Complex temporal patterns in molecular dynamics: A direct measure of the phase-space exploration by the trajectory at macroscopic time scales

Computer simulated trajectories of bulk water molecules form complex spatiotemporal structures at the picosecond time scale. This intrinsic complexity, which underlies the formation of molecular structures at longer time scales, has been quantified using a measure of statistical complexity. The method estimates the information contained in the molecular trajectory by detecting and quantifying temporal patterns present in the simulated data (velocity time series). Two types of temporal patterns are found. The first, defined by the short-time correlations corresponding to the velocity autocorrelation decay times $\left(\le 0.1\text{ps}\right)$, remains asymptotically stable for time intervals longer than several tens of nanoseconds. The second is caused by previously unknown longer-time correlations (found at longer than the nanoseconds time scales) leading to a value of statistical complexity that slowly increases with time. A direct measure based on the notion of statistical complexity that describes how the trajectory explores the phase space and independent from the particular molecular signal used as the observed time series is introduced.

Figure 1

(Color) Approximations for generating partitions obtained using the method by Buhl and Kennel [33] for the cross section of the hydrogen velocity space for the partitions corresponding to two-, three-, four-, and five-symbols alphabet.

Figure 2

(Color online) Velocity autocorrelation function for oxygen (dashed line) and hydrogen atoms of two water molecules calculated as time average over 2 ns. The curves for different atoms of the same type are practically indistinguishable.

Figure 3

Spectrum of the velocity $x$ component of a hydrogen atom in bulk SPC water at 300 K.

Figure 4

(Color online) Dipole moment orientation ($\varphi$ and $\theta$ angles in the spherical coordinates) as a function of time for the selected water molecule.

Figure 5

Diffusion coefficient of the $x$ component of a hydrogen atom in bulk SPC water at 300 K.

Figure 6

Non-Gaussianity parameter for three Cartesian components of the displacement for a hydrogen atom ($x$—circles, $y$—squares, $z$—triangles) and phase-shuffled surrogate for the $x$ component (filled squares).

Figure 7

(Color online) Statistical complexity against time for the hydrogen velocity signal and the surrogate. The curves, from bottom to top, correspond to the values of the history length $l$ from 2 to 11. The $l=11$ curve does not settle on the logarithmic part within the shown area but seems to follow the same trend. The thick line is the $C_{\mu }$ values for the phase-shuffled surrogate signal $\left(l=9\right)$. For all curves the alphabet size $K$ is equal to 3.

Figure 8

Statistical complexity vs the length of histories (dimension of the phase space) for the original data (solid line) and the surrogate (dashed line), $t=30\text{ns}$.

Figure 9

(Color online) $h_Q$ values (indicated on the right) for SPCE $\left(h_Q=0.69\right)$, TIP3P $\left(h_Q=0.78\right)$, and SPC $\left(h_Q=0.71\right)$ models of water. The values for the SPC model at 275, 300, and 380 K are labeled 0.70, 0.71, and 0.76, respectively.

Figure 10

Statistical complexity $C_{\mu }$ and number of states $n_{st}$ for the temporal pattern-shuffled surrogate data with $l=9$. From bottom to top, $n_r=3,4,10,20,50$, and the original data.

Figure 11

Clustering of the causal states for the hydrogen atom velocity time series into core (diamonds) and noncore (triangles) classes. Parameter $G$ is plotted vs occurrence rates of corresponding causal states.

Figure 12

Power spectra [(a) and (d)] and histograms of recurrence times [(b), (c), (e), and (f)] for typical causal states belonging to different types: a core state [(a)–(c)] and a noncore state [(d)–(f)]. The histograms on (c) and (f) are zoomed and smoothed fragments of those shown in (b) and (e). Spectra in (a) and (d) are the functions of inverse frequency.

Figure 13

(Color) $h_Q$ values (indicated on the right) for various observables: black—the hydrogen velocity, red—the oxygen velocity, and blue—the instantaneous temperature.