On a path integral description of the dynamics of an inextensible chain and its connection to constrained stochastic dynamics

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 Ψ[J] of this field theory contains nonholonomic constraints, which are imposed by inserting in the path integral expressing Ψ[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 re-establish this contact. For this purpose, it is shown here that the generating functional Ψ[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.

Dynamical Aspects of Inextensible Chains

“…The consistency of the approach presented in this work in order to tackle the problem of the dynamics of an inextensible chain is also confirmed by the results of Ref. [32]. In has been shown there that the generating functional of the FHC given in Eq.…”

“…in the Langevin equations (32). The solutions R i,ν i ,λ i of the new Langevin equations obtained in this way will depend also on the Lagrange multipliers λ i .…”

“…where we have used the fact that R i,ν i satisfies Eq. (32). Ignoring the constant factor N i=1 det ∂ ∂t and performing the simple integrations over the noises ν i , the generating functional Z F HC [J] takes the form:…”

“…Such connection was up to now somewhat obscure, despite the fact that in Ref. [32] it has been possible to verify that the GNLσM reproduces as expected the sum over all chain conformations satisfying a free Langevin equation and subjected to constraints that ensure the property of inextensibility.…”

From constrained stochastic processes to the nonlinear sigma model. Two old problems revisited

“…In this limit, in the path integral of the generating functional appear functional Dirac delta functions whose role is to impose the constraints. The proposed method is tested in the case of a discrete inextensible chain, for which the generating functional of the relevant correlation functions has already been derived in path integral form in [31] and has undergone several consistency checks [32,33]. A considerable advance with respect to [31] is the inclusion in the treatment of the external forces, that were previously missing.…”