A path integral approach to the dynamics of a random chain with rigid constraints

TL;DR

This paper develops a path integral method to analyze the stochastic dynamics of a constrained, flexible chain in two dimensions, linking it to a generalized nonlinear sigma model and providing explicit calculations for a ring-shaped chain.

Contribution

It introduces a novel path integral framework for the dynamics of rigidly constrained chains, connecting stochastic chain fluctuations to field theory.

Findings

Probability distribution matches a generalized nonlinear sigma model

Explicit semiclassical calculations for ring-shaped chain

Framework applicable to continuous chain limit

Abstract

In this work the dynamics of a freely jointed random chain which fluctuates at constant temperature in some viscous medium is studied. The chain is regarded as a system of small particles which perform a brownian motion and are subjected to rigid constraints which forbid the breaking of the chain. For simplicity, all interactions among the particles have been switched off and the number of dimensions has been limited to two. The problem of describing the fluctuations of the chain in the limit in which it becomes a continuous system is solved using a path integral approach, in which the constraints are imposed with the insertion in the path integral of suitable Dirac delta functions. It is shown that the probability distribution of the possible conformations in which the fluctuating chain can be found during its evolution in time coincides with the partition function of a field theory which is a generalization of the nonlinear sigma model in two dimensions. Both the probability distribution and the generating functional of the correlation functions of the positions of the beads are computed explicitly in a semiclassical approximation for a ring-shaped chain.