Add reduced diagnostic: 2d differential luminosity

[aeriforme]

Adds a luminosity diagnostic differentiated in the energies of two colliding species, called DifferentialLuminosity2D.
It is defined as follows:

$$\begin{align*} \frac{d^2\mathcal{L}}{dE_1 dE_2}(E_1, E_2, t) = \int_0^t dt'\int d\boldsymbol{x}\, & \int d\boldsymbol{p}_1 \int d\boldsymbol{p}_2\; \sqrt{ |\boldsymbol{v}_1 - \boldsymbol{v}_2|^2 - |\boldsymbol{v}_1\times\boldsymbol{v}_2|^2/c^2} \\ & f_1(\boldsymbol{x}, \boldsymbol{p}_1, t')f_2(\boldsymbol{x}, \boldsymbol{p}_2, t') \delta(E_1 - E_1(\boldsymbol{p}_1) \delta( E_2 - E_2(\boldsymbol{p}_2)) \end{align*}$$

where:

• $\boldsymbol{p}_i$ is the momentum of a particle of species $i$
• $E_i$ is the energy of a particle of species $i$, $E_i (\boldsymbol{p}_i) = \sqrt{m_1^2c^4 + c^2 |\boldsymbol{p}_i|^2}$
• $f_i$ is the distribution function of species $i$, normalized such that $\int \int f(\boldsymbol{x} \boldsymbol{p}, t )d\boldsymbol{x} d\boldsymbol{p} = N$, the number of particles in species $i$ at time $t$

The 2D differential luminosity is given in units of $\text{m}^{-2} \ \text{eV}^{-2}$.

The user must specify the minimum, maximum, and number of bins to discretize the $E_1$ and $E_2$ axes.
The computation of this diagnostic is similar to that of ParticleHistogram2D.

The output is a folder containing a set of openPMD files.
The values of the diagnostic are stored in a record labeled d2L_dE1_dE2, with axes E1 and E2.

[RemiLehe]

Is this change intentional? It seems unrelated to the PR.

[aeriforme]

It is intentional.
This example requires the compilation flag -DWarpX_FFT=ON.
Without this, the related ctest would fail.

Note that this has not been a problem in CI so far because the FFT compilation flag is on.

[RemiLehe]

Hm, a stronger test would be to compute d2L_dE1_dE2_th and to compare the values.

[aeriforme]

You are right.