calculate the entropy and the pressure.

The MF phase diagram of the system is presented in Fig. \[fig\_mf\]. The collapse and explosion energies are $\e_{coll}\approx -0.339$ and $\e_{expl}\approx 0.267$. Examples of the uniform and the core-halo density profiles for $\e=\e_{coll}$ are shown in Fig. \[fig\_rho\].

In the MD simulations we consider a system consisting of $N=125$ – 500 particles in a spherical container of the radius $R=1$. All interparticle forces are calculated directly at each time step $dt$. This is done to avoid any mean-field-like effects inevitably present in any truncated multipole or Fourier potential expansion. The particle velocities and coordinates are updated according to the velocity-Verlet algorithm which is symplectic and reversible.

The system is initialized by randomly distributing particles according to a spherically symmetric density profile; typically the appropriate MF density profile $\r(x)$ was used. The potential energy ($U$) of the initial configuration is calculated, and the target kinetic energy $E_k=E-U$ is determined. The particle velocities are randomly generated from some (usually Maxwell) distribution with the appropriate square average. Finally, the deviation of the total energy from its target value, caused by a stochasticity in velocity assignment, is determined, and the velocities are rescaled to fine-tune the total 