Solvable Models of Supercooled Liquids in Three Dimensions

We introduce a supercooled liquid model and obtain parameter-free quantitative predictions that are in excellent agreement with numerical simulations, notably in the hard low-temperature region characterized by strong deviations from mode-coupling-theory behavior. The model is the Fredrickson-Andersen kinetically constrained model on the three-dimensional $M$-layer lattice. The agreement has implications beyond the specific model considered because the theory is potentially valid for many more systems, including realistic models and actual supercooled liquids.

Figure 1

(Left to right, top to bottom) The $M$-layer procedure: (1) $M$-independent copies of a lattice (a regular 2D lattice in the figure) are stacked on top of each other (view from the top); (2) the edges corresponding to a given link in the original lattice are rewired between the $M$ copies; (3) the procedure is repeated for every link of the original lattice. For large $M$, the graph is locally a Bethe lattice, but at large distances it has the properties of finite-dimensional lattice. (4) The three-dimensional diamond cubic lattice.

Figure 2

Persistence vs time in $d=3$. Solid lines from bottom to top: data for $M=3000$, $M=50000$, $M=100000$, $M=200000$ and for the Bethe lattice curve ($M=\infty$). The data follow the Bethe curve at initial times and deviate from it at later times increasing with $M$. The dotted lines represent the corresponding SBR predictions describing the data when they start to deviate from the mean-field curve (see text). (Inset) Parametric plot of ${\chi }_4(t)$ vs $g(t)$ in $d=3$ for the above values of $M$. The SBR predictions (solid lines) are in excellent agreement with the numerical data (points) in the large-time regime where $g(t)$ deviates from MF. At small intermediate times [large $g(t)$] both SBR and the numerical data approach the MF asymptote (dashed straight line), numerical data deviate on smaller (microscopic) times.