\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.    


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

