Coarse-graining in quantum mechanics: Distinguishable and indistinguishable particles

Bottom-up coarse-grained (CG) modeling expands the spatial and temporal scales of molecular simulation by seeking a reduced, thermodynamically consistent representation of an atomistic model. Developments in CG theory have largely focused on CG modeling of atomistic systems that behave classically, while CG modeling of quantum systems has remained largely unexplored. We present in this work two fundamental advances in particle-based, bottom-up CG theory for systems obeying quantum statistical mechanics. We first expand the bottom-up CG formalism to include indistinguishable quantum particles, including bosons and fermions. We next introduce a variational optimization procedure for CG model parameterization, which is founded on the relative entropy minimization (REM) principle, and then bridge the classical and quantum REM methods through a semiclassical expansion in terms of the Feynman path centroid. We provide numerical examples of REM CG models of distinguishable and indistinguishable quantum systems, including as examples a harmonically trapped bosonic system and liquid water. The theoretical results presented here constitute a means to accelerate simulating thermal quantum systems, ranging from distinguishable particle systems at higher temperatures to quantum indistinguishable particle systems at lower temperatures.