Efficient $d$-dimensional molecular dynamics simulations for studies of the glass-jamming transition

We develop an algorithm suitable for parallel molecular dynamics simulations in $d$ spatial dimensions and describe its implementation in $\mathrm{C}++$. All routines work in arbitrary $d$; the maximum simulated $d$ is limited only by available computing resources. These routines include several that are particularly useful for studies of the glass-jamming transition, such as SWAP Monte Carlo and FIRE energy minimization. The scalings of simulation runtimes with the number of particles $N$ and number of simulation threads $n_{\mathrm{threads}}$ are comparable to popular molecular dynamics codes such as LAMMPS. The efficient parallel implementation allows simulation of systems that are much larger than those employed in previous high-dimensional glass-transition studies. As a demonstration of the code's capabilities, we show that supercooled $d=6$ liquids can possess dynamics that are substantially more heterogeneous and experience a breakdown of the Stokes-Einstein relation that is substantially stronger than previously reported, owing at least in part to the much smaller system sizes employed in earlier simulations.

Figure 1

Basic structure of the buildneighborlist() routine. Parallelization is achieved by dividing its outermost loop equally amongst the $n_{\mathrm{threads}}$ concurrent threads.

Figure 2

Scaling of performance with $N$ and $n_{\mathrm{threads}}$ for $3\le d\le 6$. Runtimes for panels (a) and (b) are for $n_{\mathrm{threads}}=8$ simulations run on an iMac with a single 10-core Intel Core i9 CPU (3.6 GHz). Results for $N<{10}^5$ and $d>4$ are not shown here because these $N$ give $n_{\mathrm{cell}}\le 4$ and thus the scaling of the time devoted to VL builds is closer to $\mathcal{O}\left(N^2\right)$ than $\mathcal{O}\left(N\right)$. Runtimes for panel (c) are for $N={10}^6$ simulations on cluster nodes with two 12-core Intel Xeon E5-2650 CPUs (2.2GHz). The larger efficiencies for $n_{\mathrm{threads}}>12$ probably arise from distributing the computational effort to more than one CPU socket. Solid lines in panels (a) and (c) are guides to the eye, and the dotted vertical line in panel (b) indicates the iMac's L3 cache size (20 MB).

Figure 3

Scaling of performance with $d$. Runtimes are for $n_{\mathrm{threads}}=8$ simulations run on an iMac with a single 10-core Intel Core i9 CPU (3.6 GHz).

Figure 4

Dynamics of supercooled $d=6$ liquids for $1.02{\varphi }_d^{*}\le \varphi \le 1.05{\varphi }_d^{*}$ and $k_{B}T=0.25\tilde{\epsilon }$. Panels (a)–(c) respectively show the mean-squared displacement, overlap parameter $f_{\mathrm{ov}}$ [Eq. (13)], and non-Gaussian parameter [Eq. (12)]. Straight gray lines in panel (a) are guides to the eye, and dotted vertical lines in panels (b) and (c) indicate $t={\tau }_{\alpha }$. All times are given in units of $\tau =\sqrt{\tilde{m}{\tilde{\sigma }}^2/\tilde{\epsilon }}$ (Sec. 2a).

Figure 5

Breakdown of the Stokes-Einstein relation in supercooled $d=6$ liquids at $k_{B}T=0.25\tilde{\epsilon }$. Symbols show MD data while lines show fits to $D{\tau }_{\alpha }\sim {\tau }_{\alpha }^{\omega }$ and $D{\tau }_{\alpha }\sim {\tau }_{\alpha }^y$. The inset contrasts results for $N={10}^5$ (same symbols shown in the main panel) to $N=5000$ results for selected $1.01{\varphi }_d^{*}\le \varphi \le 1.055{\varphi }_d^{*}$ (open circles); all results for $N=5000$ are averaged over 10 independently prepared systems. Note that the range of ${\tau }_{\alpha }$ depicted here is almost identical to that considered in Ref. [32].

Figure 6

Heterogenous dynamics of supercooled 50:50 1:1.4 bidisperse $d=6$ liquids. Following Refs. [32, 63], we calculated ${\alpha }_2\left(t\right)$ and $f_{\mathrm{ov}}\left(t\right)$ separately for small and large particles. Panel (a) shows ${\alpha }_2\left(t\right)$ for the large particles, while panel (b) shows $f_{\mathrm{ov}}\left(t\right)$ for all particles.

Figure 7

Non-Gaussian caging of the large particles in supercooled 50:50 1:1.4 bidisperse $d=6$ liquids. Panels (a) and (b) show results for $\varphi =1.03{\varphi }_d^{*}$ and $\varphi =1.06{\varphi }_d^{*}$; large particles in these systems respectively have ${\tau }_{\alpha }\simeq 7.8\times {10}^2$ and ${\tau }_{\alpha }=7\times {10}^3$. The dashed lines show fits to exponential tails with $\lambda =0.301\tilde{\sigma }$ and $\lambda =0.363\tilde{\sigma }$.