Model Comparison All Sims

I took the three models: Arber, Alpha, and Ratio and fit the Alpha and Ratio Models with all the simulation data at once. Then, I created some plots that show how well each of these models compared in comparison to results from the PIC sim.

The Alpha Model is given as a generalization of the Arber Model where

\begin{equation} \frac{dT_{eV}}{dt_{ps}} = C_1 T_{0,eV}^{1 - \alpha/2} \Delta x_{nm}^\alpha n_{23}^{(\alpha + 1)/2} \end{equation}

where $C_1$ is some proportionality constant and $\alpha$ is the power that the debye ratio is raised to in the the heating time $\tau$. The Arber Formula is when $\alpha = 2$ and $C_1 = \alpha_H / n_{ppc} = 0.08$ in my sims. The ratio model would be given by

\begin{equation} \frac{dT_{eV}}{dt_{ps}} = C_2 T_0 \sqrt{n_{23}} \frac{1}{(\lambda_D / \Delta x + 1)^2} \end{equation}

where $C_2$ is another proportionality constant that we would determine empirically.

In my simulations, I determined that the Arber Formula had an RMS error of 20.2, the Alpha Model had an error of 7.76, and the Ratio Model had an error of 16.47 which is only marginally better than the Arber Formula. The best fit hyper-parameters were $C_1 = 0.074$, $\alpha = 0.5$, and $C_2 = 0.06$.

This first plot is the amount of heating predicted during the full 1 ps of simulation. The magenta dashed line shows the result of the PIC simulation. The red, green, and blue dotted lines are the arber, alpha, and ratio models respectively. On the graph I put blue vertical lines to group the simulations by constant debye ratio with the numerical value of the debye ratio labeled in the corresponding region in the graph.

Next, I plotted the heating prediction with the PIC simulation heating subtracted out. As a result, $\Delta T = 0$ means that the model matches up exactly to the PIC simulation.

Finally, I plotted the above differences as a percentage of the PIC simulation result.

Modifying Ratio Model

To get better results, I considered comparing the effect of the ratio model where instead of the debye ratio, I have some fraction of the debye ratio.

\begin{equation} \frac{dT_{eV}}{dt_{ps}} = C_2 T_0 \sqrt{n_{23}} \frac{1}{(C_3 \lambda_D / \Delta x + 1)^2} \end{equation}

where I will fix $C_2$ in each comparison. I call $C_3$ the debye ratio multiplier. Below I have a plot where I reported the mean RMS error from this modified ratio model as a function of $C_3$ just to explore what might be an improvement (varying two parameters $C_2$ and $C_3$). I am doing this to get an estimate of what the length scale is where the model goes from an arber-like formula to that of constant heating.

I have also plotted, for comparison, the RMS error for the arber and alpha models. These results suggest that around $\lambda_D \sim 10 \Delta x$ is the best length scale for the ratio model. We can see that the ratio model performs better than the alpha model with certain parameters, which is good because they both used the same amount of free parameters: 2. Now, it would be good if we could justify why we used the multiplier of 10, so that we would only have to rely on one free parameter of the constant of proportionality $C_2$.

I also redid this plot, but instead of using the mean squared error, I took the mean squared percent error.

This has the same qualitative look, except that the optimal length scale is now around $\lambda_D \sim 5 \Delta x$. Due to the fact that the amount of heating in eV differs greatly between different simulations, this percentage diagnostic might be a better thing to look at. Regardless, it seems like there should be a small numerical factor as the debye ratio multiplier that is greater than 1 and at most 1 order of magnitude.

Things to Do

• Find out where factor of 10 comes from
• Try to use Ricky’s Sims from OSC