Constrained dynamics of a polymer ring enclosing a constant area

TL;DR

This paper investigates the constrained dynamics of a polymer ring with fixed algebraic area, revealing how the area constraint influences mode relaxation, correlation functions, and diffusive behavior.

Contribution

It introduces a detailed analysis of the time-dependent Lagrange multiplier enforcing the area constraint and its impact on the polymer ring's dynamics.

Findings

The Lagrange multiplier is explicitly evaluated at short and long times.

The first Rouse mode dominates the position correlation at long times.

The mean square displacement exhibits diffusive behavior due to rotational diffusion.

Abstract

The dynamics of a polymer ring enclosing a constant {\sl algebraic} area is studied. The constraint of a constant area is found to couple the dynamics of the two Cartesian components of the position vector of the polymer ring through the Lagrange multiplier function which is time dependent. The time dependence of the Lagrange multiplier is evaluated in a closed form both at short and long times. At long times, the time dependence is weak, and is mainly governed by the inverse of the first mode of the area. The presence of the constraint changes the nature of the relaxation of the internal modes. The time correlation of the position vectors of the ring is found to be dominated by the first Rouse mode which does not relax even at very long times. The mean square displacement of the radius vector is found to be diffusive, which is associated with the rotational diffusion of the ring.