Fluctuation-Stabilized Marginal Networks and Anomalous Entropic Elasticity

We study the elastic properties of thermal networks of Hookean springs. In the purely mechanical limit, such systems are known to have a vanishing rigidity when their connectivity falls below a critical, isostatic value. In this work, we show that thermal networks exhibit a nonzero 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 proportional to T alpha below the isostatic point, where alpha similar or equal to 0.8. We show that this anomalous T dependence is entropic in origin.