= Vector noise normalization in inertial Langevin dynamics
{title2=$\langle\mathbf A(t)\cdot\mathbf A(s)\rangle=(6\gamma k_BT/m^2)\delta(t-s)$}
Let the physical <viscous drag coefficient> be $\gamma$ in
$$
m\dot{\mathbf u}=-\gamma\mathbf u+\boldsymbol\eta(t).
$$
For isotropic three-dimensional equilibrium <Gaussian white noise>, each force component has <covariance> $\langle\eta_i(t)\eta_j(s)\rangle=2\gamma k_BT\delta_{ij}\delta(t-s)$. The random <acceleration> $\mathbf A=\boldsymbol\eta/m$ therefore satisfies
$$
\langle\mathbf A(t)\cdot\mathbf A(s)\rangle
=\frac{6\gamma k_BT}{m^2}\delta(t-s).
$$
The factor six is twice the three Cartesian components. It agrees with the <equipartition theorem>, $\langle|\mathbf u|^2\rangle=3k_BT/m$, and with damping rate $\gamma/m$. A one-component <variance> has factor two; a vector dot-covariance is the <trace> and has factor six. Stating whether a coefficient is a drag or a damping rate avoids a further factor-of-<mass> error.
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