  We focus on the response of mechanically stable (MS) packings of
frictionless, bidisperse disks to thermal fluctuations, with the aim of
quantifying how nonlinearities affect system properties at finite temperature.
Packings of disks with purely repulsive contact interactions possess two main
types of nonlinearities, one from the form of the interaction potential and one
from the breaking (or forming) of interparticle contacts. To identify the
temperature regime at which the contact-breaking nonlinearities begin to
contribute, we first calculated the minimum temperatures $T_{cb}$ required to
break a single contact in the MS packing for both single and multiple eigenmode
perturbations of the $T=0$ MS packing. We then studied deviations in the
constant volume specific heat $C_V$ and deviations of the average disk
positions $\Delta r$ from their $T=0$ values in the temperature regime $T_{cb}
< T < T_{r}$, where $T_r$ is the temperature beyond which the system samples
the basin of a new MS packing. We find that the deviation in the specific heat
per particle $\Delta {\overline C}_V^0/{\overline C}_V^0$ relative to the zero
temperature value ${\overline C}_V^0$ can grow rapidly above $T_{cb}$, however,
the deviation $\Delta {\overline C}_V^0/{\overline C}_V^0$ decreases as
$N^{-1}$ with increasing system size. To characterize the relative strength of
contact-breaking versus form nonlinearities, we measured the ratio of the
average position deviations $\Delta r^{ss}/\Delta r^{ds}$ for single- and
double-sided linear and nonlinear spring interactions. We find that $\Delta
r^{ss}/\Delta r^{ds} > 100$ for linear spring interactions and is independent
of system size.
