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Consequently,
![{\displaystyle (**)-\chi (t)=\beta {\frac {d}{dt}}(A(t)\theta (t)).}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b15fd037a55f00cc84ef579ab11b559e77a63617)
becomes
![{\displaystyle (**)-\chi (t)=\beta {\operatorname {d} A(t) \over \operatorname {d} t}\theta (t).}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1e8521370a844698feaf25ca16eda8a2be4de9da)
For stationary processes, the Wiener-Khinchin theorem states that
the power spectrum equals twice the Fourier transform of the auto-correlation
function
![{\displaystyle S_{x}(\omega )=2{\tilde {A}}(\omega ).}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d835fa1ec933959b4452515a9ffb9fabf08b74f3)
The last step is to Fourier transform equation (**) and to take the
imaginary part. For this it is useful to recall that the Fourier transform
of a real symmetric function is real, while the Fourier transform of a real
antisymmetric function is purely imaginary.
We can split
into a symmetric and an
anti-symmetric part
![{\displaystyle 2{\operatorname {d} A(t) \over \operatorname {d} t}\theta (t)\ ={\operatorname {d} A(t) \over \operatorname {d} t}+{\operatorname {d} A(t) \over \operatorname {d} t}{\rm {sign}}(t).}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8ce07030f36982e162ac9cdff222cc4c0ff00b07)
Now the fluctuation-dissipation theorem follows.
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