Initial-value solution of a damped harmonic trap
= Initial-value solution of a damped harmonic trap
For distinct roots $r_\pm=(-\gamma\pm\sqrt{\gamma^2-4mk})/(2m)$, the deterministic response is $X=(v_0-r_-x_0)e^{r_+t}/(r_+-r_-)+(r_+x_0-v_0)e^{r_-t}/(r_+-r_-)$. It satisfies $X(0)=x_0$ and $\dot X(0)=v_0$. With zero-<mean> future noise and prescribed initial data it is also the conditional ensemble <mean>, not a full stochastic trajectory.