Information-theoretic measurements of coupling between structure and dynamics in glass-formers

References

Appendix A Supporting Information

Details of the models described in the main text, and the methods used to identify locally-favored structures.

Illustrative results of mutual information between propensity and particle type

Analysis of the numerical method that we use when estimating mutual information.

The polydisperse hard sphere system consists of an equimolar mix of five particle species with diameters $(1.000,0.938,0.899,0.861,0.799)\sigma$, all with equal masses $m$. The particles interact as hard spheres and the system evolves by event-driven molecular dynamics (implemented by DynamO ). The system comprises $N=1372$ particles and the simulation box is cubic with periodic boundary conditions. The time unit in the system is $\Delta t=\sqrt{m\sigma^{2}/k_{\rm B}T}$. The structural relaxation time is evaluated at $k=2\pi/\sigma$.

A.2 Identifying locally favored structures

Here, we briefly describe the structural measurements $n_{155}$ and $n_{028}$ that we use to identify locally-favoured structures in these systems. These measurements are based on Voronoi analyses of the system. In the KA model, we perform this analysis after quenching the system to its nearest energy minimum (inherent structure). We follow in using a Voronoi analysis where faces between A and B particles are located closer to the B particles, consistent with their smaller size. In the HS system, we use a regular Voronoi analysis, in which faces are midway between neighbouring particles.

We identify $(0,2,8)$ Voronoi polyhedra in the KA system as those with ten faces, of which exactly two have four edges, and eight have five edges. This particle and its ten Voronoi neighbours form a cluster (11A in the topological cluster classification [S1]), and we set $n_{028}(i)=1$ for all particles in these clusters.

To identify the pentagonal bipyramids in which particle $i$ participates (in both HS and KA systems), we identify $n_{155}(i)$ as the number of pentagonal faces on the Voronoi cell of that particle. This gives the number of neighbours of particle $i$ that share exactly five mutual neighbours with particle $i$. The procedure is equivalent to counting the number of ‘1551’ bonds in the common neighbour analysis (CNA) , and is similar to the identification of ‘7A’ clusters in the topological cluster classification [S1].

A.3 MI between particle type and propensity

To demonstrate the physical meaning of MI, Fig. S1(a) shows $I_{t}(\mu;s)$ for the KA system, where the structural measurement $s_{i}$ is taken to be the particle type $\alpha_{i}={\rm A,B}$. The B-particles are more mobile in this system, and as shown in the inset, at large times $t\gg\tau_{\alpha}$, the propensity distributions for the two kinds of particle have almost zero overlap. Thus, for these very long times, measuring the particle type splits the propensity distribution into two distinct components: this provides $-f\log_{2}f-(1-f)\log_{2}(1-f)$ bits of information (similar to a mixing entropy), where $f$ and $(1-f)$ are the fractions of particles in each component. If the components were equal in size, the MI would be exactly 1 bit: here the B particles are less numerous ($f=0.2$) so the MI is less, approximately $0.7$ bits. For times $t$ close to the structural relaxation time $\tau_{\alpha}$, Fig. S1(b) shows that the propensity distributions of the two types differ from each other, but there is a region of significant overlap. In this case, measuring the particle type provides $0.3$ bits of information about $\mu_{it}$.

A.5 Estimating mutual information

Calculating mutual information (MI) from numerical data requires some care, since estimators are vulnerable to systematic errors if sample sizes are not sufficiently large. A variety of estimators have been developed (see for example [S1-S6]), many of which use Bayesian methods, exploiting prior knowledge (or assumptions) about the form of the underlying distributions in order to better estimate either entropies or mutual informations S 4; S 6; S 7. In this work, we use a simple method that we have tailored to the problem of interest here, based on the method of S 5.

In all measurements, we discretise the propensity, forming a histogram with bins of width $\delta\mu=\langle\mu\rangle/10$, where $\langle\mu\rangle$ is the mean propensity. The width of the bins is comparable with the numerical uncertainties in our estimates of the $\mu_{i}$, which are obtained from between 100 and 250 independent trajectories. For this reason, storing the propensities to greater accuracy than the bin width would not make our measurements of MI any more accurate – the binning does not introduce numerical artefacts, and is convenient in what follows. Further, since the same binning is used for all MI measurements, we are able to make a fair comparison between the different structural measurements shown in Figures 1 and 2 of main text. In the following, we use $m_{i}$ as an integer-valued label for the bin in which the propensity $\mu_{i}$ is located (for example, one may take $m_{i}=\lfloor\mu_{i}/\delta\mu\rfloor$, the largest integer that is less than or equal to $\mu_{i}/\delta\mu$).

