Sensible noise

[ilateur]

The Idea

The idea is to expand the sensible time step calculation with thermal noise
$dt = f / \max{(|\tau|)}$
with $f$ a sensible factor. This slightly complicates things, because the thermal noise torque depends on the time step. Take $n$ as all thermal noise and prefactors except the time step $dt$, then
$\tau \rightarrow \tau + \frac{1}{\sqrt{dt}} n$
To deal with the maximum and norm, I used
$\max{\left( |\tau + \frac{1}{\sqrt{dt}} n | \right)} \quad \le \quad \max{(|\tau|)} + \frac{1}{\sqrt{dt}} \max{(|n|)}$
This becomes a second order equation to solve.

One bug I encountered was when noise dominates over torque, which can easily happen with minimized magnetization, that the sensible time step would become identically 0 due to numerical limitations. In this limit I used a different function.

Graphing Behavior

To check this visually, we can move to dimensionless parameters. We can define dimensionless temperature
$x = N^2 / (f \mathcal{T})$
with $N$ the max norm of noise (sqrt of temperature) and $\mathcal{T}$ the max norm of torque in the system. We can also define a dimensionless time step.
$y = dt * \mathcal{T} / f$
Keeping $\mathcal{T}$ and $f$ constant (in a skyrmion system) and varying temperature, we get the following time step estimates.

At $x=0$ we get the old sensible time step formula ($y=1$). Adding any noise decreases this time step, where eventually noise will dominate.
We can see that this behaves sensibly. Inspired by all cases of this figure, I added the tests.

Consistency

One thing that bothers me is that the sensible time step is now random, because taking the max norm of the noise is random. It is possible to estimate the expected maximum norm of the thermal noise. First, you can simply take the maximum pre-factor of all physical variables. For the noise itself, each norm of 3 random Gaussians is distributed as a $\chi_3$ (or Maxwell-Boltzmann) distribution. The maximum of a collection of these numbers depends on the number of cells in the geometry. Using Extreme Value Theory, the expected value of this maximum can be calculated in the limit of a large number of cells. This works fairly well, as seen in the following figure.

The Problem

The problem in the code base, is that this would require each DynamicEquation to keep track of an extra QuantityEvaluator , simply for the Timesolver to estimate the noise about once per simulation if temperature is non-zero. This felt so stupid, that I did not implement it.