This project seeks to provide a fundamental understanding of the process of energetic particle redistribution from the perspective of thermodynamics and entropy.
Initial studies would focus on characterisation entropy changes during redistribution in a reduced 1-dimensional “bump-on-tail” code, in both the absence and presence of collisions. Is entropy maximised? Armed with this information we seek to develop a reduced model to compute final wave saturation levels. One trial approach is a variational principle in which the system entropy is maximised subject to constant energy, and resonant slices of the distribution function (a phase space island) are coarse-grained into “waterbag” functions with a set of weights to describe the magnitude of the distribution function. The candidate model is a phase space analogue of multi-region relaxed MHD (MRxMHD) with a different variational principle.
The problem of determining the water-bag weights is however non trivial, and in general involves minimise a “distance” between any given distribution function belonging to some functional spaces and the water-bag distribution function under some appropriate constraints. This kind of moment problem under constraints (or coarse graining) can be recast as a general nonlinear optimisation problem with constraints. The choice of constraints reflects the problem under study. In gyrokinetic turbulence studies the focus is typically on micro instabilities such as ion or electron temperature gradient modes.
The problem we pose is rich in mathematical complexity. As with MRxMD, we expect the formulation to reduce to evolution equations for the phase space interfaces and volumes, and solve for the weights. The problem will be cast in toroidally symmetric (tokamak) geometry, for which flux surface coordinates can always be constructed. We will explore the optimal choice of phase space discretisation in which to solve the problem.