We first describe the estimator that we use for calculating MI between two discrete-valued variables. We have in mind that $s_{i}$ is a structural observable with a discrete set of possible values, while $m_{i}$ is the propensity bin-index as described above. However, the discussion is general for joint distributions of discrete random variables. For each particle, suppose that we measure two integers $m_{i}$ and $s_{i}$. Then given data for $N_{\rm p}$ particles, let $n(m,s)$ be the number of particles with $(m_{i},s_{i})=(m,s)$; also let $n(m)=\sum_{s}n(m,s)$ be the number of particles with $m_{i}=m$, and similarly $n(s)=\sum_{m}n(m,s)$. The simplest MI estimate based on these data is the “plugin estimator”:

where the sum runs over all pairs $(m,s)$ for which $n(m,s)>0$. Given sufficient data, $I^{0}$ converges to the mutual information $I(m;s)$, however, this convergence is often quite slow, requiring very large $N_{\rm p}$ for an accurate estimate. In particular, even if the data set is constructed so that $m_{i}$ and $s_{i}$ are independent, one typically finds $I^{0}>0$, recovering $I^{0}\to 0$ only as $N_{\rm p}\to\infty$.

To see the reason for this, it is useful to write $I^{0}=H^{0}(m)-H^{0}(m|s)$ with

The key point is that for large enough data sets ($N_{\rm p}\to\infty$) one has $n(m)/N_{\rm p}\to p(m)$ by the law of large numbers, so that $H^{0}(m)$ is an entropy estimator for $H(m)=-\sum_{m}p(m)\log_{2}p(m)$. Similarly, $H^{0}(m|s)$ converges to a weighted sum of conditional entropies of the form $\sum_{m}p(m|s)\log_{2}p(m|s)$, as long as $n(s)\to\infty$ for all $s$. The difficulty is that the convergence of $H^{0}(m)$ and $H^{0}(m|s)$ to their respective limits are ruled by different large parameters ($N_{\rm p}$ and $n(s)$), and there are systematic errors associated with this convergence if these parameters are not large enough. In general, $H^{0}(m)$ and $H^{0}(m|s)$ both underestimate the relevant entropies, but the error on $H^{0}(m)$ is smaller, resulting in a positive systematic error for $I^{0}(m;s)$.

To reduce this effect, we define an alternative estimator

For each estimate of MI, we compute $I^{1}$ using several random null data sets (typically 100 realisations are sufficient). The average value of $I^{1}$ over the realisations provides our estimate of $I$ while the standard deviation among the values of $I^{1}$ gives an estimate on the uncertainty of this estimate. We therefore use this standard deviation as the error bar for the estimate of $I$. We emphasise that the $n(m|s)$ are determined by the original data and are the same for every realisation of the null data: it is the finite size of this original data set that introduces a finite uncertainty on estimates of $I$. This uncertainty is not reduced by repeated sampling over different null data sets, so it is the standard deviation of $I^{1}$ that gives the relevant error estimate, not the standard error.

Fig. S3 shows estimates for the MI between $n_{155}$ and the propensity, obtained by the estimators $I^{1}$ and $I^{0}$, for data sets of two different sizes. It can be seen that $I^{0}$ is not sufficient for the purposes used here, even for the larger data set, while $I^{1}$ gives a consistent estimate of the MI for data sets of both sizes considered. The estimate of the uncertainty based on $I^{1}$ is also self-consistent, in that error bars from independent estimates typically overlap with each other.

An alternative to (7) can be obtained by interchanging $s$ and $m$, since the MI is symmetric:

We use $I^{1}$ for all calculations of MI between discrete variables, but it is useful to define $I^{\prime 1}$ in preparation for later sections.

A.5.2 One discrete and one continuous variable

We now turn to the case where the structural variable of interest takes continuous values. The analogue of the estimator $I^{\prime 1}$ in (8) is

This converges to the required result as $n\to\infty$ because if $x^{1},x^{2},\dots x^{n}$ is an ordered list of independent random samples from $p(x)$ then, as $n\to\infty$, one has

where $\psi(n)$ is the digamma function, which satisfies $\psi(n+1)-\psi(n)=1/n$ and $\psi(1)=-\gamma$ where $\gamma=0.577\dots$ is Euler’s constant S 5.

Fig. S4 shows results using the estimator $I^{2}$. As with $I^{1}$, the uncertainty on the estimate of MI is reduced on including more data, and the systematic variation on increasing $N_{\rm p}$ is weak, indicating that the estimator is reliable.

References