Understanding and controlling regime switching in molecular diffusion

Diffusion can be strongly affected by ballistic flights (long jumps) as well as long-lived sticking trajectories (long sticks). Using statistical inference techniques in the spirit of Granger causality, we investigate the appearance of long jumps and sticks in molecular-dynamics simulations of diffusion in a prototype system, a benzene molecule on a graphite substrate. We find that specific fluctuations in certain, but not all, internal degrees of freedom of the molecule can be linked to either long jumps or sticks. Furthermore, by changing the prevalence of these predictors with an outside influence, the diffusion of the molecule can be controlled. The approach presented in this proof of concept study is very generic and can be applied to larger and more complex molecules. Additionally, the predictor variables can be chosen in a general way so as to be accessible in experiments, making the method feasible for control of diffusion in applications. Our results also demonstrate that data-mining techniques can be used to investigate the phase-space structure of high-dimensional nonlinear dynamical systems.

Figure 1

(a) The prototype system, a benzene molecule on a graphite substrate [9, 10]. The internal dynamics consist of bond stretching, bending, and torsion. (b) The 12 vibrational eigenmodes of the linearized system (numbered arbitrarily). The arrows indicate the directions of the vibrations. For modes 10, 11, and 12, which are torsion modes, the vibrations are out-of-plane. The motion for these modes (away from the reader or towards them) is indicated with respect to the center of the hexagon.

Figure 2

Detecting ballistic flights and subdiffusive sticks. (a) Example of trajectories of a simulated benzene molecule on a graphite substrate in real space including ballistic flights (long jumps) and subdiffusive sticks. In between long jumps the mean-square displacement of the center of mass of the molecule grows linearly with time and diffusion is much slower. (b) A short section of the trajectory in velocity space with the long jump highlighted. (c) A short section of the trajectory in real space with a stick highlighted.

Figure 3

Estimating the density of jump and stick time length distributions by fitting several heavy-tailed distributions. A loglikelihood ratio test revealed that the truncated power law (TPL) is the most appropriate fit among the distributions tested [power law (PL), TPL, stretched exponential, lognormal]. Note that the difference between the PL fit and the lognormal fit are not visible. Consequently only the PL fit is labeled.

Figure 4

Examples of predictor distributions for modes that are linked (a), (b), (e), (f) or not linked (others) to the occurence of events. Histograms of CPDFs $p(s_n\ge \tau |y_n)$ are plotted as lines over the marginal probability distribution $p\left(y_n\right)$ (plotted as black bars).

Figure 5

Nowcasting ballistic jumps and subdiffusive sticks. (a) An example for an ROC curve. (b) and (c) AUCs for different predictor variables ${\mu }_n^i$ and ${\sigma }_n^i$, estimated for nowcasts, i.e., lead time $l=0$. The 95% confidence intervals are shown as shaded areas in (b) and (c).

Figure 6

Forecasting long jumps and sticks. (a) and (b): Comparison of nowcasts $\left(l=0\right)$ and forecasts $\left(l>0\right)$, both made using the standard deviation as a predictor. Additionally, we separated the data set into a test and a training part (90% for training, 10% for testing) which is indicated by the factor $f_c=9.0$. Here 95% confidence intervals were also estimated through random predictions with parameters $l>0$ and $f_c=0.9$.

Figure 7

Damping mode 1 induces a long stick; i.e., the molecule remains within a region of the size of a few unit cells. The trajectory is plotted in black without damping and in red (gray) after the damping started. Note that both parts of the simulation represent the motion of the molecule during time intervals of equal length, i.e., 242 ps before the damping started and 242 ps with damping of mode 1. See Ref. [30] for this simulation in the form of a movie.

Figure 8

Damping of mode 10 does not induce any qualitative change of the molecule's motion. The molecule continues to diffuse over a wide range of the substrate, which is in contrast to the stick induced by the damping of mode 1 (see Fig. 7).

Figure 9

Diffusion versus damping in a system with a single damped mode as indicated. For degenerate pairs, only one mode is shown. There is a large drop in the diffusion coefficient when the modes with strong predictors are damped.