  Precise modeling of the magnetization dynamics of nanoparticles with finite
size effects at fast varying temperatures is a computationally challenging
task. Based on the Landau-Lifshitz-Bloch (LLB) equation we derive a coarse
grained model for disordered ferrimagnets, which is both fast and accurate.
First, we incorporate stochastic fluctuations to the existing ferrimagnetic LLB
equation. Further, we derive a thermodynamic expression for the temperature
dependent susceptibilities, which is essential to model finite size effects.
Together with the zero field equilibrium magnetization the susceptibilities are
used in the stochastic ferrimagnetic LLB to simulate a $5\times10$ nm$^2$
ferrimagnetic GdFeCo particle with 70 % FeCo and 30 % Gd under various external
applied fields and heat pulses. The obtained trajectories agree well with those
of an atomistic model, which solves the stochastic Landau-Lifshitz-Gilbert
equation for each atom. Additionally, we derive an expression for the
intergrain exchange field which couple the ferromagnetic sublattices of a
ferrimagnet. A comparison of the magnetization dynamics obtained from this
simpler model with those of the ferrimagnetic LLB equation shows a perfect
agreement.
