A bottom-up field-theoretic framework via hierarchical coarse-graining: Generalized mode theory

Multiscale simulations facilitate the efficient exploration of large spatiotemporal scales in chemical and physical systems, yet particle-based simulations become prohibitively expensive at time and length scales beyond the molecular level. Field-theoretic simulations offer an attractive alternative, but most existing formulations rely on top-down approximations and are not systematically connected to atomistic interactions. Here, we present a hierarchical bottom-up framework for constructing auxiliary field representations of molecular liquids directly from microscopic models. We introduce a hierarchical coarse-graining framework that constructs field-theoretic models directly from atomistic liquids. The method first maps atomistic interactions to coarse-grained center-of-mass potentials and regularizes short-range divergences through a perturbative expansion in reciprocal space. Building on the auxiliary field formulation developed in polymer field-theoretic simulations, we then generalize the Hubbard-Stratonovich transformation to arbitrary pair potentials by separating positive and negative Fourier modes and introducing two auxiliary fields. The resulting generalized mode theory extends bottom-up field-theoretic modeling beyond positive-definite kernels and is compatible with existing field-theoretic sampling strategies. By combining formal derivations with numerical regularization and mode-truncation procedures, this work provides the theoretical foundation for scalable, bottom-up field-theoretic simulations of molecular systems.