From statistics of regular tree-like graphs to distribution function and gyration radius of branched polymers

TL;DR

This paper derives the scaling behavior of conformational entropy for branched polymers with fixed structures, using spectral analysis of matrices associated with tree graphs, revealing mathematical properties of these structures.

Contribution

It provides a novel analytical approach to quantify the entropy and size distribution of branched polymers using eigenvalues of Kramers matrices.

Findings

Entropy scales as R^2/(a^2L) at large R

Eigenvalues of Kramers matrices are explicitly computed for regular and sparse trees

Mathematical properties of tree eigenvalues are uncovered

Abstract

We consider flexible branched polymer, with quenched branch structure, and show that its conformational entropy as a function of its gyration radius $R$, at large $R$, obeys, in the scaling sense, $\Delta S \sim R^2/(a^2L)$, with $a$ bond length (or Kuhn segment) and $L$ defined as an average spanning distance. We show that this estimate is valid up to at most the logarithmic correction for any tree. We do so by explicitly computing the largest eigenvalues of Kramers matrices for both regular and "sparse" 3-branched trees, uncovering on the way their peculiar mathematical properties.