Introducing Memory in Coarse-Grained Molecular Simulations

Abstract

Preserving the correct dynamics at the coarse-grained (CG) level is a pressing problem in the development of systematic CG models in soft matter simulation. Starting from the seminal idea of simple time-scale mapping, there have been many efforts over the years toward establishing a meticulous connection between the CG and fine-grained (FG) dynamics based on fundamental statistical mechanics approaches. One of the most successful attempts in this context has been the development of CG models based on the Mori-Zwanzig (MZ) theory, where the resulting equation of motion has the form of a generalized Langevin equation (GLE) and closely preserves the underlying FG dynamics. In this Review, we describe some of the recent studies in this regard. We focus on the construction and simulation of dynamically consistent systematic CG models based on the GLE, both in the simple Markovian limit and the non-Markovian case. Some recent studies of physical effects of memory are also discussed. The Review is aimed at summarizing recent developments in the field while highlighting the major challenges and possible future directions.

Figure 1

VACFs calculated using FG-MD, CG-MD, and MZ-DPD for various poly(2,2-dimethylpropane) systems: VACFs of the CoM of (a) monomers in a single-component system, (b) 24mers in 25% 24mer–75% dimer solution, and (c) monomers in a network of long poly(2,2-dimethylpropane) chains. The insets compare the corresponding diffusion constants. The top panel shows a representative configuration from each system where monomers are shown as blue beads and 24mers are shown as green chains. In spite of the apparent differences in the VACFs at shorter times, the long-time diffusion constants are better reproduced with MZ-DPD than CG-DPD in the first two cases. The CG-MD and MZ-DPD models fail to reproduce the FG-MD monomer diffusion coefficient in the polymer network. Adapted with permission from ref (37). Copyright 2018 AIP Publishing.

Figure 2

Non-Markovian coarse-graining procedure for a star polymer melt. (a) Illustration of the coarse-graining procedure in which each polymer is replaced by a single CG particle, which interacts with the other particles via the EoM (35). (b, c) VACF for the non-Markovian DPD (NM-DPD) model in comparison to the MD results and a Markovian DPD model for low (b) and high (c) density. Reprinted with permission from ref (98). Copyright 2015 AIP Publishing.

Figure 3

Non-Markovian coarse-graining procedure for a colloidal suspension. (a) Illustration of the coarse-graining procedure, in which every colloid is represented by a single CG particle and the interaction with the solvent is incorported purely implicitly. (b) The velocity auto-correlation function, C_vv(R, t), for colloids which have a nearest neighbor at a distance R. (c) The velocity cross-correlation function, C_vv^c(R, t), for pairs of colloids at distance R. The results in parts b and c are compared between MD results and the non-Markovian coarse-grained model (CG). (d) Comparison of the reconstructed memory kernels Γ(R, t) of the CG model (see eqs 37 and 38) with fluid dynamics (FD) theory. This also shows the importance of the introduction of distance-dependent memory kernels. Figure adapted with permission from ref (57). Copyright 2018 Royal Society of Chemistry.

Figure 4

Effect of memory on the barrier crossing dynamics of a single particle. (a) Illustration of the simulation setup. (b) The important regimes for the mean first passage time, τ_MFP: the Markovian regimes for overdamped and underdamped dynamics in which the memory has no effect, as well as the regimes in which the memory introduces a speedup or a slowdown compared to the Markovian results. τ_m and τ_D, inertial and diffusive time scales; τ_Γ, time scale of the memory kernel. Reprinted with permission from ref (47). Copyright 2018 AIP Publishing.