The passive polymer problem

In this article, we introduce a generalization of the diffusive motion of p oint-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 ab out the marginal limit of velocity-velocity correlations which are uncorrel ated in time and glow with the distance x as /x/(epsilon), and epsilon smal l. This relation gets modified in the case of polymers and membranes (the m arginal advecting now has correlations which are shorter ringed.) The const ruction is done in three steps: First, we reconsider the treatment of the p assive scaler problem using the most convenient treatment via field theory and renormalization group. We explicitly show why IR-divergences and thus t he system-size appear in physical observables. which is rather unusual in t he context of ordinary field-theories. like the phi (4)-model. We also disc uss, why the renormalization group can nevertheless be used to sum these di vergences and leads to anomalous scaling of 2n-point correlation functions as e.g., S-2n(x):= <[<Theta>(x,t) - Theta (0,t)](2n)>. In a second step. we reformulate the problem in terms of a Langevin equation. This is interesti ng in its own, since it allows for a distinction between single-particle an d multi-particle contributions, which is not obvious in the Focker-Planck t reatment. It also gives an efficient algorithm to deter-mine S-2n numerical ly, by measuring the diffusion of particles in a random velocity field. 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 less than or equal to D less than or equal to 2. These o bjects can inter sect each other. We also analyze what happens when self-in tersections are no longer allowed.