Exploring glassy dynamics with Markov state models from graph dynamical neural networks

Using machine learning techniques, we introduce a Markov state model (MSM) for a model glass former that reveals structural heterogeneities and their slow dynamics by coarse-graining the molecular dynamics into a low-dimensional feature space. The transition timescale between states is larger than the conventional structural relaxation time ${\tau }_{\alpha }$, but can be obtained from trajectories much shorter than ${\tau }_{\alpha }$. The learned map of states assigned to the particles corresponds to local excess Voronoi volume. These results resonate with classic free volume theories of the glass transition, singling out local packing fluctuations as one of the dominant slowly relaxing features.

Figure 1

(a) Markov state model relaxation times ${\tau }_{\mathrm{MSM}}$ (empty circles) and $\alpha$-relaxation times ${\tau }_{\alpha }$ (solid circles) as a function of inverse temperature. Crosses show the timescales obtained directly from the decay of autocorrelation functions of the smoothed Voronoi volume (see below). The dashed curve shows a Vogel-Fulcher-Tammann fit, ${\tau }_{\alpha }\sim exp[A/(T-T_0)]$, for $T\ge 0.45$. The obtained $T_0$ value is 0.34. Arrhenius fits at high and low temperatures are also shown with red and blue solid lines, resulting in energy barriers of 3.27 and 2.21 LJ energy units, respectively. In the gray-shaded area, the system is out of equilibrium. The inset shows the ratio of ${\tau }_{\mathrm{MSM}}$ to ${\tau }_{\alpha }$ (triangles) and also to the collective overlap time ${\tau }_O$ (circles, see text). (b) Forward and backward reaction rates, $k_{0\to 1}$ and $k_{1\to 0}$, with Arrhenius fits at high and low $T$. (c) Free-energy difference between states (circles) and ratio of mean state probabilities (crosses, see text).

Figure 2

(a) Example of a state map at $T=0.3$. Particles $i$ are colored according to their probability $p_i$ of being in state 0, as assigned by the graph dynamical network. (b) Probability distribution of the normalized probability $\delta p_i=(p_i-\langle p\rangle )/\sqrt{\langle {(p-\langle p\rangle )}^2\rangle }$ of majority type particles being in state 0 at three different temperatures. Distributions are averaged over 100 equally spaced snapshots of the trajectory. The dashed curve shows the Gaussian fit. The inset shows mean (circles) and standard deviations (crosses) as a function of temperature. (c) Normalized spatial correlation functions $C\left(r\right)=\langle \delta p\left(0\right)\delta p\left(r\right)\rangle /\langle \delta p\left(0\right)\delta p\left(r_{\min }\right)\rangle$ of the state maps at the same temperatures as in panel (b), where the average is over all particles separated by a distance $r$ and $r_{\min }=1$. Measurements are obtained using 20 different quench realizations to desired temperatures and an isothermal equilibration of 500 LJ time. The snapshot at the end of this equilibration period is then used to extract the data points in panel (c). The inset shows the correlation length of the state map $\zeta ={\sum }_{r_{\min }}^{r_{\mathrm{cut}}}rC\left(r\right)\delta r/{\sum }_{r_{\min }}^{r_{\mathrm{cut}}}C\left(r\right)\delta r$, with $\delta r=0.5$ and $r_{\mathrm{cut}}=7$. Visualizations are performed using ovito [25].

Figure 3

Comparison between (a) the state of majority particles and (b) the coarse-grained Voronoi volume for a snapshot at $T=0.4$. The Voronoi volume of particles is lightly smoothed by averaging the Voronoi volume of a particle with its neighbors within $r=\zeta$. Particles are colored using $min\left(p_i\right)=0$ and $max\left(p_i\right)=1$ in panel (a) and according to $min\left(v_{i,s}\right)=0.81$ and $max\left(v_{i,s}\right)=0.95$ in panel (b). Minority particles are shown in black. (c) Correlation between probability of $p_i$ and $v_{i,s}$ at three different temperatures. The Pearson correlation factor between $p_i$ and $v_{i,s}$ is equal to 0.88, 0.85, and 0.81 at $T=0.3, T=0.5$, and $T=0.7$ respectively. Error bars show the standard error of the mean.