Ballistic transport ($\mathrm{Var}X(t)\propto t^2$)

A trajectory with a persistent random velocity $V$ has $X(t)-X(0)=Vt$. An ensemble with finite velocity variance therefore has displacement variance $t^2\mathrm{Var}V$, contrasting with the linear-in-time variance of diffusion. In an impulsively kicked sinusoidal shear dispersion model with zero transverse diffusivity, the shear phase never changes and repeated identical kicks have this ballistic scaling. Transverse diffusion instead decorrelates the velocity and permits a long-time diffusive limit.