Linear theory and violent relaxation in long-range systems: A test case

In this paper, 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, 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.