Abstract
The Gilbert fascinating nonlinear equation provides a fundamental evolution of the magnetization dynamics, but its evaluation for the determination of the magnetic relaxation of superparamagnetic nanoparticles systems with a biaxial anisotropy still presents a formidable computational challenge. Herein, we report using Gilbert's equation an infinite hierarchy of differential recurrence relations to determine the equilibrium transverse correlation functions susceptible to evaluate the magnetic relaxation of superparamagnetic nanoparticles with a biaxial anisotropy. By solving the so-called infinite hierarchy, within the framework of the linear response theory, an ensemble of exact expressions of the transverse magnetic susceptibility is obtained. This framework yields exact expressions in terms of matrix continued fractions. To demonstrate the utility of the present approach, our results are compared and contrasted with those obtained using the numerically exact solutions. Despite some quantitative differences, a decent agreement between the approximated results and the exact solutions is elucidated. The candidacy and the accuracy of the analytical equations determined on the basis of the effective eigenvalue method, for a variety of applications, are established. The findings reported herein show that the analytical results perform quite well and are capable of capturing the expected trends of the exact solutions. These results hold promise for realistic nanoscale systems and pave the way for an accurate simulation of the magnetic relaxation of superparamagnetic nanoparticles having biaxial anisotropy.