Energies vs. time for N = 100 particles on a 256³ grid. The kinetic, potential, and total energy for the self-consistent interacting system of the particles from the PM code are shown as solid curves (red, purple, and blue). The kinetic, potential, and total energy from the ODE system of the particles in the external potential are shown as dashed curves (orange, violet, and teal). The fluctuations in the total energies are purely numerical, and the other energies have thermal fluctuations. The dashed gray curve is E_pot = ∫ _ρBϕdxdydz with the PM matter density and Eq. (26), hence substituting a finite N solution in the potential obtained in the thermodynamic limit. We subtracted the self-energies.