Cyclization of a Polymer: A First Passage Problem for a Non-Markovian Process

TL;DR

This paper models the cyclization kinetics of a polymer as a non-Markovian first passage problem, deriving an exact integral equation approach that captures the complex temporal correlations influencing reaction times.

Contribution

It introduces an exact integral equation framework for polymer cyclization kinetics based on three-time joint probability distributions, advancing understanding of non-Markovian reaction processes.

Findings

Survival probability described by Volterra integral equation

Numerical evaluation of kinetics is straightforward

Approximate schemes emerge from the exact formulation

Abstract

We discuss a problem of cyclization of a polymer molecule, which is an important example of reaction in a system showing strongly non-Markovian behavior on the timescales of interest. We show that the knowledge of the joint three-time probability distribution of the end-to-end distance is sufficient for the full description of the cyclization kinetics, so that the survival probability follows rigorously as a solution of the Volterra integral equation. The corresponding kinetics can easily be evaluated numerically. We moreover discuss how do some well-known approximations appear from this exact scheme due to decoupling.