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

We extend our previous work (Luo, S.; Thachuk, M. J. Phys. Chem. A 2021, 125, 64866497) on determining conservative potentials for lattice-like, coarse-grain (CG) mapping schemes to the case where the boundaries between different spatial regions are not sharply defined but are fuzzy. In other words, the system is divided into interpenetrating "subcells" such that atomistic particles continuously change their memberships as they move through space. This is done by using fuzzy switching functions to define overlapping regions between subcells with fractional particle occupations. In this case, a full mass matrix is required to describe the system, and its off-diagonal elements are nonzero and contribute to the CG potential. As the overlapping region increases in size, we observe the mass distribution transitions from a discrete spectrum, through an intermediate state, and finally to a continuous Gaussian-like function. We interpret this as a quantitative measure for signaling when a continuum-theory description of the system is appropriate. Nonzero correlations among all CG variables are calculated and are found to depend strongly on the degree of overlap. In particular, those for the diagonal mass elements decrease in magnitude, and there exists a specific value of the overlap for which the correlations are zero. Other correlations are strong only when the overlap is quite large, so there is a trade-off between the complexity of the interactions in the system and the degree of fuzziness between the subcells. However, if the number of particles in a subcell is large enough and the overlap is moderate, then the CG potential is found to be well-approximated by a generalized quadratic function. These results demonstrate the transition between atomistic and continuum resolutions in a system and have implications for designing CG schemes with mixed atomistic and continuum character.