= Overdamped harmonic-trap power spectrum
{title2=$S_x^{(1)}(f)=\frac{k_BT}{\pi^2\gamma(f^2+f_c^2)},\quad f_c=k/(2\pi\gamma)$}
In the classical thermal model, a coordinate in a <harmonic potential> obeys $\gamma\dot x+kx=\xi$, where $\gamma$ is the drag coefficient and $\langle\xi(t)\xi(t')\rangle=2\gamma k_BT\delta(t-t')$. Its stationary <one-sided spectral density> per hertz is the displayed <Lorentzian line shape>. The integral over positive <frequencies> equals $k_BT/k$, consistent with the <equipartition theorem>. On a <log-log plot> the spectrum is flat for $f\ll f_c$ and has slope $-2$ for $f\gg f_c$. The crossover is determined by the restoring stiffness and drag; an added detector response multiplies the mechanical spectrum by the squared magnitude of its transfer function.
Back to article page
