Dispersive Kinetics

Abstract

In this chapter we consider the decay of an optically excited state of a donor molecule in a fluctuating medium. The fluctuations are modeled by time dependent decay rates for electron transfer and its backreaction, deactivation by fluorescence or radiationless transitions and charge recombination to the groundstate . First we discuss a simple dichotomous model where the fluctuations of the rates are modeled by a random process switching between two values representing two different configurations of the environment. We solve the master equation and discuss the limits of fast and slow solvent fluctuations. In the second part, we apply continuous time random walk processes to model the diffusive motion. For an uncorrelated Markovian process, the coupled equations are solved with the help of the Laplace transformation. The results are generalized to describe the powertime law as observed for CO rebinding in myoglobin at low temperatures.

Problems

9.1

Dichotomous Model for Dispersive Kinetics

Consider the following system of rate equations

$$ \frac{d}{dt}\left( \begin{array}{c} P(D*,+)\\ P(D*,-)\\ P(D+A-,+)\\ P(D+A-,-) \end{array}\right) =\left( \begin{array}{cccc} -k_{+}-\alpha &{} \beta &{} 0 &{} 0\\ \alpha &{} -k_{-}-\beta &{} 0 &{} 0\\ k_{+} &{} 0 &{} -\alpha &{} \beta \\ 0 &{} k_{-} &{} \alpha &{} -\beta \end{array}\right) \left( \begin{array}{c} P(D*,+)\\ P(D*,-)\\ P(D+A-,+)\\ P(D+A-,-) \end{array}\right) $$

Determine the eigenvalues of the rate matrix M. Calculate the left- and right eigenvectors approximately for the two limiting cases:

(a) fast fluctuations $k_{\pm }\ll \alpha ,\beta$. Show that the initial state decays with an average rate.

(b) slow fluctuations $k_{\pm }\gg \alpha ,\beta$. Show that the decay is nonexponential.