Field-Driven Coupled Magnon--Phonon--Electron Relaxation in Magnetite Using Steepest-Entropy-Ascent Quantum Thermodynamic Formalism

A field-driven steepest-entropy-ascent quantum thermodynamic (SEAQT) formulation is developed for longitudinal nonequilibrium relaxation in magnetite (Fe$_3$O$_4$) with coupled electron, phonon, and magnon populations. Material-specific excitation spectra define the thermodynamic state space, while one relaxation parameter for each population sets its kinetic scale. A longitudinal magnetic field shifts the dressed magnon eigenenergies while the occupation basis remains fixed; irreversible redistribution conserves instantaneous energy and electron number while allowing the magnon population to vary. The formulation yields nonequilibrium subsystem temperatures, entropy production, magnetic-work identities, and a coupled small-signal susceptibility incorporating energy-conservation feedback among all three populations; the one-pole Debye response appears only as a limiting case. Numerical results under sinusoidal driving show a transition from nearly quasistatic behavior to frequency-dependent lag, finite-amplitude departure from the linear-response ellipse, and increasing higher-harmonic content. Relaxational work per cycle increases strongly with field amplitude and frequency, while the complex susceptibility is broader and shifted relative to a Debye reference. Entropy production remains positive, and the electron, phonon, and magnon temperatures show distinct excursions followed by secular heating when positive magnetic work is retained without heat rejection. The calculated work represents longitudinal magnon quasiparticle relaxation in a homogeneous single-domain model, not the total core loss of a finite ferrite specimen.