Statistical collision sampling

Given are two particle species 1, 2 with their weighting factors $N_{R1}, N_{R2}$, a cell with volume $v_{cell}$ and particle numbers per cell $N_1, N_2$.

The statistical collision treatment estimates the number of collision between particles, which belong to the same cell with cell volume $v_{cell}$. It is considered, that - within a time step $\delta t$

• two particles with a relative velocity $\vec{v}_r := \vec{v}_1-\vec{v}_2$ and a total collision cross section $\sigma_T$ put up a collision volume of $\sigma_T ||\vec{v}_r|| \delta t$ (see figure below).

The ratio of this collision volume and the cell volume can be considered as collision probability $p_{12}$ between two particles within a cell:

$$p_{12} = \frac{\sigma_T v_r \delta_t}{v_{cell}}, \, {\rm where} \, v_r := ||\vec{v}_r||$$

In the NTC method, for pre-selection of the maximum reasonable number of collision pairs, the maximum collision probability $p_{12.max}$ is obtained from

$$p_{12.max} = \frac{\left(\sigma_T v_r\right)_{max} \delta_t}{v_{cell}}.$$

Collisions between species with same weight factor

In case, both species have the same weight factor $N_R = N_{R1} = N_{R2}$ the maximum number $N_{col.max}$ of physical collisions is

$$N_{col.max} = N_R N_1 N_R N_2 p_{12.max}$$

;

however, for the number $N_{sel}$ of selected collision pairs, this number has to be divided by $N_R$ since performing a collision with the stored super particles affects in reality $N_R$ physical particles:

$$N_{sel} = N_R N_1 N_2 p_{12.max}$$

.

As a side note, in case both species are equal and $N := N_1 = N_2$, the number of pre-selected collision pairs has to be divided by factor two (since in case of equal species the order of collision partners does not count):

$$N_{sel} = \frac{1}{2}N_R N^2 p_{12.max}$$

.

For the number $N_{per}$ of actually performed collisions, each selected collision pair has to undergo a random experiment, where the following relation is checked (R# is a random number between ]0, 1[):

$$R\# < \frac{p_{12}}{p_{12.max}}.$$

Thus, the mean number $<N_{per}>$ of performed collisions is given as

$$\left<N_{per}\right> = N_{sel} \times \frac{p_{12}}{p_{12.max}}.$$

Collisions between species with different weight factors

In case, $N_{R1}, N_{R2}$ are different, let's call the low and high statistical weight factor $N_{RL}, N_{RH}$ with $N_{RL}<N_{RH}$. Then, the maximum number of physical collision is

$$N_{col.max} = N_{RH} N_{RL} N_1 N_2 p_{12.max}$$

.

In the selected collision pairs the collision shall be actually executed each time for the species with low statistical weight and only $N_{RL}/N_{RH}$ times for the other species with the high statistical weight. Under this constraint, the number of selected collision pairs has to be multiplied by $N_{RL}$ to represent the physical number of executed collisions. Therefore, $N_{col.max}$ has to be divided by $N_{RL}$ yielding

$$N_{sel} = N_{RH} N_1 N_2 p_{12.max}$$

.

In case, the collision yields reaction products of a third species with statistical weight $N_{R3}$ the number $N_{rp}$ of generated reaction products per executed collision is

$$N_{rp} = \frac{N_{RL}}{N_{R3}}.$$

Depending on whether $N_{RL}/N_{R3}$ is bigger or lower than one, either multiple reaction products must be generated per collision or the generation of a single reaction product is determined by random experiment.

If generation of multiple reaction products per time step shall be avoided, a higher number of collision pairs must be pre-selected:

$$N_{sel} = \frac{N_{RH}N_{RL}}{N_{R3}}N_1 N_2 p_{12.max}$$

In this case the number of generated reaction products after collision is one.

Collisions, where one of the collision partners is frozen

In a time-scale splitting scheme, one approach is to consider a certain species distribution as constant, and freeze all particles of this species during simulation. Practical examples are e.g.:

• Freeze the Ar background gas distribution and only consider movement of electrons and ions in electron time scale
• Freeze the charge distribution (electrons + ions) in a reactive plasma and only consider creation of radicals in neutral time scale

Let's denote $N_{RF}$ the weight factor of the frozen species and $N_{RM}$ the weight factor of the moving species in a collision. Then, for the number of pre-selected collision pairs follows

$$N_{sel} = N_{RF} N_1 N_2 p_{12.max}$$

.

The number of reaction product of a third species with weight factor $N_{R3}$ is then obtained via

$$N_{rp} = \frac{N_{RM}}{N_{R3}}.$$