Solution of the Critical Dynamics of the Mean-Field Kob-Andersen Model

TL;DR

This work provides a complete analytical solution for the critical dynamics of the Kob-Andersen kinetically constrained model on Bethe lattices, showing that the order-parameter dynamics near arrest obeys a Mode-Coupling Theory–like equation with universal exponents. By extending the cavity-method approach used for Fredrickson–Andersen models to KA with conserved dynamics, the authors derive beta- and alpha-relaxation scalings and demonstrate consistency with numerical simulations for both continuous and discontinuous transitions. The results establish a mean-field universality of MCT-like critical dynamics across RSB glasses and kinetically constrained models, and they connect to stochastic beta-relaxation frameworks for perturbations beyond mean field. Overall, the work clarifies how dynamical facilitation alone can yield MCT-like criticality without thermodynamic glass transitions, providing precise predictions for exponents and scaling laws that agree with simulations.

Abstract

We analytically solve the critical dynamics of the Kob-Andersen kinetically constrained model of supercooled liquids on the Bethe lattice, employing a combinatorial argument based on the cavity method. For arbitrary values of graph connectivity z and facilitation parameter m, we demonstrate that the critical behavior of the order parameter is governed by equations of motion equivalent to those found in Mode-Coupling Theory. The resulting predictions for the dynamical exponents are validated through direct comparisons with numerical simulations that include both continuous and discontinuous transition scenarios.

Figures (9)

• Figure 1: Double-step relaxation of the persistence of KA on RRGs with $z=6$, $m=3$. Points correspond to numerical simulations of systems with size $N=64\times 10^5$. From bottom left to top right the densities are $\rho= 0.83, 0.8335, 0.834, 0.8345, 0.835, 0.8355, 0.8365$. Continuous lines represent the numerical solution of the MCT integral equation \ref{['MCTcrit']}, with $\lambda$ and $\sigma$ computed analytically. In particular, $\sigma=c_{z,m}\,(\rho-\rho_c)$, where the constant $c_{z,m}$ is predicted by our theory (see App. \ref{['app:tdyneq']}). For $z=6$, $m=3$, $c_{6,3}\approx 0.251$. Note that the microscopic timescale $t_0$ appearing in the initial condition of Eq. \ref{['MCTcrit']} is not fixed by $\lambda$ and $c_{z,m}$. We fit $t_0$ from the data at the critical point, where $\phi(t)\approx\phi_{plat}+(t/t_0)^{-a}$, obtaining $t_0\approx 0.3$. Inset: persistence versus time for various densities ($0.7\leq\rho\leq 0.8345$).
• Figure 2: Parametric plot of the relative shift of the effective exponent $a_{eff}$ with respect to the analytical prediction $a$ vs. the shift $\delta\phi$ from the plateau. From bottom right to top right: $z=8$$m=4$, $z=6$$m=3$ and $z=10$$m=4$. Each point is obtained by performing numerical simulations on RRGs at different sizes ($10^5 < N < 10^6$), and then extrapolating to infinite volume.
• Figure 3: Difference between the persistence and its plateau value vs $t$ at the critical density. From top right to bottom right: $z=8$$m=4$, $z=6$$m=3$ and $z=10$$m=4$. Dashed lines correspond to curves proportional to $t^{-a_{z,m}}$, where $a_{z,m}$ are predicted analytically (see Table \ref{['tab:zflKA']}). The points represent numerical simulations on RRGs of size $N=4\times 10^5$.
• Figure 4: Numerical check of the $\beta$-regime scaling law for $z=6$ and $m=3$ and $\rho\lesssim \rho_c$. The persistence in this regime is expected to be described by \ref{['eq:scalinglaw']}, with $g(t)\equiv \delta\phi(t)=\phi(t)-\phi_{plat}$ and $\sigma=c_{z,m}\,(\rho-\rho_c)$, where the constant $c_{z,m}$ is predicted by our theory (see App. \ref{['app:tdyneq']}). In particular, $c_{6,3}\approx 0.251$. The continuous black line represents the master function $f_{-}$ computed by solving numerically \ref{['MCTcrit']} with $\sigma=-1$ and the value of $\lambda$ predicted in the case of $z=6$, $m=3$. The dashed line represents $t^{-a}$. The other curves correspond to numerical estimates of the persistence. From bottom left to top left $\rho=0.82, 0.83, 0.8335, 0.834, 0.8345$. The bottom left inset tests Eq. \ref{['eq:scal1']}. The dashed line represents $t_0\, \hat{t}\, \sigma^{-\frac{1}{2a}}$, with $t_0\approx 0.3$ fitted from the data at the critical point, and $a$ given by our analytical estimate. The top right inset tests Eq. \ref{['eq:scal2']}. The dashed line represents $\hat{t}\, f_{-}'(\hat{t})\, \sqrt{|\sigma|}$. Numerical simulations are obtained on RRGs with $N = 64 \times 10^5$ sites.
• Figure 5: Persistence in KA continuous models ($m=1$) on RRGs. From top to bottom: $z=3,4,5$. The dashed lines correspond to $C_z t^{-2 a}$, where $a=0.395263$ is the analytical prediction, and $C_3\approx 1.52$, $C_4\approx 0.54$ and $C_5\approx 0.305$. Numerical data correspond to the average of $20$ samples of size $N=64\times 10^6$.