Temporal Renormalization and the Critical-like Behavior in Supercooled Liquids

Inspired by the Kadanoff transformation in the standard renormalization group theory, we propose a temporal renormalization scheme. A Boltzmann factor that explicitly depends on the renormalized timescale is constructed, permitting thermodynamic quantities to be evaluated self-consistently across different timescales. By applying the scheme to the long-time dynamics of supercooled liquids, we uncover critical-like behaviors of supercooled liquid with three characteristic renormalization timescales: At the first timescale s_{\alpha}, the system appears to be "thermodynamically frozen", i.e., the energy fluctuation becomes temperature-independent throughout the supercooled regime. At the second timescale s_{\beta}, the third-order moment of energy distribution reaches a maximum, and s_{\beta} is nearly temperature-independent. At the third timescale s_{\gamma}, the third-order moment of energy distribution passes through a minimum, and s_{\gamma} diverges as a power law s_{\gamma}=(T-T_{c})^(-{\gamma}). The scaling relations may reveal an intrinsic behavior in supercooled liquids, highlighting their unique feature. The current findings also demonstrate that temporal renormalization provides a powerful lens for investigating the timescale-specific dynamics.