Validation

Besides unit tests of every kind’s kinematics (exact energy loss, energy sharing, isotropy, products), the test suite checks the operator in a 0D reactor against closed-form solutions. With Maxwell-molecule cross sections the frequency $\nu = nK$ is constant. The method then collides a marker within one step with probability $p = (1 - e^{-s\nu\Delta t})/s$, where $s$ is the bound safety factor, and the expectations are exact functions of the step count:

Test	Expectation	Tolerance
collided fraction in one step	$(1 - e^{-\nu_{\max}\Delta t})\,\nu/\nu_{\max}$	5σ
attachment, electron survival	$(1 - p)^n$	2 %
ionization, electron growth	$(1 + p)^n$, and $n_e - n_i$ constant	2 %
equal-mass elastic, mean energy	$\tfrac32kT_g + (E_0 - \tfrac32kT_g)\,e^{-\nu_{\rm eff}t/2}$	2 %
resonant charge exchange of a drifting beam	$T_g + (T_0-T_g)f + \frac{Mu_0^2}{3k}f(1-f)$, $f = (1-p)^n$	2 %
thermal ions on a gas at the same temperature	mean energy unchanged (detailed balance)	1 %
cold ions in a 300 K gas	relax to 300 K and stay there	2–3 %
density profile	collision probability proportional to the local density, zero where it is zero	5–8 %
ions in a flowing, heated gas	relax to the gas’s flow velocity and local temperature	5 %
ions in a uniform field $E$	drift $v_d = qE/(\mu\nu)$ and Wannier’s mean energy $\tfrac32kT_g + \tfrac12(m+M)v_d^2$	3 %
marker merging	weight, momentum and kinetic energy unchanged	exact
ion-impact ionization	energy and momentum conserved up to the electron’s $O(\sqrt{m_e/M})$ terms	exact
Opal–Peterson–Beaty sharing	mean secondary energy $w\ln(1+(R/2w)^2)/(2\arctan(R/2w))$	2 %

The drift velocity and the Wannier energy hold exactly for a Maxwell-molecule cross section with isotropic scattering, which makes the field test a check of the field push and the collision kinematics together: a swarm benchmark with a closed-form answer. For real gases, compare the same reactor diagnostics (mean_velocity, mean_energy_ev and the collision counts) with a Boltzmann solver such as BOLSIG+ for the same cross-section set.

The whole suite runs on NumPy and on cunumpy’s stand-in for CuPy in CI, and the combined coverage of the two runs is 100 %. The tutorials plot several of these comparisons.