Demonstrating the average dot-product of relative incident velocity with CM velocity of colliding pairs of ideal gas atoms is zero

Question

In The Feynman Lectures on Physics Vol. I Ch. 39: The Kinetic Theory of Gases an argument is given in 39–4 Temperature and kinetic energy, showing that the average value of the dot-product of the relative post-collision velocity with the common center of mass velocity of pairs of colliding gas atoms is zero. Feynman then asserts that the overall average of relative velocities dotted into corresponding CM velocities is zero.

A footnote states:

This argument, which was the one used by Maxwell, involves some subtleties. Although the conclusion is correct, the result does not follow purely from the considerations of symmetry that we used before, since, by going to a reference frame moving through the gas, we may find a distorted velocity distribution. We have not found a simple proof of this result.

Is there a "standard" derivation of equation 39-19

$$\left\langle \mathbf{w}\cdot\mathbf{v}_{CM}\right\rangle =0?$$

I am particularly interested in an argument that proceeds from the model of individual collisions. That is, one that does not appeal to such abstractions as phase space, etc.

Answer 1

I am particularly interested in an argument that proceeds from the model of individual collisions. That is, one that does not appeal to such abstractions as phase space, etc.

Other than a minor objection to the notion that phase space is an abstraction, the only thing I think you are missing is an assumption that doesn't seem to be explicitly stated. The assumption is that the number of particles, thus the rate at which they collide, is large. If this is the case, then no particular group of particles in the whole could generate a "preferred" velocity direction relative to another population. One can start with this as an initial condition or initial assumption (e.g., removal of a partition separating two gases at very different temperatures). However, if this is not imposed on a system with a lot of particles, thus a high collision rate, then there is no "preferred" velocity for any given particle.

The italics are used because these terms are relative to the system, i.e., some systems can accomplish the same outcome with different parameters like density and temperature.

In short, the implicit assumption is that with a sufficiently large number of particles in equilibrium, the probability of particle 1 with $\mathbf{v}_{1}$ striking particle 2 with $\mathbf{v}_{2}$ is uniform. That is, the probability of $\cos{\theta_{12}} = \mathbf{v}_{1} \cdot \mathbf{v}_{2}/\left( v_{1} \ v_{2} \right)$ is equal for all $\theta_{12}$ so the ensemble average over all $\cos{\theta_{ij}}$ is zero.

Note that if $\langle \mathbf{v}_{1} \cdot \mathbf{v}_{2} \rangle \neq 0$ then it could physically mean there are two particle populations streaming relative to each other and they need not have uniform temperatures. Uniform temperatures among different particle populations results when there is equipartition of random kinetic energy, i.e., when $\langle \tfrac{1}{2} m_{1} \ \mathbf{v}_{1} \cdot \mathbf{v}_{1} \rangle = \langle \tfrac{1}{2} m_{2} \ \mathbf{v}_{2} \cdot \mathbf{v}_{2} \rangle$ (for classical systems).