Jacobi stability of a many-body system with modified potential

The evolution of a system of mutually gravitating particles is considered taking into account the energy loss in collisions. Collisions can be described in various ways. One can use the theory of inelastic impact of solids with Newton's recovery coefficient for the relative velocity of bouncing particles. In numerical implementation, the main difficulty of this approach is to track and refine the huge number of time moments of particle collisions. Another approach is to supplement the gravitational potential with the potential of repulsive forces similar to the Lennard-Jones intermolecular forces. Numerical experiments show that, under the Jacobi stability condition, both models lead to a qualitatively identical evolution with the formation of stable configurations. For an infinite number of particles, the probability density function is determined by the system of Vlasov–Boltzmann–Poisson equations. Our proposed methodology corresponds to the use of the Vlasov kinetic equation with a potential of the Lennard–Jones type.