Entropy functional as a variational principle

Establish whether the Shannon-like entropy functional S_m = -∑_i p_{m,i} log p_{m,i}, defined from the measure-weighted coefficient energy contributions p_{m,i} = |c_{m,i}|^2 · |Ω_{m,i}| / ∑_j |c_{m,j}|^2 · |Ω_{m,j}| in the hierarchical Voronoi refinement transform, can serve—under suitable physical constraints on the refinement multiplicity, dispersion, and rotation parameters—as a variational principle for simulating energy-minimizing dynamics in physical systems.

Background

In the subsection on entropy dynamics, the authors define a normalized energy distribution over refinement regions via p_{m,i} = (|c_{m,i}|^2 * |Ω{m,i}|) / (∑_j |c{m,j}|^2 * |Ω{m,j}|). They then introduce a Shannon-like entropy S_m = -∑_i p{m,i} log p_{m,i} and discuss its interpretation in terms of microstates and thermodynamic analogies.

They further suggest that this entropy-based functional might underpin a variational framework guiding energy-minimizing behavior, but the validity of this proposal is left as a conjecture requiring rigorous establishment under appropriate physical constraints on the transform’s refinement parameters.