\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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