Conservative Potentials for a Lattice-Mapped Coarse-Grained Scheme

Abstract

The conservative potential, arising from a coarse-grain (CG) mapping scheme for nonbonded atomistic particles, is studied. This is a bottom-up approach from first-principles that maps atomistic particles to fluid element-like subcells whose centers lie on a regular, cubic lattice. Unlike standard CG mapping schemes, the current one uses dynamic labeling which on-the-fly changes the CG labels of the particles. The subcells can also be different sizes and shapes, in principle. Equilibrium atomistic molecular dynamics trajectories for different Lennard-Jones fluids are calculated and converted to CG ones, from which CG probability distribution functions are calculated. Correlation studies show position and mass CG variables are uncoupled in a given subcell, as are different vector components of position. Furthermore, the strongest coupling occurs with neighboring cells in specific directions, and the resulting distribution is well described by a multivariate Gaussian. This implies the CG potential has a generalized quadratic form, whose derivative can be determined analytically. A microscopic rationalization is provided for the signs and relative magnitudes of different correlation coefficients, and in some cases, a connection is made with bulk properties of the fluid. We argue the generalized quadratic form should be robust to changes in the particulars of the CG scheme, as well as the nature of the atomistic intermolecular potential. Only a few potential parameters need to be calculated from the underlying atomistic system. This is significant because it indicates the transferability of this form to other, more complex systems. This transferability will be tested in future work, where mapping schemes with fuzzy boundaries will be considered.

Bottom-up Coarse-Graining: Principles and Perspectives

“…Recently, a “dynamic mapping” scheme was developed by mapping velocities instead of configurations, which only requires the initial configuration from smoothed centroidal Voronoi tessellations . To note, this mapping scheme is based on a Lagrangian description to track individual fluid particles, but a complementary Eulerian description can also be established in a similar vein . In turn, this new approach allows stable propagation of the CG blobs over time, indicating its applicability to various fluids, e.g., heterogeneous multiphase systems.…”

Conservative Potentials for a Lattice-Mapped, Coarse-Grain Scheme with Fuzzy Switching Functions

“…Lynn and Thachuk derived the coarse-grain (CG) equations of motion (EoM) for a general, lattice-like mapping scheme using Mori–Zwanzig (MZ) theory. , Such an approach produces dynamically correct EoM because MZ theory is simply a way to partition the Liouville operator in the evolution equations of dynamical variables. Luo and Thachuk studied the behavior of the conservative terms in this CG EoM using a Heaviside switching function . They found that the CG potential has a quadratic form whereas the effect of particles crossing subcell boundaries shows up as linear correlations between CG variables .…”

“…Luo and Thachuk studied the behavior of the conservative terms in this CG EoM using a Heaviside switching function . They found that the CG potential has a quadratic form whereas the effect of particles crossing subcell boundaries shows up as linear correlations between CG variables . The CG diagonal mass elements are discrete, but their distributions can be described by a multivariate Gaussian .…”

“…Imagine that a system is divided into N cubic subcells with edge length L * and . The form of d i I for atomistic particle i and CG variable I is where γ is a Cartesian component, γ ∈{ x , y , z }. The switching function h ( s ) must change from 0 to 1 within the interval , and some choices are for the lattice, tanh, and linear mappings, respectively.…”

“…The switching function h ( s ) must change from 0 to 1 within the interval , and some choices are for the lattice, tanh, and linear mappings, respectively. The Heaviside switching function, given by h lattice ( s ), was studied previously by Luo and Thachuk . The α parameter controls the overlap or “fuzziness” of h ( s ) at the boundary of a subcell and must be bounded as 0 ≤ α ≤ 1.…”