= Integrated Ornstein-Uhlenbeck displacement
{title2=$\operatorname{Var}R_i(t)=2D[t-2(1-e^{-\zeta t})/\zeta+(1-e^{-2\zeta t})/(2\zeta)]$}
For a thermal inertial <Langevin equation> conditioned on fixed initial position and velocity, the centered displacement is $R_i=\zeta^{-1}\int_0^t[1-e^{-\zeta(t-s)}]A_i(s)ds$. With the <fluctuation-dissipation relation for a Langevin particle>, $D=k_BT/(m\zeta)$ and integrating the squared kernel gives the displayed <variance>. At short times it is $2D\zeta^2t^3/3$ per component; at long times it is $2Dt+O(1)$. Averaging initial velocity over thermal equilibrium adds a ballistic contribution and gives $2D[t-(1-e^{-\zeta t})/\zeta]$ per component.
Back to article page
