On the connection of the generalized nonlinear sigma model with constrained stochastic dynamics

TL;DR

This paper links the nonlinear sigma model with constrained stochastic dynamics, showing that the generating functional corresponds to solutions of a constrained Langevin equation, thus connecting field theory constraints with stochastic processes.

Contribution

It demonstrates that the generating functional of a constrained nonlinear sigma model matches that of a constrained Langevin equation, bridging field theory and stochastic dynamics.

Findings

The generating functional coincides with that of a constrained Langevin equation.

The approach applies to discrete bead-spring models of polymers.

It clarifies the connection between topological constraints and stochastic processes.

Abstract

The dynamics of a freely jointed chain in the continuous limit is described by a field theory which closely resembles the nonlinear sigma model. The generating functional $\Psi[J]$ of this field theory contains nonholonomic constraints, which are imposed by inserting in the path integral expressing $\Psi[J]$ a suitable product of delta functions. The same procedure is commonly applied in statistical mechanics in order to enforce topological conditions on a system of linked polymers. The disadvantage of this method is that the contact with the stochastic process governing the diffusion of the chain is apparently lost. The main goal of this work is to reestablish this contact. To this purpose, it is shown here that the generating functional $\Psi[J]$ coincides with the generating functional of the correlation functions of the solutions of a constrained Langevin equation. In the discrete case, this Langevin equation describes as expected the Brownian motion of beads connected together by links of fixed length.