\sect{Mass on a spring}

A balance for measuring weight consists of a sensitive spring which
hangs from a fixed point. The spring constant is ${K}$. The balance
is at temperature ${T}$ and gravity acceleration is ${g}$ in the ${x}$
direction. A small mass ${\mass}$ hangs at the end of the spring. There
is an option to apply an external force $F(t)$, to which $x$ is
conjugate or apply an external vector potential $A(t)$.

\begin{itemize}

\item [{{(a)}}] Find the partition function $Z$.
 
\item [{{(b)}}] Find $\langle x \rangle$ and $\langle x^{2}\rangle$ and $\mbox{Var}(x)$.

\item [{{(c)}}] Write a Langevin equation for $x(t)$, with friction $\gamma$, and a random force $f(t)$. 

\item [{{(d)}}] Assuming $\langle f(t)f(0)\rangle=C\delta(t)$, find $\mbox{Var}(x)$, 
and deduce what is $C$ by comparing with the canonical result.  

\item [{{(e)}}] 
Assuming $x$ is measured in the lab by averaging over time period $t_0$, 
what is the minimal mass that can be meaningfully measured?

\item [{{(f)}}] Describe the external force $F(t)$ by a scalar potential and demonstrate FDT. 

\item [{{(g)}}] Describe the external force $F(t)$ by a vector potential and demonstrate FDT.

\end{itemize}

\Dn

Note: $\int \frac{d\omega}{(\omega^2-\omega_0^2)^2+\gamma^2\omega^2}=\frac{\pi}{\gamma \omega_0^2}$.

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