Polymers and manifolds in static random flows: a renormalization group study

Abstract

We study the dynamics of a polymer or a D-dimensional elastic manifold diffusing and convected in a non-potential static random flow (the “randomly driven polymer model”). We find that short-range (SR) disorder is relevant for d ⩽ 4 for directed polymers (each monomer sees a different flow) and for d ⩽ 6 for isotropic polymers (each monomer sees the same flow) and more generally for d < d c ( D) in the case of a manifold. This leads to new large scale behavior, which we analyze using field theoretical methods. We show that all divergences can be absorbed in multilocal counter-terms which we compute to one loop order. We obtain the non-trivial roughness ζ, dynamical z and transport exponents | in a dimensional expansion. For directed polymersv we find ζ ≈ 0.63 ( d = 3), ζ ≈ 0.8 ( d = 2) and for isotropic polymers ζ ≈ 0.8 ( d = 3). In all cases z > 2 and the velocity versus applied force characteristics is sublinear, i.e. at small forces v(f) ∼ f | with 0 ∗ > 1 . It indicates that this new state is glassy, with dynamically generated barriers leading to trapping, even by a divergenceless (transversal) flow. For random flows with long-range (LR) correlations, we find continuously varying exponents with the ratio g L/ g T of potential to transversal disorder, and interesting crossover phenomena between LR and SR behavior. For isotropic polymers new effects (e.g. a sign change of ζ – ζ 0 result from the competition between localization and stretching by the flow. In contrast to purely potential disorder, where the dynamics gets frozen, here the dynamical exponent z is not much larger than 2, making it easily accessible by simulations. The phenomenon of pinning by transversal disorder is further demonstrated using a two monomer “dumbbell” toy model.