This should be contrasted to the displacement of a free particle with initial velocity \( v_0 \). In that case the distance from the initial position after a time \( t \) is \( x(t) = vt \) whereas for a diffusion process the root mean square value is \( \sqrt{\langle x^2\rangle-\langle x\rangle^2} \propto \sqrt{t} \). Since diffusion is strongly linked with random walks, we could say that a random walker escapes much more slowly from the starting point than would a free particle.