Predictive theory of collisionless violent relaxation
In plain words
A self-gravitating cluster or a plasma that is suddenly disturbed settles quickly into a state that is not thermal. The classic Lynden-Bell theory predicts that state from statistics but often fails, and when and why it fails is not settled.
Precise statement
For Vlasov dynamics of long-range systems (Hamiltonian mean-field model, 1D self-gravitating sheets, 3D gravitating N-body systems as $N\to\infty$), determine the conditions on initial data (virial ratio, waterbag levels) under which Lynden-Bell's maximum-entropy prediction of the quasi-stationary state is accurate, and give a predictive theory of the core-halo states produced when mixing is incomplete.
What would settle it
A theory that predicts the quasi-stationary distribution function from initial data, tested against N-body simulations over a range of initial conditions.
Status in the literature
Unverified note
Lynden-Bell theory works for virialized initial states and parametric-resonance core-halo theory describes strongly non-virialized ones; a 2025 theory replaces global ergodicity by local mixing and predicts HMF quasi-stationary states with no fitted parameters (Teles, Pakter, Levin, arXiv:2501.15593).