Using mutual information to measure order in model glass formers

Whether or not there is growing static order accompanying the dynamical heterogeneity and increasing relaxation times seen in glassy systems is a matter of dispute. An obstacle to resolving this issue is that the order is expected to be amorphous and so not amenable to simple order parameters. We use mutual information to provide a general measurement of order that is sensitive to multiparticle correlations. We apply this to two glass-forming systems (two-dimensional binary mixtures of hard disks with different size ratios to give varying amounts of hexatic order) and show that there is little growth of amorphous order in the system without crystalline order. In both cases we measure the dynamical length with a four-point correlation function and find that it increases significantly faster than the static lengths in the system as density is increased. We further show that we can recover the known scaling of the dynamic correlation length in a kinetically constrained model, the two-vacancy-assisted-hopping triangular lattice gas.

Figure 1

The kinetic constraints in the 2-TLG model: for the central particle to move into the rightmost vacancy it passes through the mutually neighboring sites. These must be empty for the move to be accepted.

Figure 2

Relaxation time against density for the 2-TLG.

Figure 3

The system for discretizing the patch configuration in the MC hard disk system (not to scale). A grid is superimposed on the patch, and pixels that include a particle center are marked 1 or 2 depending on particle type. Empty space is encoded with zero. The patches are represented by a 3-ary number. Mutual information is measured between two patches separated by a distance $d\sigma$.

Figure 4

Ornstein-Zernike fits and lengths of ${\Psi }^6$ for $R=1.2$ (a, c) and $R=1.4$ (b, d). The lower plots show the extracted lengths (${\xi }_6$) against $\phi$. There is a clear increase in the length of hexatic order for the $R=1.2$ system (c) that is absent when $R=1.4$ (d).

Figure 5

The relaxation time ${\tau }_{\alpha }$ against area fraction $\phi$ for the two values of $R$ in the hard disk system. ${\tau }_{\alpha }$ is measured as the time taken by the self-intermediate scattering function to fall to $\exp (-1)$ [45]. The lines are VFT fits: ${\tau }_{\alpha }\propto \exp [A/({\phi }_0-\phi )]$. ${\phi }_0$ equals 0.82 (0.83) for the $R=1.2$ $\left(1.4\right)$ systems.

Figure 6

${\chi }_4\left(t\right)$ for the binary hard disk system at $R=1.2$ and $R=1.4$. The time of the maximum ${\chi }_4(t={\tau }_h)$ is used when calculating the dynamic correlation length, ${\xi }_4$.

Figure 7

The dynamic correlation length ${\xi }_4$ increases with area fraction for both $R=1.2$ and $R=1.4$. As with the increase in relaxation time, the effect is greater for $R=1.2$. Lines are a guide to the eye.

Figure 8

Plots of the mutual information between patches ($I\left(d\right)$) against patch separation $d$ for the binary hard disk system with $R=1.2$ and $R=1.4$. The values have been corrected for overlapping patches and normalized so that at each distance the maximum possible mutual information is one. The left and central plots are of mutual information between patches encoding static information. The right plot shows mutual information between patches encoding dynamic information for the $R=1.4$ system. The (static) mutual information lengths (and the ${\Psi }^6$ length for $R=1.2$) are plotted in the insets.

Figure 9

A comparison of lengths in the binary hard disk $R=1.2$ and $1.4$ systems. The lines should be considered guides to the eye (they are fits of $\xi \propto {({\phi }_0-\phi )}^b$ where ${\phi }_0$ is taken from the VFT fit of ${\tau }_{\alpha }$ for the system). ${\xi }_4$ is the dynamical correlation length; ${\xi }_{\mathrm{mi}}$ is the (static) mutual information length; and ${\xi }_6$ and ${\xi }_{s^2}$ are the correlation lengths of ${\Psi }^6$ and local $s^2$, respectively. Although not obvious from the plot ${\xi }_{\mathrm{mi}}$ in the $R=1.4$ system does increase very slightly with $\phi$.

Figure 10

(a) The dynamic mutual information length [${\xi }_{\mathrm{dyn}},\mathrm{mi}\left(t\right)$] for the 2-TLG at different densities and with mobility measured over different times $t$; (b) the dynamic mutual information length at $t={\tau }_{\alpha }$ for each density [${\xi }_{\mathrm{dyn}},\mathrm{mi}\left({\tau }_{\alpha }\right),$ red triangles]. The blue circles are dynamic heterogeneity length data from Ref. [10] for the 2-TLG. For higher densities (higher ${\tau }_{\alpha }$) the lengths scale similarly. The inset shows ${\xi }_{\mathrm{dyn}},\mathrm{mi}\left({\tau }_{\alpha }\right)$ plotted against density.

Figure 11

The light gray lines in the main plot are mutual information estimates (without overlap corrections) with different sample sizes: the number of samples doubles each time from top to bottom. The red (dark) line is the estimated true entropy, $H_{\infty }(X,Y_d)$. The inset shows the first-order error fits that were used to obtain $H_{\infty }(X,Y_d)$ for various values of $d$.

Figure 12

The exponent in the error terms versus the estimated true entropy. The different coloured points represent systems with different $\phi$. Triangles are $R=1.2$ systems; circles are $R=1.4$ systems.