In this thesis, we begin by investigating a mean field hamiltonian model, exhibiting exactly solvable statistical properties, which in turn allow one to predict the asymptotical temporal behaviour of the dynamics. Starting from given initial configurations, we focus on the system's relaxation properties towards equilibrium states, and thoroughly probe the dependency of the timescales at stake here on the number of particles. The principal motivation is given by the fact that in spite of the model's simplicity, its phenomenology is reminiscent of much more complicated systems, hence providing us with a fantastic testing field for numerical and theoretical experimentations. We obtained several results tackling the interplay between the number of particles and the relaxation timescales, confirming the already existing numerical measurements as well as laying grounds for a novel approach for dealing with out-of-equilibrium states, based on a phase-space description. We then focus on the diffusion properties of heavy particles in tokamaks, motivated by the fact that the understanding of impurity diffusion is of paramount importance in fusion physics, since these can travel from the tokamak's edges towards its magnetic axis, causing a tremendous decrease in core temperature by absorbing the plasma's energy. We test the theory of stochastic diffusion during a sawtooth instability regime by following the movement of test particles, and show that the magnetic field lines' stochasticity, because of the resulting electric field, is a necessary condition to fulfill in order to reproduce the experimental results.
