Atomistic simulations become computationally expensive for large systems and long timescales, motivating the use of coarse-graining (CG) methods that reduce the number of degrees of freedom by grouping atoms into beads. This approach enables the efficient study of complex systems, particularly in biology. The challenge, however, is that once we simplify the system, we must still maintain the underlying physics by constructing accurate coarse-grained potentials.

Conventional CG methods use fixed particle groupings to define coarse-grained variables. Lynn and Thachuk\(^1\) developed a position-dependent mapping in which particles dynamically contribute to CG variables based on their positions, allowing the representation of systems where particles undergo diffusion. Building on this framework, Luo and Thachuk\(^{2}\) developed a lattice-based CG approach for fluids, in which a generalized quadratic potential with multivariate Gaussian statistics represents the interactions. In this formulation, the CG potential can be written as

\begin{equation} V(\mathbf{x}) = \frac{kT}{2} (\mathbf{x} - \boldsymbol{\mu}_x)^T \boldsymbol{\Sigma}^{-1} (\mathbf{x} - \boldsymbol{\mu}_x) \end{equation}

where \(\mathbf{x} = (\mathbf{M}, \mathbf{W})^T\), \(\boldsymbol{\mu}_x\) is its mean, and \(\boldsymbol{\Sigma}\) is the covariance matrix.

Subsequent developments extended the framework by incorporating fuzzy switching functions, which allow for smooth transitions of particles between adjacent regions.\(^{3}\) Additionally, they derived analytic expressions for CG potential parameters that can be directly derived from the fluid properties.\(^{4}\)

In this work, we study the behavior of the CG equations of motion using the expressions developed by Luo and Thachuk. The system is represented by partitioning the cubic box into \(n\) subcells of edge length \(2\ell\). A mapping was established between each CG subcell’s three-dimensional coordinates and a one-dimensional index, and vice versa. To eliminate surface effects and better approximate bulk behavior, periodic boundary conditions were applied. This introduced an indexing challenge near the boundaries, where the separation between two cells is defined as the minimum distance, which may occur within the same image or across periodic images. Using these corrected displacements, cell-cell interactions are classified by relative spatial configuration into closest, \(\sqrt{2}\), and \(\sqrt{3}\) neighbors, corresponding respectively to pairs differing by \(\pm 2\ell\) in one, two, or three Cartesian coordinates.

Using the lattice mapping framework, the elements of the \(\Sigma\) matrix are evaluated through correlations among the \(W\) variables, between \(W\) and \(M\), and among the \(M\) elements, together with the mean \(\langle M \rangle\). Each correlation function is decomposed into one-particle and two-particle contributions, which are evaluated separately and then summed to obtain the total correlation. Initial validation is performed for a single-component fluid, with extensions to mixtures left for future work.

[1] Lynn, H.; Thachuk, M. Equations of Motion for Position-Dependent Coarse-Grain Mappings Obtained with Mori–Zwanzig Theory. J. Chem. Phys. 2019, 150, 024108.
[2] Luo, S.; Thachuk, M. Conservative Potentials for a Lattice-Mapped Coarse-Grained Scheme. J. Phys. Chem. A 2021, 125, 6486–6497.
[3] Luo, S.; Thachuk, M. Conservative Potentials for a Lattice-Mapped, Coarse-Grain Scheme with Fuzzy Switching Functions. J. Phys. Chem. A 2022, 126, 4517–4527.
[4] Luo, S.; Thachuk, M. Analytic Expressions for Correlations in Coarse-Grained Simple Fluids. J. Chem. Phys. 2023, 159, 224114.