\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. % 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 ${f_B=\langle N_A\rangle/N}$. % In item (c) it is assumed that the system can be described by the rate equation % $\frac{dN_A}{dt} = k_B N_B - k_A N_A + A(t)$ % 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 $f_A$ and $f_B$ and $\alpha\mathcal{E}$. \item[(b)] Express $\mbox{Var}(N_A)$ by $\langle N_A\rangle$ 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} %%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%