A path integral approach to the dynamics of random chains

TL;DR

This paper models the dynamics of a random chain using a path integral approach, connecting microscopic joint movements to a continuum nonlinear sigma model, and explores effects of rigidity and boundary conditions.

Contribution

It introduces a stringy path integral framework for analyzing the microscopic dynamics of random chains and links it to a generalized nonlinear sigma model in the continuum limit.

Findings

Probability function matches the partition function of a nonlinear sigma model.

Analysis of open and closed chains in 2D and 3D.

Inclusion of joint rigidity constraints in 3D chains.

Abstract

In this work the dynamics of a freely jointed random chain with small masses attached to the joints is studied from a microscopic point of view. The chain is treated using a stringy approach, in which a statistical sum is performed over all two dimensional trajectories spanned by the chain during its fluctuations. In the limit in which the chain becomes a continuous curve, the probability function for such a system coincides with the partition function of a generalized nonlinear sigma model. The cases of open or closed chains in two and three dimensions are discussed. In three dimensions it is possible also to introduce some rigidity at the joints, allowing the segments of the chain to take only particular angles with respect to a given direction.