# Analytical expression for end-to-end- auto correlation function of a long chain polymer molecule in solution

## TL;DR

This paper develops an analytical model for the auto-correlation function of the end-to-end distance of a long polymer chain in dilute solution, based on a diffusion-like theory and Smoluchowski equation.

## Contribution

It provides a detailed analytical expression for the auto-correlation function, incorporating parameters like chain length, bond length, and relaxation time.

## Key findings

- Auto-correlation function depends on chain length, bond length, and relaxation time.
- Derived an explicit analytical expression for the end-to-end distance auto-correlation.
- Model enhances understanding of polymer dynamics in dilute solutions.

## Abstract

A diffusion-like theory for real time end-to-end distance of a long polymer chain in dilute solution is formulated. We give a detailed analytical expression for the end-to-end distance auto-correlation function of a long chain polymer in solution. The physical problem of dynamics of end-to-end distance can be modeled mathematically with the use of a Smoluchowski-like equation. Using this equation analytical expression for end-to-end distance auto-correlation function is derived. We find that this auto-correlation function varies with several parameters such as length of the polymer (N), bond length (b) and the relaxation time ${\tau_R}$.