Relative resolution: Multipole approximation at appropriate distances

Abstract

Recently, we introduced Relative Resolution as a hybrid formalism for fluid mixtures [1]. The essence of this approach is that it switches molecular resolution in terms or relative separation: While nearest neighbors are characterized by a detailed fine-grained model, other neighbors are characterized by a simplified coarse-grained model. Once the two models are analytically connected with each other via energy conservation, Relative Resolution can capture the structural and thermal behavior of (nonpolar) multi-component and multi-phase systems across state space. The current work is a natural continuation of our original communication [1]. Most importantly, we present the comprehensive mathematics of Relative Resolution, basically casting it as a multipole approximation at appropriate distances; the current set of equations importantly applies for all systems (e.g, polar molecules). Besides, we continue examining the capability of our multiscale approach in molecular simulations, importantly showing that we can successfully retrieve not just the statics but also the dynamics of liquid systems. We finally conclude by discussing how Relative Resolution is the inherent variant of the famous "cell-multipole" approach for molecular simulations.Over the past half of a century, invoking molecular simulations has become one of the most promising routes for studying soft matter. This computational approach is splendid for describing systems that just comprise small spatial and short temporal dimensions (i.e., < 10 −6 m and < 10 −6 s, respectively), but it is deficient in describing systems that also involve large spatial and long temporal dimensions (i.e., > 10 −3 m and > 10 −3 s, respectively); resolving such challenges is especially important for biological processes, whose particularities usually span orders of magnitude in scale. For overcoming this dimensionality issue, special attention has been given for enhancing the computational efficiency of molecular simulations, while ensuring that the phenomena of interest is still correctly retrieved. One route involves the intelligent use of statistical mechanics while designing sophisticated algorithms. For example, various successful strategies have been developed that focus on improving the computational efficiency of specific aspects of molecular simulations (e.g., free energies [2,3,4,5], reaction coefficients [6,7,8,9], etc.).Our work revolves around another set of algorithms that has received much attention in the recent couple of decades: It is commonly called the multiscale approach, and it generally takes on the in-serial or in-parallel format. Rather than focusing on the calculation of a specific feature of a molecular simulation, multiscale algorithms aim at improving the computational efficiency of the entire system, while ideally capturing all of its static and dynamic behavior. Importantly, the main signature of all multiscale simulations is that they involve two systems, one constructed of detailed Fine-Grained (FG) models with many degrees of freedom (usu...