Fluctuation-dissipation relation for a Langevin particle (${\sigma }^2=2\zeta k_{B}T$)

For a one-dimensional inertial Langevin equationlet $\zeta$ be the physical viscous drag coefficient and let $\eta$ be zero-mean Gaussian white noise. Equilibrium at temperature $T$ requiresTo see the normalization, the stationary velocity variance of this Ornstein-Uhlenbeck process is ${\sigma }^2/(2m\zeta )$. The equipartition theorem requires $⟨v^2⟩=k_{B}T/m$, giving the displayed relation. After dividing the equation by $m$, the acceleration-noise amplitude is ${\sigma }^2/m^2$ and the damping rate is $\zeta /m$; these coefficients must not be interchanged. In several independent Cartesian directions the same formula applies to each component, rather than to their sum.

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.