The passive polymer problem

Abstract

In this article, we introduce a generalization of the diffusive motion of point-particles in a turbulent convective flow with given correlations to a polymer or membrane. In analogy to the passive scalar problem we call this the passive polymer or membrane problem. We shall focus on the expansion about the marginal limit of velocity-velocity correlations which are uncorrelated in time and grow with the distance x as |x|^epsilon, and epsilon small. This relation gets modified for polymers and membranes (the marginal advecting flow has correlations which are shorter ranged.) The construction is done in three steps: First, we reconsider the treatment of the passive scalar problem using the most convenient treatment via field theory and renormalization group. We explicitly show why IR-divergences and thus the system-size appear in physical observables. In a second step, we reformulate the problem in terms of a Langevin equation. This is not only interesting in its own, but also gives an efficient algorithm to determine S^{2n} numerically. In a third and final step, we generalize the Langevin treatment of a particle to polymers and membranes, or more generally to an elastic object of inner dimension D with 0 <= D <= 2. These objects can intersect each other. We also analyze what happens when self-intersections are no longer allowed.