Energy-Casimir method for the dynamical systems with modified gravitational potential

Abstract

We discuss the evolution of an ensemble of mutually gravitating particles, which can repeatedly collide during their motion. The repulsive forces arise in the process of collision and contact interaction of the particles. This process can be described by two models. The first one is a classical model of rigid elastic or inelastic collision, using newtonian formula with a recovery coefficient. In the second model this interaction can be modeled by adding to the gravitational potential a potential of repulsive forces similar to Lennard-Jones intermolecular forces. For an infinite number of particles, the probability density function is determined by the Vlasov kinetic equation with a modified gravitational potential. This approach allows to replace the Vlasov-Boltzmann equation describing the evolution of a system with possible collisions by the Vlasov equation with a modified potential. It is useful for investigation of many-particles gravitating systems when taking under consideration the particle sizes. The 'Lennard-Jones type' potential improves the stability of computational process in comparison with newtonian gravitational potential. Using the energy-Casimir method, we justify the existence of nonlinearly stable steady states of the Vlasov equation with a modified gravitational potential.