Coloured Noise in Langevin Simulations of Superparamagnetic Nanoparticles

The coloured noise formalism has long formed an important generalisation of the white noise limit assumed in many Langevin equations. The Langevin equation most typically applied to magnetic systems, namely the Landau-Lifshitz-Gilbert (LLG) equation makes use of the white noise approximation. The correct extension of the LLG model to the coloured noise is the Landau-Lifshitz-Miyazaki-Seki pair of Langevin equations. This pair of Langevin equations correctly incorporates a correlated damping term into the equa- tion of motion, constituting a realisation of the Fluctuation-Dissipation theorem for the coloured noise in the magnetic system. We undertake numerical investigation of the properties of systems of noninteracting magnetic moments evolving under the LLMS model. In particular, we apply the model to superparamagnetic spins. We investigate the escape rate for such spins and find that departure from uncorrelated behaviour occurs as the system time approaches the bath correlation time, and we see that the relevant system time for the superparamagnetic par- ticles is the Larmor precession time at the bottom of the well, leading us to conclude that materials with higher magnetic anisotropy constitute better candidates for the exhibition of non-Markovian properties. We also model non-Markovian spin dynamics by modifying the commonly used dis- crete orientation approximation from a Markovian rate equation to a Generalised Master Equation (GME), where the interwell transition rates are promoted to memory kernels. This model makes the qualitative prediction of a frequency-dependent diamagnetic sus- ceptibility, as well as a biexponential decay profile of the magnetisation. The predictions of the GME are compared to the results of LLMS simulations, where we find a similar diamagnetic phase transition and biexponential behaviour.