Jeans instability of collisionless systems

Linear dispersion relations are solved numerically for collisionless self-gravitating systems. The results are compared to those of hydrodynamic approach. Both theories yield similar dispersion relations, with predictions of instability when the system is cold enough. In a collisionless system, however, the perturbation is found to die away without oscillations when the system is stable. A mixed system of hot-component particles and cold-component particles is also studied. The stability of such system is dominated by the temperature of the cold particles. Again, the oscillatory behaviour is not found in this case, regardless of the stability of the system.