\sect{Rate Equation with noise}

Consider molecules A and B in a box. 
The molecules are subjected to weak electric field $\mathcal{E}$ 
that modifies the binding energy of the B molecules 
such that ${\varepsilon_B = \varepsilon_B^{(0)} + \alpha \mathcal{E}}$, 
where $\alpha$ is a constant.
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In the lack of  electric field the molecules reach 
chemical equilibrium  A$\leftrightarrow$B, 
such that their fractions are ${f_A=\langle N_A\rangle/N}$
and ${f_B=\langle N_A\rangle/N}$.
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In item (c) it is assumed that the system can be described 
by the rate equation
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\[\frac{dN_A}{dt} = k_B N_B - k_A N_A + A(t)\]
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Where $A(t)$ is a stochastic term with zero average 
and correlation function ${\langle A(t)A(t')\rangle=C\delta(t-t')}$.
In item (e) assume that only~$k_B$ is affected by the weak electric field.

\begin{itemize}

\item[(a)] Express $\langle N_A\rangle,\langle N_B\rangle$ by $N$, and $f_A$ and $f_B$, and $\alpha\mathcal{E}$.

\item[(b)] Express $\mbox{Var}(N_A)$ by $N$, and $\langle N_A\rangle$ and $\langle N_B\rangle$.

\item[(c)] Determine the constants $k_A$ and $k_B$ and $D$ in the 
stochastic rate equation such that $\langle N_A\rangle$ and $\mbox{Var}(N_A)$ 
will agree with the canonical expectation of items (a) and (b).

\item[(d)] Determine how $k_B$ is modified by the weak electric field.
In other words find the coefficient in ${\delta k_B \propto \mathcal{E}}$.

\item[(e)] Find the generalized susceptibility $\chi(\omega)$ 
that relates the variation $\delta N_A$ to $\mathcal{E}$.

\item[(f)] Find the power spectrum of $\delta N_A$ in steady state.

\end{itemize}

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