Some exact results for the Smoluchowski equation for a parabolic potential with time dependent delta function sink

TL;DR

This paper presents the first exact analytical solutions for the Smoluchowski equation with a parabolic potential and a time-dependent delta function sink, simplifying the problem to an integral equation and enabling the derivation of probability distributions.

Contribution

The authors provide the first exact solutions for the Smoluchowski equation with a time-dependent sink in a parabolic potential, reducing the PDE to an integral equation.

Findings

Exact solutions for special cases of the equation.

Reduction of the PDE to an integral equation.

Analytical expressions for probability distributions.

Abstract

The Smoluchowski equation with a time dependent delta function sink is solved exactly for many special cases. In all other cases the problem can be reduced to an integral equation. It is shown that by knowing the probability distribution at the position of sink, one can derive analytical expression for probability distribution everywhere. Thus the problem is reduced from a PDE in two variables to an integral equation of one. As far as the authors knowledge, we are the first one to provide an exact analytical solution of Smoluchowski equation for a parabolic potential with time dependent sink.