Thomas-Fermi approximation
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We first define the functional $$\Omega= E - \mu_p B_p - \mu_n B_n ,
\label{Omega}$$ where $E$ is the energy, $\mu_p$ ($\mu_n$) is the proton (neutron) chemical potential and $B_p$ ($B_n$) is the proton (neutron) number. Within the semi-classical Thomas-Fermi approximation, the energy of the nuclear system with particles described by the one-body phase-space distribution function $ f({\mathbf r},{\mathbf p},t)$ at position $\mathbf r$, instant $t$ with momentum $\mathbf p$ is given by $$E= \sum_i \gamma \int \d^3r  \frac{\d^3p}{(2\pi)^3}\,
f_i({\mathbf r},{\mathbf p},t)
\left(\sqrt{{\mathbf p}^2 + {M^*}^2}
+ {\cal V}_{i}\right)$$ $$+ \frac{1}{2} \int \d^3r  \left[
(\nabla \phi)^2 + m_s^2 \phi^2 -(\nabla V_0)^2 - m_v^2 V_0^2 \right.$$ $$\left. - (\nabla b_0)^2 - m_\rho^2 b_0^2  - (\nabla A_0)^2 \right]$$ where $${\cal V}_{p}= \Gamma_v V_0  + \frac{\Gamma_\rho}{2} b_0 + e A_0\;,
\quad {\cal V}_{n}= \Gamma_v V_0  - \frac{\Gamma_\rho}{2}  b_0 \;,$$

$\gamma=2$ refers to the spin multiplicity and the distribution functions for protons and neutrons are $$f_i=\theta(k_{Fi}^2(r)-p^2), ~~~~~~~i=p,n~.$$ In this approach, the scalar, proton and neutron densities become: $$\rho_s(r)= \frac{\gamma}{2\pi^2} \sum_{i=p,n} \int_0^{k_{Fi}(r)}
p^2 \d p \frac{M^*}{\epsilon}$$ with $\epsilon=\sqrt{p^2+{M^*}^2}$ and $$B_i= \int \d^3r \rho_i, \quad
\rho_i(r)= \frac{\gamma}{6\pi^2}k_{Fi}^3(r).$$ From the above expressions we get for (\[Omega\]) $$\Omega= \int \d^3r \left( \frac{1}{2} \left [
(\nabla 