Scaling of the dynamics of homogeneous states of one-dimensional long-range interacting systems.

Quasistationary states of long-range interacting systems have been studied at length over the last 15 years. It is known that the collisional terms of the Balescu-Lenard and Landau equations vanish for one-dimensional systems in homogeneous states, thus requiring a new kinetic equation with a proper dependence on the number of particles. Here we show that the scalings discussed in the literature are mainly due either to small size effects or the use of unsuitable variables to describe the dynamics. The scaling obtained from both simulations and theoretical considerations is proportional to the square of the number of particles, and a general form for the kinetic equation valid for the homogeneous regime is obtained. Numerical evidence is given for the Hamiltonian mean field and ring models, and a kinetic equation valid for the homogeneous state is obtained for the former system.