Abstract
We develop a comprehensive scale Hamiltonian formalism for multiscale systems by recasting the renormalization group (RG) flow in a Hamiltonian framework and then reformulating it in terms of Hamilton--Jacobi theory. In this picture, an effective coupling is treated as a coordinate with a conjugate momentum, and the RG evolution is governed by a scale Hamiltonian. Under physically motivated assumptions—including a linear beta function near the glass transition and a diverging effective barrier—we derive an effective action that serves as an RG invariant. As a concrete application, we study glass-forming liquids and show that the Vogel--Fulcher--Tammann (VFT) law, $\tau(T) \sim \tau_0 \exp(D T_0/(T-T_0))$, emerges naturally from the simplest singular choice of the scale potential $F(g)$. Our framework yields the quantitative prediction $D T_0 = A/\lambda$, connecting the experimentally measurable VFT parameters ($D \approx 3$--$20$, $T_0 \approx 0.7 T_g$) to microscopic RG parameters: the linearized beta function coefficient $\lambda$ and the barrier strength $A$. For glycerol, our formalism predicts $D T_0 \approx 2055$ K, in excellent agreement with experimental values. This provides a rare example of quantitative predictions from RG methods for glassy dynamics.
Authors
- Tomer Barak
- opencode
- VScode
Keywords
cond-mat.stat-mech, cond-mat.dis-nn
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