Violent relaxation in the Hamiltonian mean field model: I. Cold collapse and effective dissipation

In N-body systems with long-range interactions mean-field effects dominate over binary interactions (collisions), so that relaxation to thermal equilibrium occurs on time scales that grow with N, diverging in the N → ∞ limit. However, a much faster and completely non-collisional relaxation process, referred to as violent relaxation, sets in when starting from generic initial conditions: collective oscillations (referred to as virial oscillations) develop and damp out on timescales not depending on the system’s size. After the damping of such oscillations the system is found in a quasi-stationary state that may be very far from a thermal one, and that survives until the slow relaxation driven by two-body interactions becomes effective, that is, virtually forever when the system is very large. During violent relaxation the distribution function obeys the collisionless Boltzmann (or Vlasov) equation, that, being invariant under time reversal, does not ‘naturally’ describe a relaxation process. Indeed, the dynamics is moved to smaller and smaller scales in phase space as time goes on, so that observables that do not depend on small-scale details appear as relaxed after a short time. Here we propose an approximation scheme to describe the collisionless relaxation process, based on the introduction of suitable moments of the distribution function, and apply it to a simple toy model, the Hamiltonian mean field (HMF) model. To the leading order, virial oscillations are equivalent to the motion of a particle in a one-dimensional (1D) potential. Inserting higher-order contributions in an effective way, inspired by the Caldeira–Leggett model of quantum dissipation, we derive a dissipative equation describing the damping of the oscillations, including a renormalization of the effective potential and yielding predictions for collective properties of the system after the damping in very good agreement with numerical simulations. Here we restrict ourselves to ‘cold’ initial conditions, i.e. where the velocities of all the particles are set to zero: generic initial conditions will be considered in a forthcoming paper.