Developing a quantum theory of multiscale coarse-graining

For the last 15 years, coarse-grained modeling of complex chemical, material and biomolecular systems has increased in popularity. This type of modeling reduces the computational resources needed for a molecular simulation by reducing the number of degrees of freedom and eliminating fine interaction details. Coarse-graining seeks to retain the system’s essential physical aspects while increasing computational efficiency.

While coarse-grained models and methods had previously been built upon classical statistical mechanics, authors of an article in The Journal of Chemical Physics developed a theory and numerical methodology of coarse-graining for the quantum mechanical regime. They describe their process of generalizing the multiscale coarse-graining method to quantum Boltzmann statistics.

Their approach begins by defining the Hamiltonian for the total energy of the high-resolution fine-grained model. The normalized thermal density matrix in quantum Boltzmann statistics is expressed in terms of the imaginary-time Feynman path integral in the coordinate representation. The coarse-grained model describing the same system, but with fewer degrees of freedom, is then mapped from the fine-grained model.

According to author Gregory Voth, they combined the Feynman path integral description of quantum statistical mechanics with a systematic variational force-matching method for the “bottom-up” development of coarse-grained models. They then formulated a fully systematic quantum mechanical theory for the coarse-graining of molecular systems. Finally, two numerical examples — a series of one-dimensional multi-body model problems, and a modified water model — were used to validate the method and demonstrate its application to molecular systems.

Voth hopes to see this methodology evolve into a practical algorithm generally applicable to the coarse-grained quantum simulation of molecular systems at equilibrium.

Source: “Quantum theory of multiscale coarse-graining,” by Yining Han, Jaehyeok Jin, Jacob W. Wagner, and Gregory A. Voth, The Journal of Chemical Physics (2018). The article can be accessed at https://doi.org/10.1063/1.5010270 .