\sect{Baruch's  D23.}

An electrical circuit has in series components with capacitance $C$,
inductance $L$, resistance $R$ and a voltage
source $V_0\cos \omega t$ with frequency $\omega$.
\begin{itemize}
\item[(a)] Identify the responsefunction $\alpha_Q(\omega)=\langle Q(\omega)\rangle /(\half V_0)$  . Use this to write the energy dissipation rate.
\item[(b)] Use the fluctuation dissipation relation to identify the
Fourier transform $\Phi_Q(\omega)$ of the charge correlation function.
Evaluate $\langle Q^2(t)\rangle$ and compare with the result from equipartition.
\item[(c)] Evaluate the current fluctuations
$\langle I^2(t)\rangle$ and compare with the result from equipartition. Under what conditions does one get Nyquist's result $\langle I^2\rangle_{\omega_1 \leftrightarrow \omega_2 }  = \frac{{2k_BT}}{{\pi R}}\left( {\omega_2  - \omega_1 } \right)$ ?

Hint: $\int_{-\infty}^{\infty} \frac{d\omega
/2\pi}{(\omega^2-\omega_0^2)^2+\gamma^2\omega^2}=\frac{1}{2\gamma
\omega_0^2}\,,\qquad\int_{-\infty}^{\infty} \frac{\omega^2 d\omega
/2\pi}{(\omega^2-\omega_0^2)^2+\gamma^2\omega^2}=\frac{1}{2\gamma}\,.$\\
\end{itemize}

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