An analytical framework is presented for determining the ground states of a collisionless plasma with a given density profile, and its associated available energy, which bounds the field energy, is calculated. We show that the bound can be tightened by enforcing that the ground state is physically realisable, i. e. that the released energy is consistently stored in the fields supported by the density profile. A simple waterbag model is employed to retrieve nonlinear bounds for particularly simple conditions, further finding a phase-transition-like behaviour at the critical energy where the constraint of non-negative number densities becomes relevant. We verify the derived bounds with one-dimensional particle-in-cell simulations, where it is found that the true energy released is typically tilde 20 percent sign ∼ 20 % 20\, \% of the bound derived. Next, we present an asymptotic framework for calculating ground states with given density profiles of distribution functions close to this ground state. This framework is employed to describe a magnetised plasma, where it is shown that the ground state seeks long-wavelength structures for large electron-to-ion temperature ratios, highlighting unfavourable transport properties in this regime. By using the corrected adiabatic response in toroidal geometry (with a flux-surface average subtracted), it is found that the ground state seeks long-wavelength zonal structures. Furthermore, a crude model of a toroidal, multispecies, quasineutral plasma highlights that there is non-trivial dependence on the impurity content with certain optimal mixing ratios. A similar model is constructed to capture the effects of a fast-ion population, showing a non-trivial dependence on the exact shape of the fast-ion distribution. These results provide rigorous, density-aware limits on energy release in collisionless plasmas, and similar methods may be used to account for electromagnetic effects.
Mackenbach et al. (Mon,) studied this question.