Temporal Renormalization and the Critical-like Behavior in Supercooled Liquids

Abstract 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 in supercooled liquids with three characteristic renormalization timescales: At the first timescale s α , the system appears to be “thermodynamically frozen”, i.e., the energy fluctuation becomes temperature-independent throughout the supercooled regime. At the second timescale s β , the third-order moment of the energy distribution reaches a maximum, and s β is nearly temperature-independent. At the third timescale s γ , the third-order moment of the energy distribution passes through a minimum, and s γ diverges as a power law, s γ ∼ | T − T c | − γ . The scaling relations may reveal an intrinsic behavior in supercooled liquids, highlighting their unique features. The current findings also demonstrate that temporal renormalization provides a powerful lens for investigating timescale-specific dynamics.