Speaker
Description
The dynamical accessibility of a quasistationary state from the infinitely many formal equilibria of the Vlasov-Poisson system is addressed. We give a first-principles asymptotic selection theory for Bernstein-Greene-Kruskal (BGK) holes produced by two-stream relaxation and cold gravitational clumps produced by collisionless collapse. The analytic construction rests on two central ideas. First, repeated shell crossing winds phase space into caustic whorls, leading to particle bunching and phase-space granulation. Nonlinear phase mixing erases the angle information but preserves the action dependence of the distribution function (DF). Following Berry and O'Dell's work on ergodic momentum density generated by caustic whorls, and Jarzynski's least-biased information-theoretic interpretation of Berry's random-wave ensemble, we show that the resulting coarse-grained DF is a circus-tent (CT) DF, now made self-consistent for Vlasov-Poisson dynamics. Second, the selected potential is constrained by the Painleve property (PP) of Poisson's equation written in Sagdeev form. PP implies that the once-integrated Poisson equation must belong to an algebraic class reducible to Riccati or Weierstrass form up to suitable variable transformations, which must be determined by the underlying physical processes. In the plasma problem this leads to the Weierstrass form, with coefficients satisfying virial identities. In the gravity problem, the regular bulk stays elliptic, while the strict cold cusp is the root-collision degeneration of the regular elliptic curve. Extensive numerical diagnostics test analytical predictions rather than fit arbitrary profiles. In the plasma case, we validate the theoretical BGK hole depth and shape, and how closely the measured Sagdeev coefficients satisfy the virial identities. In the gravity case, the Colombi-Touma cusp branch, its coefficient scale, and its regularized core follow from the measured action distribution and Poisson closure. The cold center is resolved as a physical core, the thermal de-singularization of the cold cusp rather than numerical grid rounding, while the CT controls the bulk outside it. The adiabatic theory excludes the regions where action-angle variables fail: the O-point caustic sheet in gravity and the X-point separatrix sheet in BGK. These layers mark the boundary between regular coherent self-organization and the phase-space-turbulence problem, which we leave for subsequent work.