 Fluctuation-dissipation relation for a Langevin particle

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.