Path integral approach to theories of diffusion-influenced reactions

TL;DR

This paper presents a rigorous path integral derivation of the path decomposition expansion for diffusion-influenced reactions, connecting boundary value problems with interaction potentials and deriving exact relations between Green's functions.

Contribution

It provides a novel, elementary proof of the path decomposition expansion using path integral techniques and establishes new exact identities linking different Green's functions.

Findings

Derived exact relations between Green's functions for reactive and non-reactive cases

Provided a rigorous proof of the path decomposition expansion using path integrals

Connected boundary value problems with interaction potential problems involving delta functions

Abstract

The path decomposition expansion represents the propagator of the irreversible reaction as a convolution of the first-passage, last-passage and rebinding time probability densities. Using path integral technique, we give an elementary, yet rigorous, proof of the path decomposition expansion of the Green's functions describing the non-reactive case and the irreversible reaction of an isolated pair of molecules. To this end, we exploit the connection between boundary value problems and interaction potential problems with $\delta$- and $\delta'$-function perturbation. In particular, we employ a known exact summation of a perturbation series to derive exact relations between the Green's functions of the perturbed and unperturbed problem. Along the way, we are able to derive a number of additional exact identities that relate the propagators describing the free-space, the non-reactive as well as the completely and partially reactive case.