= Stationary trapped Brownian displacement
{title2=$\langle[x(t)-x(0)]^2\rangle=2k_BT(1-e^{-t/\tau})/\kappa$}
For overdamped <Brownian motion> in potential $\kappa x^2/2$, the <relaxation time> is $\tau=\zeta/\kappa$ and the equilibrium <correlation> is $(k_BT/\kappa)e^{-|t|/\tau}$. Expanding the squared displacement between two equilibrium samples gives the title expression. Initializing the particle exactly at the centre instead gives <variance> $(k_BT/\kappa)(1-e^{-2t/\tau})$. Both are valid but concern different initial ensembles; their saturation values differ by a factor of two. These formulas neglect inertia and require times long compared with <momentum> relaxation.
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