Brownian Motion and Polymer Statistics on Certain Curved Manifolds

TL;DR

This paper analyzes the statistical behavior of Gaussian polymer chains on various curved manifolds, revealing how curvature influences end-to-end distance and localization, with explicit formulas for different geometries.

Contribution

It provides the first detailed calculation of polymer statistics on curved manifolds like spheres, cylinders, cones, and tori, including explicit crossover formulas.

Findings

Surface curvature induces a geometrical localization area.

At short lengths, the polymer behaves as if flat with (R-R')^2 = L l.

At large scales, (R-R')^2 varies depending on the geometry, being constant or linear.

Abstract

We have calculated the probability distribution function G(R,L|R',0) of the end-to-end vector R-R' and the mean-square end-to-end distance (R-R')^2 of a Gaussian polymer chain embedded on a sphere S^(D-1) in D dimensions and on a cylinder, a cone and a curved torus in 3-D. We showed that: surface curvature induces a geometrical localization area; at short length the polymer is locally "flat" and (R-R')^2 = L l in all cases; at large scales, (R-R')^2 is constant for the sphere, it is linear in L for the cylinder and reaches different constant values for the torus. The cone vertex induces (function of opening angle and R') contraction of the chain for all lengths. Explicit crossover formulas are derived.