= Velocity autocorrelation function
{title2=$\chi(u)=\mathbb E[V(t+u)V(t)]$}
The unnormalized <autocorrelation function> of a stationary <velocity>. For zero <mean>, $\chi(0)$ is the <velocity> <variance>. It is different from the normalized <autocorrelation function> $\chi(u)/\chi(0)$ often used in <time series>.
If $X(t)-X(0)=\int_0^tV(s)\,ds$, integrating the two-time <covariance> over the square $[0,t]^2$ gives
$$
\mathbb E[(X(t)-X(0))^2]=2\int_0^t(t-u)\chi(u)\,du.
$$
This identity connects a <mean squared displacement> to the <Green-Kubo formula for diffusion>.
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