Glass transition
In this article, we study the glass transition using the Frederickson Anderson (FA) model, a simple yet powerful example of a kinetically constrained model that captures the essential features of glassy dynamics. By supercooling a liquid below its freezing point, one can obtain a glass, a solid far from equilibrium that does not relax on experimentally relevant time scales. Although a characteristic temperature \(T_g\) is often introduced, the glass transition is not a true phase transition and no discontinuity appears in thermodynamic observables.
The FA model is defined on a two dimensional \(L \times L\) lattice, where each site can be either mobile or immobile. We denote these states by a binary variable \(n_i \in {0,1}\), where \(n_i = 1\) represents a mobile low density region and \(n_i = 0\) a less mobile high density one. The energy of the system is simply
\[ E = \sum_i n_i . \]
Despite this trivial Hamiltonian, the dynamics is highly non trivial. A site can flip only if at least one of its nearest neighbours is mobile. This kinetic constraint encodes the idea that local rearrangements in a glass require nearby free volume.
We simulate the dynamics using a Metropolis algorithm. At each step a random site is selected and, if the kinetic constraint is satisfied, the spin flip is accepted with probability
\[ p = \min(1, e^{-\beta \Delta E}), \]
where \(\beta\) is the inverse temperature and \(\Delta E\) is the energy difference associated with the flip. One Monte Carlo step corresponds to \(L^2\) attempted updates. After an initial thermalization stage, we measure the concentration of mobile regions
\[ C(t) = \frac{1}{L^2} \sum_i n_i(t) \]
as a function of time.
At fixed temperature, the concentration fluctuates around a well defined mean value. As the temperature is lowered, this mean concentration decreases smoothly, with no sign of a sharp transition. This is consistent with the fact that the glass transition is not a thermodynamic phase transition.
To characterize the dynamics, we study the autocorrelation function of the concentration,
\[ A(\tau) = \langle C(t) C(t + \tau)\rangle - \langle C \rangle^2. \]
At high temperatures, correlations decay rapidly, while at lower temperatures the decay becomes much slower, signaling increasingly sluggish dynamics. By fitting the autocorrelation functions with an exponential form \(A e^{-t/\tau}\), we extract a relaxation time \(\tau\).
In an intermediate temperature regime, we find that the relaxation time follows an Arrhenius law,
\[ \log \tau \propto \beta \]
indicating that the system behaves as an Arrhenius glass. At lower temperatures, this linear behavior breaks down and the relaxation time grows faster than exponentially, marking a crossover to super Arrhenius or fragile glass behavior.
The physical origin of this crossover is revealed by studying dynamic heterogeneity. By counting how often each lattice site flips after thermalization, we observe that at high temperatures the dynamics is spatially homogeneous, while at low temperatures the system separates into regions of high and low mobility. These heterogeneous dynamics provide a microscopic explanation for the dramatic slowing down observed in the glassy regime.
Overall, this work shows how a minimal kinetically constrained model reproduces many qualitative features of the glass transition, including the absence of a true phase transition, the emergence of Arrhenius and super Arrhenius regimes, and the appearance of dynamic heterogeneity at low temperatures.
For a more in-depth discussion, refer to this pdf.