A Unified Hamiltonian Formulation for Energy Loss, Entropy Evolution, and Fusion Performance in Plasmas

We develop a comprehensive Hamiltonian formulation for plasma dynamics that unifies collisionless gyrokinetic and collisional processes. Our framework rigorously describes the evolution of free energy and entropy during the transition from Maxwellian to non-Maxwellian distributions, explicitly coupling microscopic turbulent processes with macroscopic measures of energy confinement and fusion performance. Unlike standard gyrokinetic treatments that treat collisions as a minor perturbation, our approach incorporates a collision operator directly into the Hamiltonian structure, thereby accounting for irreversible dissipation and entropy production. We derive quantitative relations linking turbulence intensity, entropy production, energy confinement time and fusion yield. Our work builds on recent energetic bounds and optimal mode analyses by Helander and Plunk and on Zhdankin's generalised entropy-production framework, thus providing a bridge between microscopic kinetics and reactor-scale performance.