Fluctuation-stabilized marginal networks and anomalous entropic elasticity

TL;DR

This paper investigates the elastic behavior of thermal spring networks, revealing a non-zero shear modulus below the isostatic point with an unusual temperature dependence driven by entropy.

Contribution

It demonstrates that thermal networks maintain rigidity below the isostatic point with a sublinear temperature dependence, a novel finding in the study of elastic properties.

Findings

Shear modulus remains non-zero below the isostatic point.

Shear modulus scales as T^{0.8} below the isostatic point.

At the isostatic point, shear modulus scales as sqrt(T).

Abstract

We study the elastic properties of thermal networks of Hookean springs. In the purely mechanical limit, such systems are known to have vanishing rigidity when their connectivity falls below a critical, isostatic value. In this work we show that thermal networks exhibit a non-zero shear modulus $G$ well below the isostatic point, and that this modulus exhibits an anomalous, sublinear dependence on temperature $T$. At the isostatic point, $G$ increases as the square-root of $T$, while we find $G \propto T^{\alpha}$ below the isostatic point, where ${\alpha} \simeq 0.8$. We show that this anomalous $T$ dependence is entropic in origin.