Fractional Brownian motion and polymers: Learning from each other

L Streit,J Bornales

doi:10.1142/s2010194515600162

Abstract

Self-avoiding or self-repelling random paths, with motivation from their use in polymer physics, have been widely studied using the tools of mathematics, physics, and computer simulations. We illustrate this by three recent examples.

Highlights

Self-avoiding or self-repelling random path models for polymer configurations have been studied extensively in mathematics using combinatorics, stochastic analysis, in statistical mechanics, and in computer physics using Monte Carlo methods
Apart from self-avoiding random walks, a prominent realization is the Edwards model of self-repelling Brownian paths, an example of models where self-crossings are not strictly forbidden but where there is an exponential penalty on self-crossings.[8]
The Edwards model has been extended to fractional Brownian motion,[13] allowing for models of stiffer or curlier polymers than those described by classical Brownian motion

Summary

Introduction

Self-avoiding or self-repelling random path models for polymer configurations have been studied extensively in mathematics using combinatorics, stochastic analysis, in statistical mechanics, and in computer physics using Monte Carlo methods. Apart from self-avoiding random walks, a prominent realization is the Edwards model of self-repelling (or “weakly self-avoiding”) Brownian paths, an example of models where self-crossings are not strictly forbidden but where there is an exponential penalty on self-crossings.[8] This is an Open Access article published by World Scientific Publishing Company. The Edwards model has been extended to fractional Brownian motion (fBm),[13] allowing for models of stiffer or curlier polymers than those described by classical Brownian motion It is in this context that we shall present report on progress on three fronts, where methods from stochastic analysis, arguments from statistical physics, and numerical computation have been employed

Varadhan’s existence proof

The Flory index

Computational results