# A Bottom-Up Field-Theoretic Framework via Hierarchical Coarse-Graining: Generalized Mode Theory

## TL;DR

This paper introduces a hierarchical bottom-up framework for constructing field-theoretic models of molecular liquids directly from microscopic atomistic interactions, enabling scalable simulations beyond traditional top-down approaches.

## Contribution

It generalizes the Hubbard-Stratonovich transformation to arbitrary pair potentials and develops a systematic bottom-up method for deriving field theories from atomistic models.

## Key findings

- Hierarchical coarse-graining maps atomistic interactions to field models.
- Generalized mode theory extends field-theoretic modeling to non-positive-definite kernels.
- Provides a foundation for scalable, bottom-up molecular simulations.

## Abstract

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.

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Source: https://tomesphere.com/paper/2508.20025