\sect{Stochastic rate equation}
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.
%
In items (3-6) 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 $A_0$ and a power spectrum $\phi(\omega)$,
it follows that $n$ relaxes to an average value $\langle n \rangle$,
with fluctuations that are characterized by a power spectrum $C(\omega)$.
\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 $\langle n \rangle$ and Var$(n)$.
Additionally provide approximations for small $\varepsilon$.
\Dn
(3)
Determined what is $A_0$ such that $\langle n \rangle$
would be consistent with the canonical result.
Assuming small $\varepsilon$ deduce that $A_0 \propto \varepsilon$,
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?
Assume that the dynamics is described
by the stochastic rate equation;
care to identify correctly the conjugate variables;
and take into account your answer to item~(3).
\Dn
(5)
Deduce from the fluctuation-dissipation relation
what is the power spectrum $C(\omega)$.
Care to use the appropriate definition for $\chi(\omega)$,
else the result will come out wrong.
\Dn
(6)
Deduce what is the power spectrum $\phi(\omega)$
that is required in order to reproduce $C(\omega)$
from the stochastic rate equation.
\Dn
{\bf Advice:} In item (5) verify that your result is consistent
with the answer to item (2). Likewise you can debug the
numerical pre-factor in your answer to item (6).
Care about factors of "2" in your answers. Failure to
provide strictly correct pre-factors will be regarded
as an essential error.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%