= Inertial Brownian displacement with zero initial velocity
{title2=$\langle x^2\rangle=2D[t-2\tau(1-e^{-t/\tau})+\tau(1-e^{-2t/\tau})/2]$}
A free particle following <Langevin dynamics> initialized at zero position and velocity has relaxation time $\tau=m/\zeta$ and thermal <diffusivity> $D=k_BT/\zeta$. Its position is the convolution of the random force with $[1-e^{-t/\tau}]/\zeta$. The <fluctuation-dissipation relation for a Langevin particle> gives force covariance amplitude $2\zeta k_BT$, and integrating the squared kernel yields the displayed <variance>. At short times it grows as $2Dt^3/(3\tau^2)$ because the initial velocity is cold; at long times it grows as $2Dt$. A Maxwellian initial velocity produces a different short-time transient.
Back to article page
