Statistical Ensemble Modelling of Dynamic Hysteresis Loops in Single-Domain and Non-Interacting Magnetic Nanoparticles by Using a Double-Well Rate Equation Approach

Magnetic hyperthermia relies on the ability of magnetic nanoparticles (MNPs) to dissipate heat under alternating magnetic fields, with the heating efficiency commonly quantified through the specific loss power (SLP). Accurate estimation of SLP requires realistic modeling of the dynamic magnetic response of nanoparticle ensembles, particularly in the ferromagnetic single-domain regime where hysteresis losses dominate. In the present work, we developed a computational framework in Mathematica based on the thermally activated Stoner–Wohlfarth model to simulate dynamic hysteresis loops and estimate SLP in ensembles of non-interacting magnetic nanoparticles. The model incorporates experimentally relevant distributions of particle diameter, magnetic anisotropy, and easy-axis orientation, enabling realistic representation of nanoparticle polydispersity and orientation disorder. Thermal activation was introduced through Arrhenius-type Néel switching probabilities, while the dynamic magnetization evolution was obtained numerically through solution of the corresponding rate equations. The framework was tested for magnetite nanoparticles, one of the most widely used materials in magnetic hyperthermia, considering typical single-domain ferromagnetic particle sizes in the range of 15–30 nm and effective anisotropy Keff values representative, 3 kJ/m3 < Keff < 20 kJ/m3 of experimentally reported systems. Simulations were performed under clinically relevant alternating magnetic fields with amplitudes up to 24 kA/m and frequencies ranging from 100 to 765 kHz. The model successfully reproduced dynamic hysteresis loop evolution and enabled systematic investigation of the influence of nanoparticle size, anisotropy, and orientation distributions on loop shape, symmetry, and SLP. The developed code provides a computationally accessible tool for researchers working in magnetic hyperthermia, allowing direct connection between microscopic nanoparticle properties and macroscopic heating performance. By enabling parametric mapping of dynamic hysteresis behavior and SLP dependence, the framework may support the rational optimization of magnetic nanoparticle systems for biomedical hyperthermia applications.