= Fluctuation-dissipation relation for a Langevin particle
{title2=$\sigma^2=2\zeta k_BT$}
For a one-dimensional inertial <Langevin equation>
$$
m\dot v=-\zeta v+\eta(t),\qquad
\langle\eta(t)\eta(s)\rangle=\sigma^2\delta(t-s),
$$
let $\zeta$ be the physical <viscous drag coefficient> and let $\eta$ be zero-mean <Gaussian white noise>. Equilibrium at <temperature> $T$ requires
$$
\sigma^2=2\zeta k_BT.
$$
To see the <normalization>, the stationary <velocity> <variance> of this <Ornstein-Uhlenbeck process> is $\sigma^2/(2m\zeta)$. The <equipartition theorem> requires $\langle v^2\rangle=k_BT/m$, 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.
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