Painleve property: from dynamical accessibility of phase-space holes and gravity cusps in 1D Vlasov-Poisson dynamics to stellarator optimization for quasisymmetry
Abstract: We address the dynamical accessibility of a quasistationary state from the infinitely many formal equilibria of the Vlasov-Poisson system. 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 based on two central ideas. First, repeated shell crossing leads to particle bunching and phase-space granulation through caustic formations. 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 caustic whorls, and Jarzynski's least-biased information-theoretic interpretation, we show that the resulting coarse-grained DF is a circus-tent DF, 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. The adiabatic theory describes the regular coherent self-organized state in both BGK and gravity. The excluded regions where action-angle variables fail: the O-point caustic sheet in gravity and the X-point separatrix sheet in BGK, require a phase-space-turbulence analysis, which we leave for subsequent work. Finally, we show that PP is also directly relevant to a very different area of plasma physics: the neoclassical optimization of stellarators. PP enables the construction of a reduced phase space for quasisymmetry where the well known quasisymmetric configurations such as the Landreman-Paul are constrained to live. Analytical predictions are thoroughly benchmarked against existing configurations such as those from the QUASR database.