Arber Formula from Hockney’s Empirical Investigations

Under the section Modified Arber Formula here, I show how Arber comes up with his formula for the amount of heating. This applies in d=2 dimensions, but if we want to consider d=1 or d=3 dimensions, we would modify this to be what I called the Alpha Model shown here. From now on, I will just refer to it as the Generalized Arber Model. The Generalized Arber Model Predicts the Amount of Heating as

\begin{equation} \frac{dT_{eV}}{dt_{ps}} \equiv \Delta T_{GA}(d) = C_H \frac{T_{0,eV}^{1 - d/2} \Delta x_{nm}^d n_{23}^{(d + 1)/2}}{n_{ppc}} \label{eq:arber} \end{equation}

To be more explicit, this would be

\[\begin{equation} \Delta T_{GA} = C_H \times \begin{cases} \frac{T_{0,eV}^{1/2} \Delta x_{nm} n_{23} }{n_{ppc}} &\mbox{ if } d = 1 \\ \frac{\Delta x_{nm}^2 n_{23}^{3/2} }{n_{ppc}} &\mbox{ if } d = 2 \\ \frac{\Delta x_{nm}^3 n_{23}^2 }{T_{0,eV}^{1/2} n_{ppc}} &\mbox{ if } d = 3 \end{cases} \end{equation}\]

Deriving Heating Model from Electric Field Fluctuations

Here I considered the effect of the electric field fluctuations in time and how it would impact the amount of numerical heating. I derived a formula for the 2D case there, but more generally for d dimensions, we get a formula very similar to the Generalized Arber Formula, but with an extra multiplicative factor dependent on some dimensionless quantity $u_{max} = k_{max} \lambda_D$.

\[\begin{equation} \Delta T_{EF}(d) \equiv \Delta T_{GA}(d) \times \begin{cases} \arctan(u_{max}) &\mbox{ if } d = 1 \\ \log(1 + u_{max}^2) &\mbox{ if } d = 2 \\ (u_{max} - \arctan(u_{max})) &\mbox{ if } d = 3 \end{cases} \end{equation}\]

where $k_{max}$ is the maximum relevant wave number in the problem of interest. In the finite mesh PIC model, Hockney claims that $k_{max} = \frac{\pi}{\Delta x}$. More generally, I can define $k_{max} = R \frac{\pi}{\Delta x}$ to allow for some multiplicative constant $R$ to be another hyper parameter.

Model Comparisons

For all my comparisons, I generated the simulations in the same manner described here. I created a uniform volume of protons and electrons with equal number. The protons are initialized at 1eV, while the electrons are initialized at a higher temperature: (1/9, 1/4, 1, 4, 9)$\times T_0$ where $T_0$ = 100eV. I choose the number density and grid spacing $\Delta x$ in such a way to keep the debye ratio of $\lambda_D / \Delta x$ constant for a given set of simulations. I varied over 7 different ratios to explore regimes with different resolutions.

The simulations were ran for a total of 1ps and for the measurement of heating, I took the amount of heating (average of electron and proton temp) in the last 1/2 of the sim and multiplied by two to avoid any weird fluctuations shortly after starting the simulation.

I considered the Generalized Arber Model (which has one free parameter of the heating constant) and the Hockney Model (which again has the one free parameter of the heating constant in addition to a multiplicative ratio constant that I allowed to be 1, 2, 4, or 8).

1D

For my 1D sims, I used a 64 cell grid with $n_{ppc} = 25$ and ran the simulations for 500 fs. I attempted to run the simulations the same way, however, the temperature did not linearly increase over time. For most of the simulations, the temperature stayed roughtly constant and oscillated a bit up and down. It could be that 500 fs is not a long enough timescale in 1D sims.

2D

For my 2D sims, I used a 32x32 grid with $n_{ppc} = 25$ and ran the simulations for 1 ps.

The Best Arber Model here was the 1D model with an error of 66.4%. The Best Hockney Models used a multiplier of 8.0 and the 2D model had 40.2% and the 3D model had 33.5%. Below, I have a visual depicting the generalized Arber Model Errors and the 2D Hockney Model Errors

We see that the Arber Models perform similarly. Here the default 2D Arber Model performs just as well as the others, which is to be expected because Arber’s Paper was primarily designed for 2D sims. In this case, the Hockney Models seem to perform better than the Generalized Arber Model for higher multipliers $R$.

