Dynamical properties of cluster in the Hamiltonian Mean Field model

We investigate some properties of the trapping/untrapping mechanism of a single particle into/outside the cluster in the Hamiltonian Mean Field model. Particle are clustered in the ordered low-energy phase in this model. However, when the number of particles is finite, some particles can acquire a high energy and leave the cluster. Hence, below the critical energy, the fully-clustered and excited states appear by turns. First, we show that the numerically computed time-averaged trapping ratio agrees with that obtained by a statistical average performed for the Boltzmann-Gibbs stationary solution of the Vlasov equation. Second, we found numerically that the probability distribution of the lifetime of the fully-clustered state is not exponential but follows instead a power law. This means that the excitation of a particle from the cluster is not a Poisson process and might be controlled by some type of collective motion with long memory. Therefore, although an average trapping ratio exists, there appear to be no typical trapping ratio in the probabilistic sense. Finally, we discuss the dynamical mechanism using a modified model.