Bottom-up Coarse-Graining of Fluid Particles via Time-Evolving Fuzzy Clustering: A Pressure-Correction from Membership Evolution

Coarse-graining is a powerful tool for bridging atomistic and mesoscopic scales in fluid particle systems. However, fixed coarse-grained (CG) mappings do not account for the fact that fluid particles do not form persistent groups. Here we propose an entropy-regularized fuzzy clustering approach with temporal smoothness constraints, and then examine in detail the role of the time-evolving fuzzy membership degrees throughout the coarse-graining process. Entropy regularization controls the level of membership fuzziness, while the temporal smoothness constraints enhance the continuity of cluster trajectories. Within a bottom-up force-matching framework, the interactions between clusters are decomposed into a particle-interaction term (the weighted sum of particle-particle interactions) and a membership-evolution term (arising from the temporal variation of membership degrees). Analyses based on a Lennard–Jones (L-J) fluid particle system show that an intermediate fuzziness best preserves local structural properties, and that the membership-evolution term provides a repulsive contribution. Moreover, CG dynamics simulations demonstrate that including the membership-evolution term effectively restores the system pressure, which can be interpreted as a pressure-correction mechanism. This finding provides a physical perspective on how microscopic interactions transform into mesoscopic effective interactions between fluid particles, which could be beneficial for modelling atmospheric dynamical processes.