  The emergence of the solid state in glass-forming materials upon cooling is
accompanied by changes in both thermodynamic and viscoelastic properties and by
a precipitous drop in fluidity. Here, we investigate changes in basic elastic
properties upon cooling in a family of simulated polymer fluids, as
characterized by a number of stiffness measures. We show that $\tau_{\alpha}$
can be expressed quantitatively both in terms of measures of the material
``stiffness'', $G_p$ and $\langle u^2 \rangle$, and the extent $L$ of
cooperative particle exchange motion in the form of strings, establishing a
direct relation between the growth of emergent elasticity and collective
motion. Moreover, the macroscopic stiffness parameters, $G_p$, $B$, and $f_{s,
q^*}$, can all be expressed quantitatively in terms of the molecular scale
stiffness parameter, $k_{\mathrm{B}}T / \langle u^2 \rangle$ with
$k_{\mathrm{B}}$ being Boltzmann's constant, and we discuss the thermodynamic
scaling of these properties. We also find that $G_p$ is related to the cohesive
energy density $\Pi_{\mathrm{CED}}$, pointing to the critical importance of
attractive interactions in the elasticity and dynamics of glass-forming
liquids. Finally, we discuss fluctuations in the local stiffness parameter as a
quantitative measure of elastic heterogeneity and their significance for
understanding both the linear and nonlinear elastic properties of glassy
materials.
