Once again, we start with the Landau-Lifshitz-Gilbert (LLG) equation
where is an unit vector describing the orientation of the magnetic moment, is time, is the gyromagnetic ratio, is the effective field and is a damping constant. Landau added the second term in order that the magnetization moment would come into equilibrium (e.g. and therefore ) eventually. The equation has been derived in multiple theories for particular cases. In particular,  recovered LLG from a quantum macrospin in the limit of constant coupling and uniform density of states of the thermal bath.
For small magnetic particles, Brown  derived the required stochastic field from the Fokker-Planck equation. The resulting stochastic field, , is a Gaussian with mean of zero and variance, in each dimension where is temperature, is the saturation magnetization and is the volume of the magnetic grain.
The stochastic field will cause a diffusion of from its equilibrium. Then, since the precession term will introduce a rotational flux of . Ignoring this rotational flux, we can define a diffusion coefficient of the magnetization as
where the last equality is calculated by replacing with Brown’s stochastic fields in the LLG and integrating over all . Reasonable values would give rad/s. Roughly speaking this would mean in 1ns would have diffused ~ – from equilibrium. This does not include the additional processional term which also depend on this angle.
Instead of diffusion of magnetization, one could look at diffusion of free energy, , of the magnetic moment. In this case,
and one can show Einstein’s relation holds, , where is advective flux due to in the absence of stochastic fields, namely
From above, any analysis should include since it’s a consequence of damping (). We will show its effect during conventional magnetic recording.
For conventional writing, as discussed previously, the applied write field goes as such that is a slowly varying function of time (i.e. medium motion). On the other hand, is quickly varying in ~ 0.1ns with a peak large enough to eliminate the free energy barrier (e.g ). Therefore, one would expect the diffusive flux (e.g. ) would lower required .
We first confirm does, in fact, result in a Boltzmann distribution of the orientation of .
For micro-magnetic simulations, we have identical square prisms (extruded 10nm) grains of volume 503nm packed in a periodic square lattice with kOe, () is normal to media plane, emu/cc and .
For simplicity, we take 65536 grains, K and exclude applied and magnetostatic fields. We start with all the grains along and integrate in time until equilibrium is achieved.
We see not only is the standard deviation of is as expected at equilibrium but the time evolution to equilibrium is also correct. Further, the distribution of as a function of energy is, in fact, a Boltzmann distribution (thereby ensuring the diffusive and drift flux will exactly balance). Of course, without , the variance would be identically zero, the distribution function would be a delta function at zero energy and the phase space plot would show all 65,536 grain orientations at the north pole (i.e. ).
We next use 1024 identical square prism grains with full magnetostatic fields and apply an uniform field of linearly increasing from zero to in 0.1ns. For kOe9.6 kOe () no grains should flip. We confirm this by running with :
We use the same identical set up, but now :
We also note that in the absence of , kOe was sufficient to flip all 1024 grains. However, since breaks the symmetry, one requires ~ kOe to switch all grains in 1ns. Therefore we see does lower for both first grain flip (~ 7kOe) and all grains flip.
For completeness, we repeat the previous calculations but we ramp in 0.1ns (instead of 5ns) and include :
|microstructure||magnetostatic field||initial orientation|| to flip first (kOe)|
| to flip all (kOe)|
|1024 grains, square prism, h=10nm, any packing fraction||none||any||13.9|
|1024 grains, square prism, h=10nm, any packing fraction||self demagnetization||any||14.4|
|1024 grains, square prism, h=10nm, 100% packing||full||up||9.5|
|1024 grains, square prism, h=10nm, 100% packing||full||random||11.4|
|849 grains, square prism, h=10nm, 83% packing||full||up||9.5|
|849 grains, square prism, h=10nm, 83% packing||full||random||11.6|
|481 grains, nm, nm, 83% packing||none||any||13.9|
|481 grains, nm, nm, 83% packing||self demagnetization||any||14.1|
|481 grains, nm, nm, 83% packing||full||up||9.8|
|481 grains, nm, nm, 83% packing||full||random||11.7|
The one clear conclusion we see is that the ‘switching field distribution’ (the difference between for switching all grains to switching one grain) increases when . This is usually due to the reduction of needed to switch the first grain. When , the results above depend on the time that is applied. For longer times (0.5ns used here), would decrease.
This, once again, shows the importance of including the full magnetostatic and the stochastic fields during conventional writing. Be aware that many simulations reported in the literature don’t included either one or both (i.e. “effective write field” for switching)- they will lead you to wrong conclusions if you’re trying to design a new storage system. A. Rebei and G.J. Parker, Fluctuations and dissipation of coherent magnetization, Phys. Rev. B, 67, 104434, (2003).  W.F. Brown, Thermal fluctuations of a single-domain particle, Phys. Rev., 130, 1677 (1963).