Ballistic-to-diffusive Brownian displacement

With thermal initial velocities and relaxation time $\tau =m/\zeta$, the one-dimensional displacement variance is $2D[t-\tau (1-e^{-t/\tau })]$. It approaches $(k_{B}T/m)t^2$ at short times and $2Dt$ at long times. Initial velocity preparation changes the short-time result.