Below, I have the model prediction of the 1D Arber Model and 2D Hockney Model

3D

For my 3D sims, I used a 16x16x16 grid with $n_{ppc} = 20$ and only ran the simulation for 500fs and for only 5 of the 7 ratios to save time. The simulations all took less than 24 hours on Unity, so I could easily run it longer if I switched to OSC.

The Best Arber Model was actually the 1D Model for these 3D Simulations with a error of 22.9%. The Best Hockney Models used a multiplier of 8.0 and the 2D model has 11.4% and the 3D model had 17.0%. Below, I have a visual depicting the generalized Arber Model Errors and the 3D Hockney Model Errors

From these results, it does not appear that the 3D Hockney Model has a lot of utility. Additionally, the default Arber Model has an error of 110.1% which is more than a model that would just predict 0 eV of heating for every simulation (100%).

Below, I have the model prediction of the 1D Arber Model and 3D Hockney Model

Keeping Ions Immobile

I ran some simulations keeping the Ions (Protons) Immobile (using an EPOCH input.deck specification). I did these simulations for two resolutions, $\lambda_D / \Delta x = {0.156, 0.026}$. There are a total of 26 different simulations this way (because each debye ratio had 13 sims). Below, I have plots of the average temperature, electron temperature, and proton temperature for each simulation.

Average Temperature

Electron Temperature

Proton Temperature

From these plots, we can see that the average temperature is mostly unchanged over the course of the 1 ps simulation. The electron temperature goes up more in the immobile case while the proton temperature is forced to be constant. From these plots, it seems like treating the protons immobile vs mobile wouldn’t change the results of my simulation too much if I’m just looking at the average temperature.

Summary of Project

In short, it seems like the model for predicting the amount of heating is complicated. In my previous simulations, it was made clear that the number of particles per cell has the most direct impact on the observed heating. However, the other factors are not as clear. In the summer, I tried to quantify these dependencies with log-log plots and I found the the density dependence of $n$ varied from the 1st to the 2nd power. Also, I found that the temperature dependence was approximately to the 1/2 power, and the dependce with cell size increased depending on the resolution.

Surprisingly the 1D Arber Model actually worked best for both the 2D and 3D case. This is actually consistent with the work I did in the summer here when I initially just wanted to create a new arber-like model that would take into account the initial temperature. The 1D Arber Model has a less strong dependence on the initial electron density and cell size while also keeping a relationship with initial temperature which is something we want.

I believe that Arber’s Model worked well for his set of simulations because he was operating in a regime where the debye length was only slighlty under-resolved. However, oftentimes it is necessary to severely under-resolve the debye length to get a feasible computation and his formula definitely seems inadequate for those types of estimations. Although the 1D Generalized Arber Formula works better, it is not perfect by any means, there is still a sizeable percent error and we don’t have a good way of theoreticaly motivating the heating constant out front. The Hockney Model was capable of better predictions when increasing $R$ to 8, but I still do not have a theoretical justification for it which is a big drawback. If I did have a more theoretical justification for it, then I think the Hockney Model may be a better replacement for the Arber Formula

Additionally, the 3D Models don’t work that well. One thing that is clearly wrong with the 3D Models is the higher temperature gain for lower initial temperatures which does not make any intuitive or physical sense. However, in doing this, we realize that Arber’s chosen power depence of 2 is not necessarily because 2D models were being considered, it was just what empirically fit to the data the best. So, I do not think that this generalized arber model would be a good way to predict the temperature gain obtained in 1D or 3D simulations.

In addition to my simulations, I also looked at a subset of Ricky’s Simulations without the laser. He had a simulation that had multiple ion species (Carbon, Oxygen, Hydrogen) where ionization now becomes relevant. He also had a target that did not fill up the entire simulation volume. These complications made the analysis of his simulations very confusing. I only had around 15 simulations I could look at, so the sample size was not large enough to generate a decent model. The only useful thing I could extract from this was that the finite size of the target does decrease the amount of heating in the simulation which is also consistent with some of the simulations I did with finite size targets.

To conclude, there are many complications from a more realistic laser-plasma simulation that would make this formula made by Arber useless. I am not convinced that there exists a simple formula for the amount of heating because it depends on many things: the resolution of the cell size, the target size, the initial temperature regime, and possibly other factors that I didn’t even explore due to focusing on simplistic simulations. And it may be the case that my idealized case of only considering a plasma of protons and electrons does not generalize to a more real plasma with heavier ions and multiple species.

Things to Do

• Resume working through Chen Plasma Physics
• Get ready for Candidacy by emailing potential committee members
• Try to learn how to use WarpX and document installation