Kinetic Models for Semiflexible Polymers in a Half-plane

TL;DR

This paper derives a kinetic Fokker-Planck-type equation for semiflexible polymers in a half-plane, analyzing its well-posedness, regularity, and asymptotic behavior for long chains.

Contribution

It introduces a continuum limit derivation of a kinetic model for semiflexible polymers with non-local boundary conditions and studies its mathematical properties.

Findings

Existence of a unique measure-valued solution.

The equation is hypoelliptic with locally Hölder continuous solutions.

Asymptotic behavior characterized for large polymer chains.

Abstract

Based on a general discrete model for a semiflexible polymer chain, we introduce a formal derivation of a kinetic equation for semiflexible polymers in the half-plane via a continuum limit. It turns out that the resulting equation is the kinetic Fokker-Planck-type equation with the Laplace-Beltrami operator under a non-local trapping boundary condition. We then study the well-posedness and the long-chain asymptotics of the solutions of the resulting equation. In particular, we prove that there exists a unique measure-valued solution for the corresponding boundary value problem. In addition, we prove that the equation is hypoelliptic and the solutions are locally H\"older continuous near the singular boundary. Finally, we provide the asymptotic behaviors of the solutions for large polymer chains.