-------
               $\e$                $-0.3 \pm 5\times 10^{-7}$   -0.3

            $\e_{kin}$                  $0.66 \pm 0.05$         0.644

               $\s$                       $0\pm 0.03$           0.012

   $ 19 \langle v^2 \rangle ^2 /         $1.01\pm0.04$            1
      5 \langle v^4 \rangle $                                  
  --------------------------------------------------------------------

  : Equilibrium MD and MF results for a uniform state for $\e=-0.3$, $N=250$, and $0\leq \t \leq 5000 $

  ------------------------------------------------------------------------
            \[tab\_ch\]                         MD                   MF
  ------------------------------- ------------------------------- --------
               $\e$                $-0.3392 \pm 2\times 10^{-4}$   -0.339

            $\e_{kin}$                    $2.9  \pm 0.1$            2.94

               $\s$                       $-1.5 \pm 0.1$           -1.46

   $ 19 \langle v^2 \rangle ^2 /          $0.99 \pm 0.03$            1
      5 \langle v^4 \rangle $                                     

            $N_{core}$                       $48\pm 2$              47.6
  ------------------------------------------------------------------------

  : Equilibrium MD and MF results for a core-halo state at $\e=-0.339 $, $N=250$, and $0\leq \t \leq 1500 $

To obtain the expression for the MF virial variable $$\label{vir}
\s_{MF}=\e+\e_{kin}(1-8\pi\r(1)/3)$$ we write for the pressure at the container wall $P=2\rho(x=1)\e_{kin}/3$ implying that the system is isothermal. Since the interparticle potential is not pure Coulombic, the virial variable is non-zero. The difference is especially prominent for the core-halo states where more particles ”probe” the short-range part of the potential.

To evaluate the core radius and number of core particles of the core-halo system, we considered an integrated MF density profile, $f(x)=\int_0^x 4 \pi y^2 \r(y) dy$ 