The passive polymer problem
Kay Jörg Wiese
Fachbereich Physik, Universität GH Essen, 45117 Essen, Germany
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|ε, and ε 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,
which is rather unusual in the context of ordinary field-theories, like
the φ4-model. We also discuss, why the renormalization group
can nevertheless be used to sum these divergences and leads to
anomalous scaling of 2n-point correlation functions as e.g.
S2n(x) := <[ Θ(x,t) - Θ(0,t) ]2n>.
In a second step, we reformulate the problem in terms of a
Langevin equation. This is interesting in its own, since it allows
for a distinction between single-particle and multi-particle
contributions, which is not obvious in the Focker-Planck treatment.
It also gives an efficient algorithm to determine S2n numerically,
by measuring the diffusion of two particles in a random velocity
field. In this formulation S2n is given by the n-th moment
of the time which one and only one of the two particles spends
inside a box of size L.
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.
chao-dyn/9911005 [pdf]
J. Stat. Phys. 101 (2000) 843-891 [pdf]
Copyright (C) by Kay Wiese. Last edited March 17, 2008.