  The influence of the finite number N of particles coupled to a monochromatic
wave in a collisionless plasma is investigated. For growth as well as damping
of the wave, discrete particle numerical simulations show an N-dependent long
time behavior resulting from the dynamics of individual particles. This
behavior differs from the one due to the numerical errors incurred by Vlasov
approaches. Trapping oscillations are crucial to long time dynamics, as the
wave oscillations are controlled by the particle distribution inhomogeneities
and the pulsating separatrix crossings drive the relaxation towards thermal
equilibrium.
