Vector noise normalization in inertial Langevin dynamics ($⟨\mathbf{A}(t)\cdot \mathbf{A}(s)⟩=(6\gamma k_{B}T/m^2)\delta (t-s)$)

Divide the inertial Langevin equation by mass and let $\zeta =\gamma /m$. An isotropic three-dimensional equilibrium velocity has $⟨|\mathbf{u}{|}^2⟩=3k_{B}T/m$ by the equipartition theorem. Its white acceleration dot-covariance amplitude is therefore $6\zeta k_{B}T/m$, while the amplitude per Cartesian component is $2\zeta k_{B}T/m$. A scalar moment formula must use the component variance rather than the trace.