Localization and freezing of a Gaussian chain in a quenched random potential

TL;DR

This paper investigates the localization and freezing behavior of a Gaussian polymer chain in a quenched random potential using the replica variational method, revealing critical disorder thresholds and metastable frozen states.

Contribution

It derives a systematic free energy expression within the 1-RSB framework and identifies the conditions for chain localization and internal freezing in a quenched disorder environment.

Findings

Critical disorder for localization scales as Δ_c ≈ b^d N^{-2 + d/2}

Gyration radius scales as R_g ≈ b (b^d/Δ)^{1/(4 - d)}

Internal degrees of freedom freeze into metastable states at Δ = Δ_A

Abstract

The Gaussian chain in a quenched random potential (which is characterized by the disorder strength $\Delta$) is investigated in the $d$ - dimensional space by the replicated variational method. The general expression for the free energy within so called one - step - replica symmetry breaking (1 - RSB) scenario has been systematically derived. We have shown that the replica symmetrical (RS) limit of this expression can describe the chain center of mass localization and collapse. The critical disorder when the chain becomes localized scales as $\Delta_c \simeq b^d N^{-2 + d/2}$ (where $b$ is the length of the Kuhn segment length and $N$ is the chain length) whereas the chain gyration radius $R_{\rm g} \simeq b (b^d/\Delta)^{1/(4 - d)}$. The freezing of the internal degrees of freedom follows to the 1-RSB - scenario and is characterized by the beads localization length $\bar{{\cal D}^2}$. It was demonstrated that the solution for $\bar{{\cal D}^2}$ appears as a metastable state at $\Delta = \Delta_A$ and behaves similarly to the corresponding frozen states in heteropolymers or in $p$ - spin random spherical model.