  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.
