Rate Equation

In pump prove time resolved spectroscopy, we assume reaction occurs just after pump pulse. So, for 1st order dynamics, what we should to solve is

\[\begin{equation*} y'(t) = \begin{cases} 0& \text{if $t < 0$}, \\ Ay(t)& \text{if $t>0$}. \end{cases} \end{equation*}\]

with, \(y(0)=y_0\).

Ususally, y is modeled as sum of the exponential decays, so we can assume the matrix A could be diagonalizable.

Then,

\[\begin{equation*} y(t) = \begin{cases} 0& \text{if $t < 0$}, \\ \sum_i c_i \exp(\lambda_i t) v_i& \text{if $ t \geq 0$} \end{cases} \end{equation*}\]

Where \(\lambda_i\) is \(i\)th eigenvalue of the matrix \(A\), \(v_i\) is the eigenvector corresponding to \(\lambda_i\) and coefficient \(c_i\) are chosen to satisfy \(y(0)=y_0\).

To model experimental signal corresponding to the dynamics, we convolve our model \(y(t)\) to IRF, then we can model experimental signal as

\[\begin{equation*} signal(t) = \sum_i c_i (\exp * {irf})(\lambda_i t) v_i \end{equation*}\]