The real and apparent convergence of N-body simulations of the dark matter structures: is the Navarro-Frenk-White profile real?

We consider the reasons why a cuspy NFW-like profile persistently occurs in N-body simulations, in contradiction to some astronomical observations. The routine method of testing the convergence of N-body simulations (in particular, the negligibility of two-body scattering effect) is to find the conditions under which the shape of the formed structures is insensitive to numerical parameters. The results obtained with this approach suggest a surprisingly minor role of the particle collisions: the central density profile remains untouched and close to NFW, even if the simulation time significantly exceeds the collisional relaxation time $\tau_r$. We analyze the test body distribution in the halo center with help of the Fokker-Planck equation. It turns out that the Fokker-Planck diffusion transforms any reasonable initial distribution into NFW-like profile $\rho\propto r^{-1}$ in a time shorter than $\tau_r$. On the contrary, profile $\rho\propto r^{-1}$ should survive much longer, being a sort of attractor: the Fokker-Planck diffusion is self-compensated in this case. Thus the test body scattering may create a stable NFW-like pseudosolution that can be mixed up with the real convergence. This fact might help to eliminate the well-known 'cusp vs. core' problem.