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Background Let's say I have the $i$'th gas molecule with velocity $\vec v_i(t)$ at time $t$. To find the net displacement $s_i$ we integrate with respect to $t$: $$\vec s_i = \int_{0}^{t} \vec v_i(t') dt'$$ Since, the gas molecule under goes collisions. Initially the $i$'th gas molecule has ve...

Welcome to the site! While I think users here can provide some insight into whether the derivation is correct, I'm not sure if you will get an answer about experimental verification (and it may make the question too broad anyway). The site is most composed of computationalists/modelers, so unless you mean a computational experiment (e.g do molecular dynamics and compare the collision rate with your formula), that question may be better suited for another site. — Tyberius ♦ 9 hours ago
In that case I'll add computational experiment ... ;) — More Anonymous 9 hours ago
The second formula seems to be wrong. If you discretize the time line 0, t1, t2, ..., t, then the RHS should be v(i,1)t1 + v(i,2)(t2-t1) + v(i,3)(t3-t2)..., or something like that. The RHS you wrote, however, is a quite different thing and diverges to infinity when the time discretization interval tends to zero. — wzkchem5 5 hours ago
@wzkchem5 I've edited to emphasize that $t_1$, $t_2$ , $\dots$, $t_n$ are all time intervals and not coordinate time. — More Anonymous 4 hours ago
@MoreAnonymous Thanks for the clarification, and sorry for my misunderstanding. But there still seems to be a problem when going from the fourth equation to the fifth one. Instead of the average time interval, you should have the sum of the time intervals instead. — wzkchem5 4 hours ago
@wzkchem5 The equation can be derived as shown here: math.stackexchange.com/questions/2888976/… there (I believe I linked this) ... The misunderstanding might be how does this equation: $\lim_{s \to 1^+} (s-1) \sum_{n=1}^\infty t_n^s n^{-s} = \lim_{N \to \infty} \sum_{i}^N t_i/N$ is a result proved by mathematicians (see comments here) math.stackexchange.com/questions/3630694/…More Anonymous 2 hours ago
Welcome to our community! — Nike Dattani 4 mins ago

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@NikeDattani thanks :)))