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.
Categories: cond-mat.stat-mech cond-mat.soft