  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.
