Realization of the thermal equilibrium in inhomogeneous magnetic systems by the Landau-Lifshitz-Gilbert equation with stochastic noise, and its dynamical aspects

TL;DR

This paper investigates how the stochastic Landau-Lifshitz-Gilbert equation models thermal equilibrium in inhomogeneous magnetic systems, demonstrating the importance of the fluctuation-dissipation relation for accurate equilibrium and analyzing different relaxation dynamics.

Contribution

It systematically shows the conditions for the stochastic LLG equation to reproduce true thermal equilibrium in inhomogeneous magnetic systems, including the role of the fluctuation-dissipation relation.

Findings

Agreement between stochastic LLG steady state and Monte Carlo equilibrium.

Violations of the fluctuation-dissipation condition lead to deviations from true equilibrium.

Different parameter choices affect relaxation processes but not the equilibrium state.

Abstract

It is crucially important to investigate effects of temperature on magnetic properties such as critical phenomena, nucleation, pinning, domain wall motion, coercivity, etc. The Landau-Lifshitz-Gilbert (LLG) equation has been applied extensively to study dynamics of magnetic properties. Approaches of Langevin noises have been developed to introduce the temperature effect into the LLG equation. To have the thermal equilibrium state (canonical distribution) as the steady state, the system parameters must satisfy some condition known as the fluctuation-dissipation relation. In inhomogeneous magnetic systems in which spin magnitudes are different at sites, the condition requires that the ratio between the amplitude of the random noise and the damping parameter depends on the magnitude of the magnetic moment at each site. Focused on inhomogeneous magnetic systems, we systematically showed agreement between the stationary state of the stochastic LLG equation and the corresponding equilibrium state obtained by Monte Carlo simulations in various magnetic systems including dipole-dipole interactions. We demonstrated how violations of the condition result in deviations from the true equilibrium state. We also studied the characteristic features of the dynamics depending on the choice of the parameter set. All the parameter sets satisfying the condition realize the same stationary state (equilibrium state). In contrast, different choices of parameter set cause seriously different relaxation processes. We show two relaxation types, i.e., magnetization reversals with uniform rotation and with nucleation.