High temperature laboratory and rarefied space plasmas can be considered in many situations as collisionless and their dynamics as
Hamiltonian. The capability of generating, during the time evolution, smaller and smaller scales in the velocity space is a characteristic of the majority of the processes developing in such plasmas. Since the non collisional dynamics, especially during the non linear regime, is often driven by kinetic effects, to describe theoretically its evolution a kinetic approach is necessary. The analytical model used for the study of small scales non collisional dynamics is based on the Vlasov equation coupled, in the electrostatic limit, to the Poisson equation. However, this is a non linear system of equations usually very difficult to solve analytically. For this reason, in this context numerical simulations are one of the most effective tools of investigation. However, the necessity of a numerical approach to discretize the equations on a numerical mesh introduces a strong limit. In fact, in Vlasov simulations, when a process develops scales so small to be comparable to the size of the numerical mesh, the algorithm becomes necessarily unable to describe correctly the plasma dynamics, with the consequent effect of introducing in the system numerical dissipation and dispersion and consequently of breaking the Hamiltonian character of the system. Here we have studied the problem of the role of small scales on the evolution of the non linear, collisionless plasma dynamics by considering several specific physical phenomena of basic interest in plasma physics described by the Vlasov Poisson mode
