\sect{Stochastic picture of sweep in 2-site system}

Consider $N$ classical particles in a two site system. 
The two sites are subjected to a potential difference $\varepsilon$.
The temperature of the system is~$T$.
Define $n\in[-N,N]$ as the occupation difference. 
%
Assume that the thermalization process 
can be described by a stochastic rate equation 
%
\[ \frac{dn}{dt} \ = \ -\gamma n + A(t) \]
% 
where $A(t)$ is a noisy term that reflects 
the fluctuations of the potential difference. 
Assuming that it has an average value $A_{\varepsilon}$ and a power spectrum $\phi(\omega)$,  
it follows that $n$ relaxes to an average value $\langle n \rangle_{\varepsilon}$, 
with fluctuations that are characterized by a power spectrum $\tilde{C}(\omega)$
and intensity ${\nu \equiv \tilde{C}(0)}$.


\Dn

(1) 
Write what is the interaction energy $H_{\text{int}}$ of $n$ with the field $\varepsilon$. 
Later you will have to be careful with the identification of the conjugate variables. 

\Dn

(2) 
Using the canonical formalism find what 
are $\langle n \rangle_{\varepsilon}$ and Var$(n)$.
Additionally provide approximations for small $\varepsilon$. 

\Dn

(3)
Determine what is $A_{\varepsilon}$ such that $\langle n \rangle_{\varepsilon}$ 
would be consistent with the canonical result. 
Assuming small $\varepsilon$ deduce that $A_{\varepsilon} \propto \varepsilon$, 
and find the pre-factor. 

\Dn

(4)
What is the $\chi(\omega)$ that characterizes the  
response of $n$ to the applied potential 
in the linear-response regime?
Care to identify correctly the conjugate variables; 
and take into account your answer to item~(3).

\Dn

(5)
Consider a quasi-static sweep process, namely, a process during which $\varepsilon$ is varied slowly with constant rate $\dot{\varepsilon}$. Use your result for $\chi(\omega)$ 
in order to express $\langle n \rangle$ in terms of $\langle n \rangle_{\varepsilon}$ and $\dot{\varepsilon}$. 
%Identify the disspation coefficient $\eta$.  

\Dn

(6)  
Deduce from the fluctuation-dissipation relation
what is the correlation function $C(\tau)$ that describes the fluctuations.  
Explain how your answer in item~(5) is related to the fluctuation intensity~$\nu$. 
%Identify the intensity $\nu$, and verify that $\eta = \nu/(2T)$.  

   
\Dn

{\bf Advice:} 
Care about factors of "2" in your answers. Failure to 
provide strictly correct pre-factors will be regarded 
as an essential error.  
Exploit item~(6) in order to double check your answer in~(5).  


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