Path-integral approach to the dynamics of a random chain with rigid constraints

Abstract

In this work the dynamics of a chain consisting of a set of beads attached to the ends of segments of fixed lengths is investigated. The chain fluctuates at constant temperature in a viscous medium. For simplicity, all interactions among the beads have been switched off and the number of spatial dimensions has been limited to two. In the limit in which the chain becomes a continuous system, its behavior may be described by a path integral, in which the rigid constraints coming from the infinitesimally small segments are imposed by means of a functional delta function. In this way a model of the dynamics of the chain is obtained, which closely resembles a two-dimensional nonlinear sigma model. The partition function of this generalized nonlinear sigma model is computed explicitly for a ring-shaped chain in the semiclassical approximation. The behavior of the chain at both long and short scales of time and distances is investigated. The connection between the generalized nonlinear sigma model presented here and the Rouse model is discussed.

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.…”

“…They make the computation of the generating functional extremely complicated. Usually, the constrained path integral is reduced to a gaussian one at the price of drastic simplifications [31,[37][38][39]. In the second part of this article it is shown in the case of the inextensible chain that it is possible to go beyond the gaussian approximation.…”

“…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.…”

“…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. While with the usual approaches the path integral of the generating functional contains extra degrees of freedom (the Lagrange multipliers) [8,9,34,35] or, alternatively, is complicated by functional determinants of block matrices [8,9,36], in the present approach the functional determinants are relatively simple and can be computed analytically.…”