Summary of Last Week

Below, I am going to summarize some of the results I obtained in the last two weeks. I used a uniform density plasma of $n = 10^{23}$ 1/cc electrons with an equal amount of protons. I made the simulation grid 32 cells in both x and y directions and used 50 computational particles per cell. I started the protons at a cold temperature of 1eV and varied the initial electron temperature: $T_e = {10, 100, 1000, 10000}$ eV. Additionally, I varied the cell size as $\Delta x = {1, 4, 7, 10}$ nm. These parameters explored a region where the Debye Length is under-resolved where $\lambda_D / \Delta x \in (0.007, 0.74)$ and the number of particles contained in a Debye Sphere is $N_D \in (0.17, 5463)$.

Below are results when the full simulation volume is filled with plasma.

Since we wish to study the effects of the vacuum on the heating, I did another simulation that is restricted to a target with size of half the simulation length in each direction (a square that is 1/4 of the total area).

Target Shape Comparison

The issue with the previous simulation is the initial drop in electron temperature at the first few hundred femto seconds of the simulation. We think this is probably happening because the fast moving electrons will want to move away from the target, but are being pulled back by the stationary (by comparison) protons and so will get slowed down. However, the average energy between the protons and electrons seems to rise nearly linearly which is what we can use for a comparison. I redid these graphs for a variety of different target shapes and just plotted the average temperature over time.

Below, I organized the plots by comparing the effect of different target size (32 x 32 refers to no vacuum) for a given grid spacing and initial temperature (where I also added in what the estimated heating would be from the formula in the Arber Paper, labeled AS).

In these simulations, it seems that the larger target sizes exhibit more numerical heating. For some of the sims with smaller targets, the electrons may be going fast and crossing the periodic boundaries before the plasma gets a chance to equilibrate. However, the simulations with lower temperature and higher grid spacing do not follow this trend. In these sims, the electrons may not reach the boundary that quickly and the smaller target sizes would be able to expand faster, causing a higher change in temperature.

Below, I organized the plots by comparing the effect of different grid size for a given target size and initial temperature.

Here, it can be seen that there is more heating as $\Delta x$ increases. This is due to the Debye Length being further under-resolved which we know results in more heating from the Arber Paper. However, when the electron temperature is 10eV, this does not seem to be true. In this regime, $N_D < 1$, so the particles do not exhibit collective behavior of a normal plasma which may factor into play. Additionally, at really high electron temperature of 10keV, we see that the heating curves are not as smooth. This may be because the electrons are moving so much faster than the protons.

Particles Per Cell Comparison

From the Arber paper, we expect the heating rate to be inversely proportional to the number of computational particles per cell. I did some simulations with a full target (32x32) and half target (16x16) and varied the number of particles per cell as PPC= 10, 20, 40, 70, 100, 150. I used grid spacings of $\Delta x = {1, 5}$ nm and initial temperatures of $T_0 = {10, 100, 1000}$ eV.

Here are the results with the full target

where I’ve additionally plotted a log-log plot to determine the relationship between $\frac{dT}{dt}$ and PPC

and additionally, the results with half target (dotted lines)

Qualitatively, both the full and half targets look very similar, the only difference is that the full target nearly always has slightly more heating. We see that the Arber formula relationship holds up (slope of -1) for 4 out of the 6 different cases. When $\Delta x = 5$ nm and $T_0 = {10, 100}$ eV, we get a slope that is more negative than -1. These are the cases which are the most under-resolved. $\lambda_D / \Delta x = {0.01, 0.04}$ respectively. Even in the case where $T_0 = 10$ eV and $\Delta x = 1$ nm where $N_D < 1$, we still get a slope of approximately -1. This shows that $N_D > 1$ may not be an important factor when using Arber’s Formula.

Density Comparison

The Density comparison tests look similar to the PPC tests, except, we see that higher densities experience more heating. I will refer to densities in units of 1e22 1/cc. (Also, I ignored the n = 1 simulations for the log-log plots later because had so little heating that they would mess up the slope calculation).

The one set of simulations with 10eV and 5nm looks very pecular though. This may because this has the smallest ratio of $\lambda_D / \Delta x$ (0.007 - 0.05). Here, the higher densities (10,20,40) do not increase the heating as much as you’d expect from the Arber formula.

Similarly, here are the results for the half target:

An additional problem is that the heating for $T_0 = 1000$ eV $\Delta x = 1$ nm is neglible for low densities, which screws up my log-log plot.

Overall, it seems harder to predict the density dependence of the heating rate from these log-log plots. From the Arber Formula, the exponent of n is 3/2, which means we should expect a positive slope of 1.5 in the plot. We see some of the plots have slopes roughly close to 1.5 (maybe smaller than that on average), however this is not as clear as the particles per cell comparison.

Things to Do

1. Parse through this data more and try to get more refined conclusions
2. Compare my results with Ricky’s Sims
3. Redo Sims for Best Settings (Spline, Current Smoothing)
4. Look at Machine Learning Framework that Chris Sent me.
5. Can type of summary of Tom’s Synthetic Data Generation and Neural Network Creation.