title: Development of Bottom-up and Iterative Methods for Non-Markovian Coarse-Grained Modeling of Molecular Dynamics
Creators: name: Klippenstein, Viktor
Publisher: name: UNSPECIFIED
Year: 2024

description: While computational power has been steadily increasing throughout the recent decades, computational modeling in terms of atomistic molecular dynamics simulations is still limited in scope with respect to length and time scales. Thus, there is a need for the development of coarser models which allow to study molecular systems for which relevant length and time scales are not accessible with typical atomistic models with a reasonable expenditure of computational resources. In the quite well established field of systematic bottom-up coarse-graining many methods have been developed for the derivation of effective coarse-grained interactions from fine-grained reference models. On the one hand, these models, used in conjunction with standard molecular dynamics simulations, allow to study structural and thermodynamic properties of the underlying reference more efficiently. On the other hand, the correct representation of dynamic properties with coarse-grained models poses additional distinct challenges. The reduction of complexity due to the reduced number of modeled degrees of freedom inevitably leads to a loss of friction in the coarse-grained representation. Subsequently, this leads to a speed-up of dynamics and thus to a misrepresentation of time scales. In particular, whenever the chosen level of coarsening is kept moderate the speed-up can affect different processes to different degrees. This is due to the fact that whenever the time scales of the dynamics of fine-grained and coarse-grained degrees of freedom are not sufficiently separated, friction due to the fine-grained degrees of freedom, acting on the coarse-grained degrees of freedom, becomes time (frequency) dependent and thus non-Markovian. To correct for these effects in coarse-grained models, modified equations of motion, which explicitly incorporate friction or, if required, time dependent friction have to be utilized in coarse-grained simulations. A common class of such equations of motion are generalized Langevin equations, which are non-Markovian stochastic integro-differential equations explicitly incorporating time dependent friction via, the so called, memory
kernels. While the field of structural coarse-graining is quite well established, methodological development for deriving dynamically consistent coarse-grained models falls behind in comparison. 

The focus of the work reported within this thesis is the furthering of methods for parametrizing optimal memory kernels for generalized Langevin equations to achieve dynamic consistency in coarse-grained simulations. Simultaneously this endeavor allows to further the understanding of the role of memory effects in coarse-grained modeling. 

This thesis is structured as follows. Chapter 1 gives a short introduction on and motivation of non-Markovian models in coarse-graining. In chapter 2, a peer-reviewed review article is presented, which provides a detailed overview on Markovian and non-Markovian coarse-grained modeling and numerical evaluation of memory kernels. Chapter 3 summarizes the key theoretical background, relevant for the understanding of the following chapters.
Chapters 4-6 contain the main results of this thesis, based on 3 peer-reviewed research articles. In all three chapters, coarse-grained models are developed to yield consistent dynamic properties. 

In chapter 4, a novel route for a-priori bottom-up estimation of memory kernels is presented. In this study it is shown that by splitting the memory kernel of an exactly solvable single-particle interpretation of a coarse-grained degree of freedom into different contributions stemming from conservative coarse-grained interactions and residual fluctuating interactions, a good choice for the memory kernel in many-body oarse-grained generalized Langevin simulations can be estimated. In particular, it is found that the contributions from coarse-grained conservative and fluctuating interactions are strongly correlated. These cross-correlations have a strong impact on the overall dynamics and thus have to be considered in the parametrization of coarse-grained models. This method is developed and tested on a prominent test case for coarse-grained
methods: a generic star-polymer melt. While this approach has shown to be quite accurate, small deviations between fine-grained and coarse-grained dynamics persist. The data presented in this chapter indicates that these residual deviations might mainly stem from modeling errors in the coarse-grained conservative interactions.

To improve the understanding of the role of the accuracy of coarse-grained conservative interactions, the work presented in chapter 5 aims at illuminating the origin of the remaining discrepancies by deliberately choosing a test case, the well known Asakura-Oosawa model, for which it is a-priori known that very accurate coarse-grained conservative interactions can be derived. By doing so, one generally hard to control error source is removed and the previously proposed methodology is examined under idealized
circumstances. The data presented in this chapter validate the general soundness of the proposed approach and confirms that inaccuracies in conservative interactions are likely to be the main error source in non-idealized applications. At the same time, subtle but relevant limitations of using a simple isotropic thermostat in many-body simulations are discussed. The chosen equation of motion includes hydrodynamic interactions only in an averaged sense. This can can yield to errors in the hydrodynamic scaling of e.g. velocity autocorrelation functions, in particular, in systems with low viscosity. Thus, a key insight from this study is that bottom-up informed approaches for the derivation of memory kernels are useful for a better interpretation of the origin of memory effects. At the same time, for the practical development of coarse-grained models, even subtle error sources due to both modeling errors in coarse-grained potentials and general limitations of the chosen coarse-grained equation of motion can not always be safely ignored. As a way to circumvent these limitations, three novel, simple, iterative optimization procedures are presented for optimizing the generalized Langevin thermostat parametrization to match the references velocity autocorrelation function exactly. 

In chapter 6, the newly developed methods are applied for coarse-graining a realistic molecular system, namely water. The complex local structuring and strong electrostatic interactions in water poses additional hurdles in developing consistent coarse-grained models. Despite that, it is demonstrated
that bottom-up informed memory kernels can correct dynamic properties quite well. To get an exact match between fine-grained and coarse-grained velocity autocorrelation functions, the most promising of the newly developed iterative methods (in this chapter referred to as iterative optimization of memory kernels (IOMK)) is successfully employed. Also, it is demonstrated that the IOMK method is roughly ten times more efficient than the best alternative previously proposed optimization scheme in literature. By studying dynamic properties beyond single-particle time correlation functions, the distinct Van Hove function as a measure of the relaxation of pair structure, it is found that the IOMK method yields very accurate results and, in particular, that Markovian models are not sufficient to achieve comparably good results. By furthermore comparing different coarse-grained conservative interactions, the role of multi-body interactions on structural relaxations is discussed.

A few preliminary and unpublished results and ideas, which can serve as motivation for further research, are discussed in chapter 7. In section 7.1, the challenges in a naive application of the IOMK method to systems which include bonds are discussed. Such mapping schemes give rise to high frequency modes in mapped trajectories. A smoothing approach is proposed, to access the most relevant features and thus to obtain memory kernels which yield comparably good results as in single-bead mapping schemes. In section 7.2 an implementation of the IOMK method as a Gauss-Newton method is proposed, which allows to directly optimize a few parameters which serve as input of the auxiliary variable generalized Langevin thermostat. Such an approach would further simplify the procedure of the IOMK method and thus improve its applicability as a out of box tool for dynamically consistent coarse-graining. 

A summary and an outlook into future research is provided in the final chapter.