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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...