Gas particle collision rate dependence on particle velocity/ Hard sphere simulation From a basic kinetic theory perspective the A-B collision rate $\theta_{AB}$ is given as a function of the mean speed and the number density.$$\theta_{AB} \propto \bar{C}_A n_B$$
If we are considering particles moving in 3D space, with velocity components ($C_1,C_2,C_3$) described by a Maxwellian distribution. Then what is the mean collision rate of the particles as a function of their $C_1$ velocity?
As a starting point, I derived an equation for the mean speed of particles with one given velocity component $C_1 = a$
$$ \bar{C}(a) = \frac{1}{\int_a^\infty \chi(C)dC} \int_a^\infty C\chi(C)dC$$
Where $\chi(C)$ is the speed distribution. This gives the following relation:
$$ \bar{C}(a) = a+\frac{\sqrt{\pi}}{2} \big(\frac{2kT}{m}\big)^{1/2} e^{(ma^2/2kT)}\Big[1 - erf\sqrt{\frac{ma^2}{2kT}}\Big]$$
Using the basic model of the collision rate developed from kinetic theory suggests that the collision rate would be proportional to this mean speed value. However, from my data this seems to significantly overestimate the number of collisions. Results seem to agree better with  a $\sqrt{\bar{C}_A^2+ \bar{C}_B^2}$ scaling. Does anyone have a 3D hard sphere particle simulation that could be used to shed any light on this?
 A: UPDATE: (Let me know what you think)
I made the mistake of using $\theta_{ab}$ when I should have used $Z_{ab}$ as shown in my other question here.
$$Z_{ab} = n_an_b\sigma_{ab} \Bigg(\frac{8 k T}{\pi \mu}\Bigg)^{1/2}=  n_an_b\sigma_{ab} \sqrt{\bar{C}_a^2 + \bar{C}_b^2}$$
Where $\mu = m_am_b/(m_a+m_b)$ is the reduced mass. $\theta$ and $Z$ give the same pressure and temperature scaling of collisions (when Z is divided by n$_a$), but when looking at 2 species with different mean speeds (or in this case a subset of the same species), it is important to use the form which takes into account the mean speed of both species.
The picture below shows the results of a single species, hard-sphere particle collision simulation I made which seems to confirm this scaling. The simulation compared the distribution of velocities to the distribution of velocities present in collisions. The "relative collision probability" was calculated by taking the ratio of these distributions
$$ \textrm{Relative Collision Probability} = \frac{\#\textrm{ of collisions per velocity group}}{\#\textrm{ of particles per velocity group}}$$

