User:Psh07733/NewPage

This is a test to create NewPage

Consequently,


 * $$ (**) -\chi(t) = \beta \frac{d}{dt} ( A(t) \theta(t) ) . $$

becomes


 * $$ (**) -\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. Add a few random words