Geometric ergodicity of a bead-spring pair with stochastic Stokes forcing

TL;DR

This paper proves geometric ergodicity for a model of a flexible polymer in a stochastic fluid, extending previous models by including nonlinear spring forces and singular potentials, with implications for understanding polymer dynamics.

Contribution

It introduces a novel analysis of a polymer model with nonlinear and singular interactions, demonstrating exponential convergence using advanced probabilistic techniques.

Findings

Proves geometric ergodicity for the polymer model.

Handles singular potentials like Lennard-Jones.

Extends previous linear models to nonlinear and singular cases.

Abstract

We consider a simple model for the fluctuating hydrodynamics of a flexible polymer in dilute solution, demonstrating geometric ergodicity for a pair of particles that interact with each other through a nonlinear spring potential while being advected by a stochastic Stokes fluid velocity field. This is a generalization of previous models which have used linear spring forces as well as white-in-time fluid velocity fields. We follow previous work combining control theoretic arguments, Lyapunov functions, and hypo-elliptic diffusion theory to prove exponential convergence via a Harris chain argument. In addition we allow the possibility of excluding certain "bad" sets in phase space in which the assumptions are violated but from which the system leaves with a controllable probability. This allows for the treatment of singular drifts, such as those derived from the Lennard-Jones potential, which is a novel feature of this work.