
\sect{FDT for velocities}

%Baruch's  D20

Fluctuation Dissipation Theorem (FDT) for velocities:
Consider an external $F(t)=\half f_0\eexp{-i\omega t}+\half f_0^*\eexp{i\omega t}$ coupled to
the momentum as
\[
H=\frac{p^2}{2M}+V(x;\text{env})-\frac{1}{M}F(t)p
\]
where "env" stands for the environment's coordinates and momenta.
\begin{itemize}

\item[(a)] Define the velocity response function by $\langle v(\omega)\rangle =\alpha_v(\omega)F(\omega)$ and show that the average dissipation rate is
\[   \overline {\frac{dE}{dt}}=\half \omega |f_0|^2\im \alpha_v(\omega)\,.\]

\item[(b)] Construct a Langevin's equation with $F(t)$ and identify $\alpha_v(\omega)$. [Identify also $\alpha_{p/M}(\omega)$ and show that $\im\alpha_v(\omega)=\im\alpha_{p/M}(\omega)$.]

Using the known velocity correlations $\phi_v(\omega)$ (for $F=0$) show the FDT
\[\phi_v(\omega)=\frac{2k_BT}{\omega}\im \alpha_v(\omega)\,.\]

\end{itemize}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
