### Project Summary

Molecular Dynamics (MD) has become a powerful tool supporting our understanding and interpretation of microscopic properties of matter. Despite its success, its computational efficiency still leaves ample room for improvement. In particular, when several scales are tightly connected and cannot be separated, as in the case of local interactions of a solvent with a complex solvated molecule, then the treatment with MD can become very expensive. Aiming for enhanced computational efficiency one would want to interface a highly resolved atomistic model for the detailed physico-chemical interactions of solvent and solvated molecules with a coarse-grained solvent model in such a way that the expensive detailed model has to be invoked only in the very vicinity of the solvated molecule. With proper coupling strategies such an approach would yield efficiency as well as an accurate description of the pertinent micro-macro interactions.

Related approaches have been proposed in the chemistry and physics literature in recent years. Here we focus on the Adaptive Resolution Simulation method (AdResS), which has shown convincing robustness technically and conceptually. AdResS interfaces given spatial regions in which different levels of coarse-graining are invoked with each other by changing the resolution of molecules according to their position in space. The original scheme is based on empirical, but well justi.ed, physical/chemical principles, yet the reason for its performance remains understood merely at an intuitive level.

An AdResS simulation involves interactions across a cascade of length and time scales. The former include the overall system dimensions, the sizes of the coarse-grained and atomistic regions, the thickness of the transition zone, and the respective particle interaction lengths and mean free paths. Versions of molecular dynamics equations serve as both the root and reduced models, however the techniques for coarse-graining and, in part, the coarse-grained models themselves still lack mathematical rationalization.

One goal of this project is a mathematical rationalization of the approximations invoked in AdResS. The typical set-up for AdResS applications is very similar in some respect to partial differential equation (PDE) problems amenable to matched asymptotic expansions: The local importance of particular detailed processes in a core region diminishes at farther distances, and far from this core one can safely describe the system by a reduced equation set, [KC96], [10]. Although MD is not phrased in terms of PDEs, we aim at substantiating and exploiting these apparent similarities of the two problem settings for the formulation of a mathematical framework for AdResS.

Potential gains from a successful mathematical analysis of this type include a systematic quality control for AdResS as well as hints at possible improvements or enhancements of the technique, e.g., to non-equilibrium situations and to continuum models. The development of a well-founded AdResS technique for non-equilibrium situations is a long-term goal of the project, with a formulation for a Grand Canonical Ensemble as an intermediate step.

### Project publications

Junghans, C. and Agarwal, A. and Delle Site, L.
(2017)
*Computational Efficiency and Amdahl's law for the Adaptive Resolution Simulation Technique.*
Computer Physics Communications, 215
.
pp. 20-25.
ISSN 0010-4655

Seshaditya, A. and Ghiringhelli, L. M. and Delle Site, L.
(2017)
*Levy-Lieb-based Monte Carlo study of the dimensionality behaviour of the electronic kinetic functional.*
Computationa, 5
(2).
pp. 1-10.
ISSN 2079-3197

Agarwal, A. and Clementi, C. and Delle Site, L.
(2017)
*Path Integral-GC-AdResS simulation of a large hydrophobic solute in water: A tool to investigate the interplay between local microscopic structures and quantum delocalization of atoms in space.*
Physical Chemistry Chemical Physics, 19
.
pp. 13030-13037.
ISSN 1463-9084

Winkelmann, S.
(2017)
*Markov Control with Rare State Observation: Average Optimality.*
Markov Processes and Related Fields, 23
.
pp. 1-34.
ISSN 1024-2953

Zhu, J. and Klein, R. and Delle Site, L.
(2016)
*Adaptive Molecular Resolution Approach in Hamiltonian
Form: An Asymptotic Analysis.*
Physical Review E, 94
(043321).

Delle Site, L.
(2016)
*Formulation of Liouville's theorem for grand ensemble molecular simulations.*
Physical Review E, 93
(022130).

Enciso, M. and SchÃ¼tte, Ch. and Delle Site, L.
(2015)
*Influence of pH and sequence in peptide aggregation via
molecular simulation.*
Journal of Chemical Physics, 143
(24).
p. 243130.
ISSN 0021-9606

Agarwal, A. and Delle Site, L.
(2015)
*Path Integral Molecular Dynamics within the Grand
Canonical-like Adaptive Resolution Technique: Simulation of
Liquid Water.*
Journal of Chemical Physics, 143
(9).
ISSN 0021-9606

Hartmann, C. and Delle Site, L.
(2015)
*Scale Bridging in Molecular Simulation.*
The European Physical Journal Special Topics, 224
(12).
pp. 2173-2176.
ISSN 1951-6355

Klein, R.
(2015)
*Comments on "Open Boundary Molecular Dynamics " by R. Delgado-Buscalioni, J. Sablic, and M.
Praprotnik.*
The European Physical Journal, 224
(12).
pp. 2509-2510.
ISSN Online: 1951-6401 Print: 1951-6355

Klein, R.
(2015)
*Comments on "Advantages and challenges in coupling an ideal gas to atomistic models in adaptive resolution simulations" by K. Kreis, A.C. Fogarty, K. Kremer R. Potestio.*
The European Physical Journal, 224
(12).
pp. 2503-2504.
ISSN Online: 1951-6401 Print: 1951-6355

Agarwal, A. and Zhu, J. and Wang, H. and Hartmann, C. and Delle Site, L.
(2015)
*Molecular dynamics in a Grand Ensemble: Bergmann-Lebowitz model and adaptive resolution simulation.*
New Journal of Physics, 17
(083042).
ISSN 1367-2630

Klein, R.
(2015)
*Comments on "Adaptive Resolution Simulation in Equilibrium
and Beyond" by H. Wang and A. Agarwal.*
The European Physical Journal, 224
(12).
pp. 2497-2499.