Information Loss in Coarse Graining of Polymer Configurations via Contact Matrices

TL;DR

This paper investigates how contact matrices, as coarse-grained representations of polymer configurations, relate to the underlying walks, revealing differences between simple and self-avoiding random walks in terms of enumeration and entropy.

Contribution

It provides theoretical analysis of contact matrices for polymers, showing asymptotic enumeration equivalence for SRW and differences for SAW, and compares coarse and fine grained entropies.

Findings

Number of contact matrices matches SRW walk count asymptotically.

Coarse grained entropy aligns with fine grained for SRW when n <= 2.

Differences in entropy emerge for SAW and higher dimensions.

Abstract

Contact matrices provide a coarse grained description of the configuration omega of a linear chain (polymer or random walk) on Z^n: C_{ij}(omega)=1 when the distance between the position of the i-th and j-th step are less than or equal to some distance "a" and C_{ij}(omega)=0 otherwise. We consider models in which polymers of length N have weights corresponding to simple and self-avoiding random walks, SRW and SAW, with "a" the minimal permissible distance. We prove that to leading order in N, the number of matrices equals the number of walks for SRW, but not for SAW. The coarse grained Shannon entropies for SRW agree with the fine grained ones for n <= 2, but differs for n >= 3.