A Solvable Model for Homopolymers and the Critical Phenomena

TL;DR

This paper models the distribution of long homopolymers near a critical point using self-adjoint extensions of the Laplacian, analyzing phase transitions between globular and extended phases through explicit resolvent formulas.

Contribution

It introduces a solvable model for homopolymers with a zero-range potential, providing detailed analysis of behavior near the critical transition point using spectral theory.

Findings

Characterization of phase transition at critical parameter 

Explicit resolvent formulas for the associated operators

Analysis of polymer behavior near criticality

Abstract

We consider a model for the distribution of a long homopolymer with a zero-range potential at the origin in $\mathbb{R}^3$. The distribution can be obtained as a limit of Gibbs distributions corresponding to properly normalized potentials concentrated in small neighborhoods of the origin as the size of the neighborhoods tends to zero. The distribution depends on the length $T$ of the polymer and a parameter $\gamma$ that corresponds, roughly speaking, to the difference between the inverse temperature in our model and the critical value of the inverse temperature. At the critical point $\gamma_{cr} = 0$ the transition occurs from the globular phase (positive recurrent behavior of the polymer, $\gamma > 0$) to the extended phase (Brownian type behavior, $\gamma < 0$). The main result of the paper is a detailed analysis of the behavior of the polymer when $\gamma$ is near $\gamma_{cr}$. Our approach is based on analyzing the semigroups generated by the self-adjoint extensions $\mathcal{L}_\gamma$ of the Laplacian on $C_0^\infty(\mathbb{R}^3 \setminus \{0\})$ parametrized by $\gamma$, which are related to the distribution of the polymer. The main technical tool of the paper is the explicit formula for the resolvent of the operator $\mathcal{L}_\gamma$.