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The dynamics of gyrotropic vortex motion in a thin circular nanodisk of softferromagnetic material is considered. The demagnetization field is calculatedusing two-dimensional Green's functions for the thin film problem and fastFourier transforms. At zero temperature, the dynamics of theLandau-Lifshitz-Gilbert equation is simulated using fourth order Runge-Kuttaintegration. Pure vortex initial conditions at a desired position are obtainedwith a Lagrange multipliers constraint. These methods give accurate estimatesof the vortex restoring force constant $k_F$ and gyrotropic frequency, showingthat the vortex core motion is described by the Thiele equation to very highprecision. At finite temperature, the second order Heun algorithm is applied tothe Langevin dynamical equation with thermal noise and damping. A spontaneousgyrotropic motion takes place without the application of an external magneticfield, driven only by thermal fluctuations. The statistics of the vortex radialposition and rotational velocity are described with Boltzmann distributionsdetermined by $k_F$ and by a vortex gyrotropic mass $m_G=G^2/k_F$,respectively, where $G$ is the vortex gyrovector.

Thermal vortex dynamics in thin circular ferromagnetic nanodisks