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## Implementation of temperature in micromagnetic simulations

### Micromagnetic theory

The usual way to simulate the evolution of a magnetic system is the cellwise numerical integration of
the Landau-Lifshitz differential equation (LL)

where *H _{eff}* 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

*h*representing the irregular influence of temperature. The ordinary LL then becomes the stochastic Landau-Lifshitz differential equation (SLL).

_{fluct}The most important parameters to be specified now, are the first two moments (mean and variance) of the fluctuating field

*h*. The first moment <

_{fluct}*h*> 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

_{fluct}*h*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

_{fluct}For most systems

*h*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

_{fluct}*H*. 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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