A Field-Theoretical Paradigm via Hierarchical Coarse-Graining: I. Generalized Mode Theory

Multiscale computer simulations facilitate the efficient exploration of large spatiotemporal scales in chemical and physical systems. However, molecular simulations predominantly rely on particle-based representations, which become computationally prohibitive when applied beyond the molecular scale. Field-theoretical simulations have emerged as an alternative to overcome these limitations. Despite their applicability in mesoscopic simulations, these approaches are typically derived from top-down principles, leaving a gap in the bottom-up derivation of field-theoretical models for targeted molecular systems. This work presents a systematic strategy for constructing statistical field models for molecular liquids from the atomistic level, marking a critical step toward bridging particle-based and field-theoretical modeling techniques. By introducing molecular coarse-grained models as an intermediate step, this hierarchical approach reduces the complexity of the atomistic-to-field-theoretical transformation while preserving important microscopic structural correlations. We systematically derive field-theoretical models in reciprocal space for both canonical and grand canonical ensembles using the Hubbard-Stratonovich transformation. By incorporating additional fields for both positive and negative Fourier modes, our generalized mode theory extends the applicability of bottom-up field-theoretical models beyond the conventional Hubbard-Stratonovich transformation, which can be limited to positive Fourier modes. Furthermore, we introduce an efficient perturbative approach for approximating Fourier modes of molecular interactions, reducing computational cost and allowing for integrating complex molecular interactions into the field-theoretic description.