Insights into the Kramers' flux-over-population rate for chemical reactions in liquid phases through the matrix transport equation

Abstract

Kramers' equation models a chemical reaction as a Brownian particle diffusing over a potential barrier under the influence of medium viscosity. In the case of high viscosity, the equation reduces to a simpler Smoluchowski equation. In this report, we have contrived an equivalent matrix-transport equation that relates the ordered pair (activity, flux) of the output (activated complex) to that of the input (reactant). With an initial condition of the Dirac delta type placed at the location of the reactant, and a reflecting boundary condition set on the reactant state, and an absorbing boundary condition on the activated complex state, we are able to prove the equality relation between the mean first passage time, t over bar fp, for the diffusion and the inverse of the rate constant, k(-1), for the reaction counterpart. We have also derived t over bar fp= n-ary sumation i lambda i-1, where lambda(i) is the ith eigenvalue of the Smoluchowski differential operator stipulated with the above-mentioned boundary conditions. We have also deduced that, in the long time limit, the number of particles remaining inside the diffusion domain decays exponentially with a relaxation time tau=t over bar fp=k-1 just the same as the concentration of the reactant does for a first-order reaction system.