Coarse-Grained Mapping of Fluid Particles via Evolutionary Fuzzy Clustering: Membership-Evolution Term as a Pressure Correction Mechanism.

Coarse-graining is an effective approach for bridging atomistic and mesoscopic descriptions of fluid particle systems. However, fixed coarse-grained (CG) mappings do not account for the unbundled nature of fluid particles. We propose an entropy-regularized fuzzy clustering method with temporal smoothness constraints, examining in detail the role of the evolution of fuzzy particle-cluster membership degrees throughout the coarse-graining process. Entropy regularization controls the level of spatial fuzziness, while the temporal smoothness constraints enhance the continuity of cluster position evolution. Within a bottom-up force-matching framework, the interactions between clusters are decomposed into two contributions: a particle-interaction term, which is the weighted sum of interactions between particles, and a membership-evolution term, which originates from the temporal variation of membership degrees. Analyses based on the Lennard-Jones (L-J) fluid particle system and the water molecule system show that an intermediate level of fuzziness yields the most pronounced structural features in the radial distribution functions. The particle-interaction term exhibits system-dependent characteristics, whereas the membership-evolution term consistently provides a repulsive contribution across different systems. Moreover, CG dynamics simulations of the L-J fluid demonstrate that including the membership-evolution term effectively restores the system pressure, which could be interpreted as a pressure correction scheme. This finding provides a physical perspective on the transition from microscopic particle interactions to macroscopic fluid pressure constraints and reveals a bottom-up origin for incorporating additional pressure corrections into fluid CG dynamics, which could be beneficial for the future design of coarse-graining strategies.