Is an average of an inner product of relative velocity and center-of-mass velocity zero?

According to a textbook in my hand, when there are two particles and no interaction, and if the system is in equilibrium,

$< \vec{a} \cdot \vec{b} > = 0$

where $\vec{a}$ is the relative velocity and $\vec{b}$ the center-of-mass velocity.

Is this true? Any hints will be helpful. I welcome not only a mathematical proof, but also an intuitive, physical understanding.

• It would be better if you could provide more details. Average over what? Time? Are the particles confined to a box? Is it thermal equilibrium or ..? There is no interaction between the particles or no interaction with other particles? Which book is this? Since I don't have a clear understanding of the assumptions, it's hard to say anything. But notice that, while the relative velocity is frame invariant, the velocity of the center of mass is not. So in the frame that is at rest with respect to the center of mass this statement would be trivial but then what is there to say about other frames? – secavara Apr 15 '18 at 18:50
• This would be assuming there are no interactions over the particles and then their combined system is free, so that we can use the center of mass frame as inertial frame. Which makes me think that there is context missing in your statement, as I mentioned. – secavara Apr 15 '18 at 19:03
• Oh you might mean this: link ? – secavara Apr 15 '18 at 19:26
• @secavara Thanks to you, now I've just got the intuitive understanding for the equation. The link you wrote was one I've already visited but it is not "my textbook". Actually feynmanlectures.caltech.edu/I_39.html#Ch39-S4 was very helpful. Thank you again. – ynn Apr 15 '18 at 19:28