Normalized Comparison of Simulation

Below, I take all the simulations where I kept PPC = 50, $n_{23} = 1$ constant and varied the initial temperature and cell width and show the PIC heatings vs the amount predicted by Arber.

The above plot shows the Log10 of the amount of heating from the PIC simulation (I extrapolated from the last 20% of the simulation) vs the amount of heating from the Arber Formula. If the Arber formula was somewhat accurate, the results would be between -1 and 1 (indicating 1 order of magnitude below or above). We see that many of these data points lie outside this region.

Additionally, there seems to be a clear relationship between the initial temperature and the amount of heating, which is not predicted from the Arber Formula.

Comparing Effect of Number of Cells

In all of these types of simulations, I have used a 32 x 32 grid. Just to be safe, I did some simulations with a 16x16 grid and 64x64 grid to make sure that this choice does not affect our results.

From the above plot, it is clear that the amount of heating does not change for different simulation sizes when we keep the cell size constant.

Additionall, we can see that all of these simulations have the same qualitative relationship when we vary the cell size. As expected, when we use fewer cells, our heating curve looks more rough and when we use more cells, the heating curve looks smooth.

Exploring Peculiarity of 10eV Simulations

From our sims, we should expect that larger cell sizes contribute to more heating. However, when we use 10eV electrons, we don’t seem to get that result. Below, I have a plot of the heating for a range of cell sizes.

For these sims, it seems like the heating increases from 1 to 3 nm, but then decreases after that. I tried to plot the Electric Fields on the grid as a function of time as an animation, but since there are only 32 cells in each direction, it may be difficult to actually extract information from this.