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

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.