Linear Theory and violent relaxation in long-range systems: a test case

Overview & Synopsis

In this article, several aspects of the dynamics of a toy model for long-range Hamiltonian systems are tackled focusing on linearly unstable unmagnetized (i.e. force-free) cold equilibria states of the Hamiltonian Mean Field (HMF). For special cases, exact finite-$N$ linear growth rates have been exhibited, including, in some spatially inhomogeneous case, finite-N corrections. A random matrix approach is then proposed to estimate the finite-N growth rate for some random initial states. Within the continuous, infinite N, approach, the growth rates are finally derived without restricting to spatially homogeneous cases. Then, these linear results are used to discuss the large-time nonlinear evolution. A simple criterion is proposed to measure the ability of the system to undergo a violent relaxation that transports the mean field modulus in the vicinity of its equilibrium value within some linear e-folding times.