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The `Xf_ThermHeunEvolve` OOMMF extension is a combination of the `Oxs_RungeKuttaEvolve` evolver that comes default with OOMMF, and the thermal fluctuation model that is in the `UHH_ThetaEvolve` evolver. The `Xf_ThermHeunEvolve` extension was created to allow the simulation of thermal fluctuation effects on a magnetic ensemble using Heun’s method, which was shown to be more stable in [1].

The thermal fluctuations may be modeled as an effective field derived by Brown [2]:

According to the OOMMF user guide:

has a default value of . is the saturation magnetization in , and is the permeability of free space with a value of .

The Specify block for the Xf_ThermHeunEvolve class has the form

Most of these options appear also in the `Oxs_EulerEvolve` class. The repeats have the same meaning as in that class, and the same default values except for `relative_step_error` and `error_rate`, which for `Xf_ThermHeunEvolve` have the default values of 0.01 and -1.0, respectively. Additionally, the ` alpha`,

`and`

**gamma_LL**`options may be initialized using scalar field objects, to allow these material parameters to vary spatially.`

**gamma_G**Simulations with thermal fluctuations are enabled using the parameters ** temperature** and

**. The parameter**

`tempscript``temperature`sets the temperature of thermal fluctuations in units of Kelvin. Heating effects may be simulated using the

`tempscript`, which behaves similarly to

`Jprofile`from

`Oxs_SpinXferEvolve`. The Tcl procedure defined by

`tempscript`should return a scalar and is used to give a time varying profile to

`temperature`. Only spatially uniform temperature is allowed in the simulation, and

`temperature`should be given as a scalar constant. The inputs given to the Tcl procedure defined in

`tempscript`may be from the list of arguments specified in

`. The list must be from the following:`

**tempscript_args**`stage`,

`stage_time`,

`total_time`.

The parameter ** uniform_seed** sets up the random number generator used to generate the thermal fluctuation field for thermal simulations. (Developmental note: thermal simulations seem to have random errors if this is not set during simulations with non-zero temperature.)

The ** allow_signed_gamma** parameter is for simulation testing purposes, and is intended for advanced use only. There is some lack of consistency in the literature with respect to the sign of . For this reason the Landau-Lifshitz-Gilbert equations are presented above using the absolute value of . This is the interpretation used if

**is 0 (the default). If instead**

`allow_signed_gamma`**is set to 1, then the Landau-Lifshitz-Gilbert equations are interpreted without the absolute values and with a sign change on the terms, i.e., the default value for in this case is (units are ). In this setting, if is set positive then the spins will precess backwards about the effective field, and the damping term will force the spins away from the effective field and increase the total energy. If you are experimenting with , you should either set to force spins back towards the effective field, or disable the energy precision control (discussed below).**

`allow_signed_gamma`The two controls ** min_step_headroom** (default value 0.33) and

**(default value 0.95) replace the single**

`max_step_headroom``step_headroom`option in

`Oxs_EulerEvolve`. The effective

`step_headroom`is automatically adjusted by the evolver between the

`min_headroom`and

`max_headroom`limits to make the observed reject proportion approach the

**(default value 0.05).**

`reject_goal`The ** method** entry selects a particular Runge-Kutta implementation. It should be set to one of

`rk2`,

`rk2heun`,

`rk4`,

`rkf54`,

`rkf54m`, or

`rkf54s`; the default value is

`rkf54`. The

`rk2`and

`rk4`methods implement canonical second and fourth global order Runge-Kutta methods [1], respectively. For

`rk2`, stepsize control is managed by comparing at the middle and final points of the interval, similar to what is done for stepsize control for the

`Oxs_EulerEvolve`class. One step of the

`rk2`method involves 2 evaluations of . The

`rk2heun`method implements Heun’s method and is essentially a modified version of the forward Euler method. Heun’s method calculates at the initial and the predict step, and uses the average of the two as the actual for calculating the next step.

In the `rk4` method, two successive steps are taken at half the nominal step size, and the difference between that end point and that obtained with one full size step are compared. The error is estimated at 1/15th the maximum difference between these two states. One step of the `rk4` method involves 11 evaluations of , but the end result is that of the 2 half-sized steps.

The remaining methods, `rkf54`, `rkf54m`, and `rkf54s`, are closely related Runge-Kutta-Fehlberg methods derived by Dormand and Prince [2, 3]. In the nomenclature of these papers, `rkf54` implements RK5(4)7FC, `rkf54m` implements RK5(4)7FM, and `rkf54s` implements RK5(4)7FS. All are 5th global order with an embedded 4th order method for stepsize control. Each step of these methods requires 6 evaluations of if the step is accepted, 7 if rejected. The difference between the methods involves tradeoffs between stability and error minimization. The RK5(4)7FS method has the best stability, RK5(4)7FM the smallest error, and RK5(4)7FC represents a compromise between the two. The default method used by `Oxs_RungeKuttaEvolve` is RK5(4)7FC.

The remaining undiscussed entry in the `Xf_ThermSpinXferEvolve` Specify block is ** energy_precision**. This should be set to an estimate of the expected relative accuracy of the energy calculation. After accounting for any change in the total energy arising from time-varying applied fields, the energy remainder should decrease from one step of the LLG ODE to the next.

`Xf_ThermSpinXferEvolve`will reject a step if the energy remainder is found to increase by more than that allowed by

`eprecision`. The default value for

`eprecision`is 1e-10. This control may be disabled by setting

`eprecision`to -1.

The `Xf_ThermHeunEvolve` module provides the same five scalar outputs and three vector outputs as `Oxs_RungeKuttaEvolve`, plus the scalar output “Temperature.”

[1] W. Scholz et al., “Micromagnetic simulation of thermally activated switching in fine particles,” Thesis, 1999

[2] W. F. Brown, “Thermal fluctuations of a single-domain particle,” Physical Review vol. 130, no. 5, pp. 1677-1686, Jun. 1963.

Is it OK to realize the thermal fluctuation field by an energy class, not evolver.

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Yes and no. The issue is that all evolvers pick up energy classes. However, it does not make sense for the CGEvolve evolver to pick up energy classes such as Xf_STT or thermal fluctuation. CGEvolve is optimized for finding energy minimum near a initial point. A thermal fluctuation field will constantly change the minimum and hence, CGEvolve will not be guaranteed to reach convergence. On the other hand, the time-based evolvers such as Oxs_SpinXferEvolve are not looking for the energy minimum. So it is fine for things like thermal fluctuation and spin-transfer torque to be captured as an energy class. However, the energy calculated by the time evolver is not guaranteed to be correct in this case.

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Thank you. I saw you implement the thermal fluctuation within RungeKutta Time evolver. Is it equivalent to do it in an energy class? What I wanna do is similar to thermal fluctuation, but I don’t wanna modify the evolver class. It’s kind of difficult.

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