Dynamical Aspects of Inextensible Chains

Abstract

In the present work the dynamics of a continuous inextensible chain is studied. The chain is regarded as a system of small particles subjected to constraints on their reciprocal distances. It is proposed a treatment of systems of this kind based on a set Langevin equations in which the noise is characterized by a non-gaussian probability distribution. The method is explained in the case of a freely hinged chain. In particular, the generating functional of the correlation functions of the relevant degrees of freedom which describe the conformations of this chain is derived. It is shown that in the continuous limit this generating functional coincides with a model of an inextensible chain previously discussed by one of the authors of this work. Next, the approach developed here is applied to a inextensible chain, called the freely jointed bar chain, in which the basic units are small extended objects. The generating functional of the freely jointed bar chain is constructed. It is shown that it differs profoundly from that of the freely hinged chain. Despite the differences, it is verified that in the continuous limit both generating functionals coincide as it is expected.

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

“…It remains to perform the limitD α → 0. To this purpose it is possible to apply the formula [31,33]:…”

“…Up to now, the generating functional Z[J ] has been computed in the semiclassical approximation [31] or it has been linearized using a variational method [33]. In the rest of this Section an attempt to go beyond these approximations will be presented.…”

“…(32) to be those of Eq. (33). For this reason, at the instants t = 0 and t = t f we may chose for k(t, s) Dirichlet boundary conditions, i. e.…”

“…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.…”