Theory of wavelet-based coarse-graining hierarchies for molecular dynamics

We present a multiresolution approach to compressing the degrees of freedom and potentials associated with molecular dynamics, such as the bond potentials. The approach suggests a systematic way to accelerate large-scale molecular simulations with more than two levels of coarse graining, particularly applications of polymeric materials. In particular, we derive explicit models for (arbitrarily large) linear (homo)polymers and iterative methods to compute large-scale wavelet decompositions from fragment solutions. This approach does not require explicit preparation of atomistic-to-coarse-grained mappings, but instead uses the theory of diffusion wavelets for graph Laplacians to develop system-specific mappings. Our methodology leads to a hierarchy of system-specific coarse-grained degrees of freedom that provides a conceptually clear and mathematically rigorous framework for modeling chemical systems at relevant model scales. The approach is capable of automatically generating as many coarse-grained model scales as necessary, that is, to go beyond the two scales in conventional coarse-grained strategies; furthermore, the wavelet-based coarse-grained models explicitly link time and length scales. Furthermore, a straightforward method for the reintroduction of omitted degrees of freedom is presented, which plays a major role in maintaining model fidelity in long-time simulations and in capturing emergent behaviors.

Figure 1

Example of a weighted graph Laplacian derived from a simple weighted graph.

Figure 2

Dyadic trees generated by multiresolution formalism using a filter $T$. Shaded spaces are final subspaces of the multiresolution. All other spaces are intermediates of the construction. Shown on top is the generic scheme and on bottom is an example using butane. The final spaces are in gray; diamonds (red) and squares (blue) denote opposite signs of weights in the construction from the finer scale.

Figure 3

Chemical structure of polytetrazine.

Figure 4

Simulated 3D structure of a 100 800-atom polyethylene single crystal.

Figure 5

Fourier transform of the $y$ component [$\text{FT}\left(y\right)$] of 1000 atoms in crystalline PE (100 800 atoms) with frequency $\omega$, excluding zero frequency to allow detail at other frequencies. The MD simulations were at $500\mathrm{K}$ and $1\mathrm{atm}$. Shown on the left are individual power spectra per atom and on the right is the power spectrum of magnitude of optimal representation.

Figure 6

Fourier transform of the $y$ component [$\text{FT}\left(y\right)$] of a 100 800-atom crystalline PE sampled at $1\mathrm{fs}$ with frequency $\omega$. Three scales are shown: (a) the first (finest) scale, (b) the fifth scale, and (c) the 12th scale (out of 25).

Figure 7

Dyadic tree generated by separable coarsenings. Shaded spaces are final subspaces of the multiresolution. All other spaces are intermediates of the construction with ${\gamma }_1={\mu }_A\oplus {\mu }_A^{\perp }$ and ${\gamma }_2={\mu }_A\oplus {\mu }_B$.