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Equation (**) implies a discontinuous step-function response for $$ \langle x(t) \rangle - \langle x \rangle_0 $$ at t=0, which is incorrect. As t approaches zero from minus infinity, $$ \langle x(t) \rangle - \langle x \rangle_0 $$ adiabatically approaches $$\ \beta f_0 A(0) $$. Equation (**) can be corrected by removing the outer parentheses on the right-hand side. As a consequence of this change, the outer parentheses in the second term on the right-hand side in the final equation of the derivation must also be removed. Thus the derivation beginning with equation (**) should read:


 * $$ (**) -\chi(t) = \beta {\operatorname{d}A(t)\over\operatorname{d}t}

\theta(t). $$

For stationary processes, the Wiener-Khinchin theorem states that the power spectrum equals twice the Fourier transform of the auto-correlation function


 * $$ S_x(\omega) = 2 \tilde{A}(\omega). $$

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 $$ {\operatorname{d}A(t)\over\operatorname{d}t} \theta(t)   $$   into a symmetric and an anti-symmetric part


 * $$ 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). $$

Now the fluctuation-dissipation theorem follows.

(Remark to reviewer, not to be included in the actual article: Observe that the corrected derivation above does not change the imaginary (absorptive) part of chi, but it does modify the real (dispersive) part.)