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

Let the physical viscous drag coefficient be $\gamma$ inFor isotropic three-dimensional equilibrium Gaussian white noise, each force component has covariance $⟨{\eta }_i(t){\eta }_j(s)⟩=2\gamma k_{B}T{\delta }_{ij}\delta (t-s)$. The random acceleration $\mathbf{A}=\mathbf{\eta }/m$ therefore satisfiesThe factor six is twice the three Cartesian components. It agrees with the equipartition theorem, $⟨|\mathbf{u}{|}^2⟩=3k_{B}T/m$, 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.