Helmholtz-decomposition-code.md

@@ -16,8 +16,18 @@ From this, we can apply the Helmholtz Decomposition theorem to obtain the compre
 
 <a href="https://www.codecogs.com/eqnedit.php?latex=\omega_c(\textbf{r},t)=\sum_{\textbf{k}\neq0}^{}&space;\frac{\textbf{k}\cdot&space;\widetilde{\omega}(\textbf{k},t)}{k^2}\;&space;\textbf{k}e^{i\textbf{k}\cdot\textbf{r}}" target="_blank"><img src="https://latex.codecogs.com/gif.latex?\omega_c(\textbf{r},t)=\sum_{\textbf{k}\neq0}^{}&space;\frac{\textbf{k}\cdot&space;\widetilde{\omega}(\textbf{k},t)}{k^2}\;&space;\textbf{k}e^{i\textbf{k}\cdot\textbf{r}}" title="\omega_c(\textbf{r},t)=\sum_{\textbf{k}\neq0}^{} \frac{\textbf{k}\cdot \widetilde{\omega}(\textbf{k},t)}{k^2}\; \textbf{k}e^{i\textbf{k}\cdot\textbf{r}}" /></a>
 
+<a href="https://www.codecogs.com/eqnedit.php?latex=\omega_i(\textbf{r},t)=\sum_{\textbf{k}\neq0}^{}&space;\left&space;\{&space;\widetilde{\omega}(\textbf{k},t)&space;-&space;\frac{\textbf{k}\cdot&space;\widetilde{\omega}(\textbf{k},t)}{k^2}\;&space;\textbf{k}\right&space;\}e^{i\textbf{k}\cdot\textbf{r}}" target="_blank"><img src="https://latex.codecogs.com/gif.latex?\omega_i(\textbf{r},t)=\sum_{\textbf{k}\neq0}^{}&space;\left&space;\{&space;\widetilde{\omega}(\textbf{k},t)&space;-&space;\frac{\textbf{k}\cdot&space;\widetilde{\omega}(\textbf{k},t)}{k^2}\;&space;\textbf{k}\right&space;\}e^{i\textbf{k}\cdot\textbf{r}}" title="\omega_i(\textbf{r},t)=\sum_{\textbf{k}\neq0}^{} \left \{ \widetilde{\omega}(\textbf{k},t) - \frac{\textbf{k}\cdot \widetilde{\omega}(\textbf{k},t)}{k^2}\; \textbf{k}\right \}e^{i\textbf{k}\cdot\textbf{r}}" /></a>
+
+Note that in our specific case, we assume the component <a href="https://www.codecogs.com/eqnedit.php?latex=\omega_0(t)" target="_blank"><img src="https://latex.codecogs.com/gif.latex?\omega_0(t)" title="\omega_0(t)" /></a> to be zero across the whole lattice.
+
 # Usage
 TODO: explain how to use...
+The cast line consists of few specific entries. After the executable name, call the .wtxt file that needs to be analyzed, together with a number indicating the spin component you wish to inspect:
+* 0 is for spin a (up)
+* 1 is for spin b (down)
+* 2 is for spin a+b 
+
+Here is an example of how it should look like:
 ```bash
 [gabrielw@node2066 tmp]$ mpirun -np 5 wslda-hdc-3d test2.wtxt 0          
 
@@ -35,11 +45,16 @@ TODO: explain how to use...
          ....
         Completed in    4s ( 0m 4s)! 
 ```
+For large lattices, the code might take longer than it shows due to MPI memory cleaning routines.
+
+The code outputs four files:
+* Two of them are `.wdat`, and they contain data about <a href="https://www.codecogs.com/eqnedit.php?latex=\omega_0(t)" target="_blank"><img src="https://latex.codecogs.com/gif.latex?\omega_c(r,t)" title="\omega_0(t)" /></a> and <a href="https://www.codecogs.com/eqnedit.php?latex=\omega_0(t)" target="_blank"><img src="https://latex.codecogs.com/gif.latex?\omega_i(r,t)" title="\omega_0(t)" /></a> in binary form. These can be read via VisIt by selecting the corresponding quantity from the VisIt HUD (wc*/wi*);
+* One `.txt` file, containing **time, compressive energy, rotational energy, compressive energy calculated via spectral decomposition and rotational energy calculated via spectral decomposition** in this order;
 
 TODO: Explain what are output files... their meaning and content, in what units are columns
 
 # Limitations
-Presently the code supports only 3d data. However, you can apply it to 2D as well, but before that, you need to increase data dimensionality. You can do it using wdata tool `wdata-datadim-up`. For example:
+Presently the code supports only 3d data. However, you can apply it to 2D as well. Before that, you need to increase data dimensionality of the datasets you wish to analyze. You can do it using wdata tool `wdata-datadim-up`. For example:
 ```bash
 [gabrielw@node2066 tmp]$ wdata-datadim-up test1.wtxt test2 3
 # WDATA TOOL: DATADIM MODIFIER