL. There exists
$k_{0}\geq 2$ such that the languages $L_{0}$ and
$L_{0}$-$\left\\{\varepsilon\right\\}$ are accepted by nondeterministic two-
way automata provided with $k_{0}$ pebbles. Let $L$ be a context-free
language. We have that $L$ is an inverse homomorphic image of $L_{0}$ or
$L_{0}$-$\left\\{\varepsilon\right\\}.$ We claim that:

  1. 1.

Language $L$ can be accepted using a nondeterministic two-way finite state
automaton provided with $k_{0}+1$ pebbles (Theorem 1.1).

###### Remark 2

It can be proved that $L$ can be accepted with $k_{0}$ pebbles, since the
reduction of $L$ in $L_{0}$ is a syntactic morphism.

  2. 2.

Let $k\geq 1,$ there exists a deterministic context-free language $L_{k}$ such
that $L_{k}$ can be accepted with $k+1$ pebbles but $L_{k}$ cannot be accepted
with $k$ pebbles (Lemma 1).

It suffices if we pick, for the sake of contradiction, the language
$L_{k_{0}+1}.$

Let us prove the lemma

###### Lemma 1

Let $k\geq 1,$ there exists a deterministic context-free language $L_{k}$ such
that $L_{k}$ can be accepted with $k+1$ pebbles but $L_{k}$ cannot be accepted
with $k$ pebbles.

###### Proof

Let $k\geq 1.$ We use the symbol $\mathcal{REG}_{k}$ to denote the set of
languages that can be accepted by two-way finite state automata that use no
more than $k$ pebbles. We prove by induction that there exists a deterministic
context-free language $L_{k}$ such that
$L_{k}\in\mathcal{REG}_{k+1}$-$\mathcal{REG}_{k}.$

  1. 1.

$k=1.$ The set $\mathcal{REG}_{1}$ equals the set of regular languages [1].
Let $c\neq 0,1,$ and let

$pal=\left\\{wcw^{R}:w\in\left\\{0,1\right\\}^{\ast}\right\\},$

where $w^{R}$ denotes the reverse of $w$. Language $pal$ is a deterministic
context-free language that can be accepted with $2$ pebbles. We get that
$pal\in\mathcal{REG}_{2}$-$\mathcal{REG}_{1}.$ We set $L_{1}=pal.$

  2. 2.

Let $k\geq 1,$ we suppose that there exists a deterministic context-free
language $L_{k}$ that belongs to $\mathcal{REG}_{k+1}$-$\mathcal{REG}_{k}.$
Let $\Sigma_{k}$ be the alphabet of $L_{k}.$ We pick $a,b\notin\Sigma_{k}$ and
we set

$L_{k+1}=\left\\{a^{i}bw_{1}bw_{2}b\cdots bw_{n}:\left|\left\\{j\leq
n:w_{j}\in L_{k}\right\\}\right|\geq i\right\\}$

We claim that $L_{k+1}$ is a deterministic context-free language that belongs
to $\mathcal{REG}_{k+2}$-$\mathcal{REG}_{k+1}.$ Let us check this:

    1. (a)

We prove that $L_{k+1}$ is a deterministic context-free language. Let
$\mathcal{N}$ be a deterministic pushdown automaton accepting the language
$L_{k}.$ We use the symbol $\mathcal{N}^{\ast}$ to denote a modified automaton
whose input alphabet is $\Sigma_{k}\cup\left\\{a,b\right\\}.$ Let
$a^{i}bw_{1}bw_{2}b\cdots bw_{n}$ be an input of $\mathcal{N}^{\ast}$.
Automaton $\mathcal{N}^{\ast}$ works, on this input, as follows:

       * •

$\mathcal{N}^{\ast}$ scans $a^{i}$ and push $i$ special symbols into the
pushdown store. We call those special symbols stones.

