Simulation data processing

During DSMC / PIC-MC runs, averaged values are stored for each simulation cell. The average is taken over the ensemble of particles within the respective cell. If time averageing is enabled (which is usually the case), the ensemble averages are additionally time averaged. The following averaged quantities are continuously recorded during simulation:

Quantity	Formula
Number	$N = \sum\limits_{i\,\epsilon\,\textrm{cell}}{}{1}$
Density	$\langle n\rangle= \dfrac{N_\textrm{R}\,N}{V_\textrm{cell}}$
Velocity component	$\langle v^{(\alpha)}\rangle = \frac{1}{N}\sum\limits_{i\,\epsilon\,\textrm{cell}}v_i^{(\alpha)}\qquad\alpha=\{x,y,z\}$
Square velocity	$\langle v^2\rangle = \frac{1}{N}\sum\limits_{i\,\epsilon\,\textrm{cell}}\vec{v}_i\!^2$
Velocity vector	$\langle \vec{v}\rangle = \{\langle v_x\rangle, \langle v_y\rangle, \langle v_z\rangle\}$
Velocity	$\langle v\rangle = \sqrt{\langle v_x\rangle^2 + \langle v_y\rangle^2 + \langle v_z\rangle^2}$

The parameter $N_\textrm{R}$ is the weighting factor of the respective species. From these averaged quantities, the following macroscopic results can be obtained:

Quantity	Formula
Pressure	$p = \dfrac{1}{3}m\langle n\rangle\{\langle v^2\rangle - \langle v\rangle^2\}$
Temperature	$T = \dfrac{m}{3\,k_{\scriptscriptstyle\textrm{B}}}\{\langle v^2\rangle - \langle v\rangle^2\}$
Energy	$E = \dfrac{m}{2\,q}\langle v^2\rangle$

Here, $m$ is the mass of one particle, $k_B$ is the Boltzmann constant, and $q$ is the elemental charge (in order to yield the averaged energy in [eV]).