##manager.scheduler.building##: Edificio San Jose
##manager.scheduler.room##: Auditorio 1
Date: 2019-07-11 03:15 PM – 03:30 PM
Last modified: 2019-06-10
Abstract
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 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 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. 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 at a coarse-grained level [1], introducing suitable moments of the distribution function, and apply it to the Hamiltonian Mean Field (HMF) model. To the leading order, collective oscillations are equivalent to the motion of a particle in a one-dimensional potential. Considering higher-order contributions in an effective way we derive a dissipative equation describing the damping of the oscillations, including a renormalization of the effective potential and yielding predictions in very good agreement with numerical simulations. We then show that such an approach can be seen as a particular instance of a general coarse-grained dynamics: we derive an effective equation that describes the collisionless dynamics at a given scale for a generic long-range-interacting system and discuss its predictions in some particular cases [2].
[1] G. Giachetti and L. Casetti, "Violent relaxation in the Hamiltonian Mean Field model: I. Cold collapse and effective dissipation", arXiv:1902.02436, JSTAT (2019, to appear)
[2] G. Giachetti, A. Santini, and L. Casetti, "Coarse-grained collisionless dynamics with long-range interactions", in preparation