# What is the PDF of relative collision speeds in an ideal gas?

In the Kinetic theory of gases, the speeds of particles follow a Maxwell-Boltzmann distribution.

However, what if one is interested in the distribution of relative collision speeds, aggregated over collisions?

Faster particles are more likely to collide during a given time window, because they sweep over a larger volume that can contain slow particles. I don't know whether they are more likely to collide with other fast particles. (I guess I can say that my intuition and knowledge fail me.)

Having the distribution I propose, along with a statistic on the number of collisions per time window, I could answer questions such as "What is the probability that a least one collision during time window T happens with a relative velocity above v?".

Is there a solution to this?

• Are you asking about collision rates as in that discussed at: physics.stackexchange.com/a/268594/59023? – honeste_vivere May 15 at 19:42
• My question is about distribution (not only average) of collision speeds. But I shared some intuition regarding how collision rates depend on particle speeds, and thus why I don't think the distribution of speeds (or velocities) can be easily converted to a distribution of collision speeds. Thanks for the link, which sheds some light on that "intuition" area. – Elias Hasle May 16 at 10:36
• I think if you left things in terms of impact parameters and then constructed a distribution of those based upon a distribution of particle velocities, you could, in principle, construct an impact velocity distribution for each particle. The problem is that you'd need to create one for each particle at each time step, since any advance in time would alter all the speeds due to frame changes from collisions. – honeste_vivere May 16 at 14:24
• @honeste_vivere I am not sure I follow. I find it hard to understand why the distribution I request would have to be updated at any time, as long as the macroscopic temperature and pressure are constant. – Elias Hasle May 19 at 10:20
• Then you'd need to do some ensemble averaging beforehand anyways because temperature and pressure are thermodynamic concepts and/or velocity moments, i.e., you're integrating over the velocity distribution. If you start in LTE and the system is isolated from any external forces, then you know the velocity distribution and the mean free path already, i.e., it's easy to construct collision rates and velocities. – honeste_vivere May 19 at 17:14

If one approximates the probability of two particles having momenta $$p_1$$ and $$p_2$$ with their product, then simply $$\text{Prob}(p_r) = \sum_p\text{Prob}(p)\text{Prob}(p+p_r)$$ (where the sum is to be understood as an integral)