[Paper Review] Theory of Kinetically-Constrained-Models Dynamics

Gianmarco Perrupato, Tommaso Rizzo|arXiv (Cornell University)|Dec 9, 2022

Theoretical and Computational Physics4 citations

TL;DR

This paper derives exact Mode-Coupling-Theory (MCT)-like dynamical equations for Kinetically-Constrained-Models (KCMs) on the Bethe lattice by leveraging numerical observations of power-law decay in spin persistence. It analytically computes dynamical exponents that match simulations across diverse models, validating the theory for continuous, discontinuous, and higher-order critical points with logarithmic decay.

ABSTRACT

The mean-field theory of Kinetically-Constrained-Models is developed by considering the Fredrickson-Andersen model on the Bethe lattice. Using certain properties of the dynamics observed in actual numerical experiments we derive asymptotic dynamical equations equal to those of Mode-Coupling-Theory. Analytical predictions obtained for the dynamical exponents are successfully compared with numerical simulations in a wide range of models, including the case of generic values of the connectivity and the facilitation, random pinning and fluctuating facilitation. The theory is thus validated for both continuous and discontinuous transitions and also in the case of higher order critical points characterized by logarithmic decays.

Motivation & Objective

• To develop a mean-field theory for Kinetically-Constrained-Models (KCMs) that bridges numerical observations with analytical predictions.
• To resolve the longstanding gap in analytical mean-field treatments of KCMs by formulating exact dynamical equations on the Bethe lattice.
• To validate the theory across a wide range of models, including those with generic connectivity, random pinning, and fluctuating facilitation.
• To establish that the dynamic arrest in KCMs follows Mode-Coupling-Theory behavior, even in cases with logarithmic decay or higher-order critical points.
• To provide fully parameter-free analytical predictions by computing all model-dependent constants analytically, eliminating reliance on numerical estimates.

Proposed method

• Derives asymptotic dynamical equations for spin persistence using probabilistic arguments based on observed numerical behavior in KCMs.
• Applies the Fredrickson-Andersen (FA) KCM on the Bethe lattice, mapping the dynamics to bootstrap percolation to determine critical temperature and plateau values.
• Uses the critical behavior of the persistence function, $φ(t) - \phi_{\text{plat}} \sim t^{-a}$, as a starting point to derive MCT-like scaling laws.
• Introduces a parameter exponent $λ$ via second derivatives of the cavity function $F(P,k,f_b)$, which determines the dynamical exponent $a = 1/\lambda$.
• Validates the derived equations by comparing analytical predictions of $a$ and $b$ with numerical simulations for various $(z,f)$ values and $p$-values.
• Considers the hierarchy of time scales and shows that corrections to the leading $t^{-a}$ behavior are negligible at large times, supporting the validity of the asymptotic approximation.

Experimental results

Research questions

• RQ1 Can exact Mode-Coupling-Theory (MCT)-like dynamical equations be derived for Kinetically-Constrained-Models (KCMs) on the Bethe lattice using only numerical observations as input?
• RQ2 Do the analytically derived dynamical exponents $a$ and $b$ match numerical simulations across different KCMs, including those with random pinning or fluctuating facilitation?
• RQ3 Is the theory valid for both continuous and discontinuous dynamical transitions, and for higher-order critical points with logarithmic decay?
• RQ4 Can all model-dependent constants, including the dynamical exponents, be computed analytically without relying on numerical estimates?
• RQ5 Why is the correction term $\Delta\phi_b(t)$ negligible at large times, and what does this imply for the hierarchy of decay rates in the persistence function?

Key findings

• The persistence function $\phi(t)$ for the FA model on the Bethe lattice follows the MCT scaling law $\phi(t) = \phi_{\text{plat}} + |\sigma|^{1/2} g_\pm(t/\tau_\beta)$, confirming MCT-type critical dynamics.
• The dynamical exponent $a = 1/\lambda$ is analytically computed as $a = \frac{2k}{1+k}$ for $f=2$, with $\lambda = \frac{1+k}{2k}$, and matches numerical simulations for $z=4, f=2$.
• For $z=4, f=2$, the plateau value is exactly $\phi_{\text{plat}} = 21/32$, and the dynamical exponent is $a \approx 0.8$, consistent with numerical data.
• The theory successfully predicts the $\beta$-regime behavior near $T_c$, including the divergence of $g_\pm(s)$ as $s^{-a}$ for small $s$ and the power-law decay $g_-(s) \sim -s^b$ for large $s$.
• The correction term $\Delta\phi_b(t)$, representing the difference between full and blocked persistence, decays as $t^{-3a}$, confirming its negligible contribution at large times.
• All model-dependent constants, including $\lambda$, $p_c$, $\phi_{\text{plat}}$, and $a$, are computed analytically, enabling fully parameter-free predictions for the first time in KCMs.