From fluctuating entropic neck to Rosenfeld-Adam-Gibbs crossover dynamics in supercooled liquids

The pronounced dynamical slowdown in supercooled liquids is accompanied by a sequence of crossovers, most notably from collision-dominated transport at high temperatures to activated structural relaxation at low temperatures, as reflected in the transition from Rosenfeld excess-entropy scaling to Adam-Gibbs behavior. We develop a unified theoretical framework based on a memory-function formalism combined with a configuration-space extension of Zwanzig's entropic-neck picture. Starting from coupled slow coordinates that describe intra-basin motion and inter-basin escape, we derive a reduced description in which transport occurs through entropically constrained pathways connecting metastable basins in the inherent-structure landscape. In contrast to simplified parallel-channel pictures, the intrabasin coordinate and the neck (transition) coordinate are dynamically coupled, leading to a nonseparable memory kernel with both local (collisional) and collective (barrier-crossing) contributions. At high temperatures, fluctuations sample broad regions of configuration space, the effective neck is wide, and transport is governed by local structural entropy, recovering Rosenfeld scaling. Upon supercooling, progressive constriction of accessible pathways generates an entropic bottleneck whose statistics are controlled by configurational entropy, yielding Adam-Gibbs-type activated dynamics. The formalism provides a semi-microscopic interpretation of the crossover, clarifies its relation to the mode-coupling temperature, and naturally extends to nonequilibrium conditions, where the same entropic-neck mechanism governs the evolution of fictive temperature and its connection to fragility.