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
$N\to\infty$ limit. However, a faster and 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 survives 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.
We propose an approximation scheme to describe collisionless relaxation, based
on the introduction of moments of the distribution function, and apply it to
the Hamiltonian Mean Field (HMF) model. To the leading order, virial
oscillations are equivalent to the motion of a particle in a one-dimensional
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; generic initial conditions will be considered in a forthcoming
paper.
