Description
We develop an analytical framework to determine the ground states—distribution functions of minimal thermal and thus maximal field energy—of collisionless plasmas with prescribed density profiles, and derive upper bounds on the field energy. Enforcing physical realisability, e.g. that the ground state supports an electric field with the correct energetic content or is quasineutral, significantly sharpens the bound compared to the classical estimate of Gardner (1963, Phys. of Fluids). A simple waterbag model reveals a fivefold reduction relative to the classical bound (for equal-temperature species) and exhibits phase-transition-like behaviour once non-negative number densities become constraining. One-dimensional Particle-In-Cell simulations validate the theory, showing that the realised energy release is typically about 20% of the bound. Applying this density-constrained perspective to magnetised plasmas, we find that large electron-to-ion temperature ratios favour long-wavelength structures, suggesting unfavourable transport properties. Using this framework, we finally evaluate the available energy of curvature-driven ion-temperature-gradient turbulence, with adiabatic electrons meaning the density profile has to remain fixed, and the parallel dynamics is neglected. Conditions for vanishing available energy closely mirror gyrokinetic stability thresholds, including strong stabilisation when $0 \leq \mathrm{d} \ln T/ \mathrm{d} \ln n \leq 2/3$. Comparisons with large gyrokinetic databases for tokamaks and stellarators show that available energy correlates well with ion heat flux when parallel dynamics is weak, capturing variability driven by density and temperature gradients more reliably than that arising from geometry. Together, this framework establishes the importance of constraints from the field equations on the ground-state, significantly affecting nonlinear stability criteria and estimations of turbulent transport trends.