Thermal vortex dynamics in thin circular ferromagnetic nanodisks
  The dynamics of gyrotropic vortex motion in a thin circular nanodisk of soft ferromagnetic material is considered. The demagnetization field is calculated using two-dimensional Green's functions for the thin film problem and fast Fourier transforms. At zero temperature, the dynamics of the Landau-Lifshitz-Gilbert equation is simulated using fourth order Runge-Kutta integration. Pure vortex initial conditions at a desired position are obtained with a Lagrange multipliers constraint. These methods give accurate estimates of the vortex restoring force constant $k_F$ and gyrotropic frequency, showing that the vortex core motion is described by the Thiele equation to very high precision. At finite temperature, the second order Heun algorithm is applied to the Langevin dynamical equation with thermal noise and damping. A spontaneous gyrotropic motion takes place without the application of an external magnetic field, driven only by thermal fluctuations. The statistics of the vortex radial position and rotational velocity are described with Boltzmann distributions determined by $k_F$ and by a vortex gyrotropic mass $m_G=G^2/k_F$, respectively, where $G$ is the vortex gyrovector. 
G. M. Wysin and W. Figueiredo
cond-mat.stat-mech