
\sect{Site occupation during a sweep process}

Consider the occupation $n$ of a site whose binding energy~$\varepsilon$ 
can be controlled, say by changing a gate voltage. 
The temperature of the environment is $T$ and its chemical potential is $\mu$.
Consider separately 3 cases: \\

(a) The occupation $n$ can be either $0$ or $1$.   

(b) The occupation $n$ can be any natural number $(0,1,2,3,...)$

(c) The occupation $n$ can be any real positive number $\in [0,\infty]$ \\

We define $\bar{n}$ as the average occupation at equilibrium.
The fluctuations of $\delta n(t) = n(t)-\bar{n}$ are 
characterized by a correlation function $C(\tau)$. 
Assume that it has exponential relaxation with time constant $\tau_0$.   
Later we define $\langle n \rangle$ as the average occupation during 
a sweep process, where the potential is varied with rate $\dot{\varepsilon}$. 

\begin {itemize}

\item[(1)]
Calculate $\bar{n}$, express it using $(T, \varepsilon, \mu)$. 

\item[(2)]
Calculate $\mbox{Var}(n)$, express the result using $\bar{n}$.  

\item[(3)]   
Write an expression for the $\omega{=}0$ intensity $\nu$ of the fluctuations.

\item[(4)]
Write an expression for $\langle n \rangle$ during a sweep process. 

\end {itemize}

Irrespective of whether you have solved (1) and (2), 
in item (3) express the result using $\mbox{Var}(n)$.  
In item (4) use the classical version of the fluctuation-dissipation relation, 
and express the result using $(T, \tau_0, \bar{n}, \dot{\varepsilon})$, 
where $\bar{n}$ had been given by your answer to item (1).
Note that the time dependence is {\em implicit} via $\bar{n}$.

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