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Implementation of temperature in micromagnetic simulations
The usual way to simulate the evolution of a magnetic system is the cellwise numerical integration of
the Landau-Lifshitz differential equation (LL)
where Heff is an effective field, which represents all forces acting on the magnetic moment. A popular method to take temperature into account is adding a highly irregular fluctuating field hfluct representing the irregular influence of temperature. The ordinary LL then becomes the stochastic Landau-Lifshitz differential equation (SLL).
The most important parameters to be specified now, are the first two moments (mean and variance) of the fluctuating field hfluct. The first moment <hfluct> of course must be zero, since we shall not allow temperature to drive our system in a specific direction. Also it can be assumed, that the values of hfluct follow a gaussian distribution because they arise from numerous interactions with the underlying crystal lattice. So for temperatures not too low the central limit theorem can be applied. Assuming a Boltzmann distribution in thermal equilibrium, the variance can be calculated to
For most systems hfluct can be looked upon as white noise with no correlation in time, space or axis direction. This will no longer hold for extremely low temperatures and/or extraordinary strong fields Heff. For temperatures below 1K this model surely becomes questionable. Another important point is that stochastic functions are to be multiplied by not like normal functions, when integrated numerically.
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