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

TL;DR

This paper clarifies the correct scaling law for the dynamics of homogeneous states in one-dimensional long-range interacting systems, emphasizing the quadratic dependence on particle number and providing a validated kinetic equation.

Contribution

It identifies the proper scaling law for these systems and derives a corresponding kinetic equation, correcting previous misconceptions in the literature.

Findings

Previous scalings were due to finite-size effects or improper variables.

The correct scaling is proportional to the square of the number of particles.

Numerical evidence supports the derived kinetic equation for specific models.

Abstract

Quasi-Stationary States of long-range interacting systems have been studied at length over the last fifteen 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 previous scalings described in the literature are due either to small size effects or the use of improper variables to describe the dynamics. The correct scaling is proportional to the square of the number of particles and deduce the kinetic equation valid for the homogeneous regime and numerical evidence is given for the Hamiltonian Mean Field and ring models.