Brownian and fractional polymers with self-repulsion

TL;DR

The paper develops a general Gaussian-process framework for modelling linear homopolymers in solvents, where monomers are timelike indices of a Brownian or fractional Brownian path and interactions are incorporated via Gibbs factors. It derives a universal formula for the average squared length $\mathbb{E}\left[x^{2}(L)\right]$ under arbitrary potentials, enabling efficient computation through Gaussian integrals and auxiliary sources. The authors compare several self-avoidance factors, notably showing Edwards and step factors are effectively equivalent in the small-cutoff limit, and provide exact results for a nontrivial Gaussian factor (Eq. GF2) along with scaling analyses that reveal how the exponent $\nu$ depends on the Hurst parameter $H$ and potential range. The framework yields insights into how solvent quality and polymer-polymer interactions shape polymer size, connecting Flory-like intuition with rigorous Gaussian-measure calculations and offering practical computation routes for related observables.

Abstract

Brownian and fractional processes are useful computational tools for the modelling of physical phenomena. Here, modelling linear homopolymers in solution as Brownian or fractional processes, we develop a formalism to take into account both the interactions of the polymer with the solvent as well as the effect of arbitrary polymer-polymer potentials and Gibbs factors. As an example the average squared length is computed for a non-trivial Gaussian Gibbs factor, which is also compared with the Edwards' and a step factor.

Figures (3)

• Figure 1: Potential $\phi$ and Gibbs factor $\exp \left( -\phi \right)$ for several values of $\beta$ ($\beta =1,3,10$)
• Figure 2: The power law dependence of the average squared length for H=0.5
• Figure 3: The scaling index