       * •

Let $j\leq n.$ When $\mathcal{N}^{\ast}$ scans $bw_{j}b,$ it simulates the
computation of $\mathcal{N}$ on input $w_{j}.$ If $\mathcal{N}$ rejects
$w_{j},$ the automaton $\mathcal{N}^{\ast}$ pops all the symbols in the
pushdown store that are not stones and continues with the scanning of
$bw_{j+1}b.$ On the other hand, if $\mathcal{N}$ accepts $w_{j},$ the
automaton $\mathcal{N}^{\ast}$ pops all the symbols in the pushdown store that
are not stones and it also pops exactly one stone. If the pushdown store
becomes empty the automaton halts and accepts the input, otherwise it
continues with the scanning of the input string. It is easy to check that
$\mathcal{N}^{\ast}$ accepts $L_{k+1}.$

    2. (b)

Let us check that $L_{k+1}$ belongs to $\mathcal{REG}_{k+2}$. Let
$\mathcal{M}$ be a $k+1$-pebble automaton that accepts $L_{k}.$ We can assume
that $\mathcal{M}$ is an automaton that halts after $O\left(n^{k+2}\right)$
time units. We use the symbol $\mathcal{M}^{\ast}$ to denote a modified
automaton that uses $k+2$ pebbles and whose input alphabet equals
$\Sigma_{k}\cup\left\\{a,b\right\\}.$ Let $a^{i}bw_{1}bw_{2}b\cdots bw_{n}$ be
an input of $\mathcal{M}^{\ast}.$ Automaton $\mathcal{M}^{\ast}$ uses $k+1$
pebbles to simulate the computations of $\mathcal{M}$ on the strings
$w_{1},...,w_{n}.$ Automaton $\mathcal{M}^{\ast}$ uses its additional pebble
to count the $w_{j}$’s that get accepted by $\mathcal{M}$. This pebble is
placed, at the beginning of the computation, on the leftmost character of
$a^{i}.$ Automaton $\mathcal{M}^{\ast}$ moves this pebble one position to the
right each time $\mathcal{M}$ accepts a string $bw_{j}b.$ Automaton
$\mathcal{M}^{\ast}$ halts and accepts when this pebble gets to be placed on
the letter $b.$ The automaton rejects its input if the $k+2^{th}$-pebble, the
counter pebble, does not reach the letter $b.$

    3. (c)

We prove that $L_{k+1}$ cannot be accepted with $k+1$ pebbles. Suppose, for
the sake of contradiction, that $\mathcal{N}$ is a $k+1$-pebble automaton that
accepts $L_{k+1}.$ Let $a^{i}bw_{1}bw_{2}b\cdots bw_{n}$ be an input string.
Suppose that $i<n$ are very large integers. Suppose also that

$\left|\left\\{j\leq n:w_{i}\in L_{k}\right\\}\right|$

is a large integer that is close to $i.$ Automaton $\mathcal{N}$ cannot use
its finite memory to count the $j$’s. Automaton $\mathcal{N}$ has to use one
of its $k+1$ pebbles as a counter. We can assume that this pebble stays, along
the whole computation, on the substring $a^{i}b.$ Automaton $\mathcal{N}$ is
forced to use no more than $k$ pebbles to detect the $j$’s for which $w_{j}\in
L_{k}.$ This latter task is as hard as accepting $L_{k},$ and we know that
this latter task cannot be done with $k$ pebbles. We get a contradiction.

We get as an easy corollary the following theorem.

###### Theorem 2.2

The following assertions hold true:

  1. 1.

NL $\neq$ logCFL.

  2. 2.

L and NL do not include the class DTIME($n^{3}$).

  3. 3.

L $\neq$ P.

  4. 4.

NL $\neq$ P.

  5. 5.

The problems in P that are P-complete cannot be solved with logarithmic space.

  6. 6.

The problems is P that are logCFL-complete cannot be solved with logarithmic
space.

Acknowledgement. The authors thank the support provided by Universidad
Nacional de Colombia and Centro de investigaciones matematicas Antonio Sanchez
de Cozar, vereda los Egidos, San Gil, Santander.

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