Cooperative elastic mechanism of activated structural relaxation in glassy liquids

The dramatic slowdown of dynamics in deeply supercooled liquids remains a central problem in condensed matter physics, chemistry, and materials science. Within the inherent-structure framework, structural relaxation may be viewed as activated transitions between basins of the potential energy landscape via high-energy bottleneck configurations. The intermediate state can be modeled as a transient entropy droplet, representing a locally reconfigured region embedded in an amorphous solid, whose formation is driven by configurational entropy and opposed by elastic mismatch with the surrounding matrix. In this study, we develop an elasticity-based theory for the activation free energy associated with the formation of such a droplet. By incorporating intrinsic elastic heterogeneity and non-affine strain redistribution, we show that the mismatch energy is reduced through cooperative pathways involving softer regions of the material. This leads to an effective interfacial penalty that scales as R3/2 with droplet size, in contrast to the conventional R2 scaling expected for a homogeneous medium. The resulting free-energy balance yields activation barriers inversely proportional to the configurational entropy, thereby recovering the Adam-Gibbs relation for structural relaxation. While the same scaling is invoked in random first-order transition theory, it is derived here from elastic heterogeneity and non-affine deformation, providing a complementary mechanical interpretation of entropy-controlled dynamics in glassy systems.