= Mean displacement
{title2=$\langle\Delta\mathbf r(t)\rangle=\langle\mathbf r(t)-\mathbf r(0)\rangle$}
The mean displacement is the <expected value> of a particle's <displacement> over a time interval. It describes systematic transport, while the <mean squared displacement> also measures fluctuations. For one component with finite second moment,
$$
\langle(\Delta x)^2\rangle=\operatorname{Var}(\Delta x)+\langle\Delta x\rangle^2.
$$
For constant drift speed $u$ and ordinary <diffusion> of coefficient $D$, the mean displacement is $ut$ and its centered <variance> is $2Dt$, so the uncentered <mean squared displacement> is $u^2t^2+2Dt$. With fixed initial velocity $v_0$ and free inertial <Langevin dynamics>, the mean displacement is $v_0\tau_v(1-e^{-t/\tau_v})$, where $\tau_v$ is the <velocity relaxation time>. Its fluctuation variance is given by the <integrated Ornstein-Uhlenbeck displacement>.
Back to article page
