Advances in Hamiltonian Hybrid Particle–Field Theory: Improving the description of interfacial systems

Abstract

Laws of physics govern the motion of molecules that constitute all materials, living and non-living, in a highly hierarchical organisation, across tremendous scales of size and time. Molecular dynamics provides a direct way of operating these laws to reproduce the natural behaviour of molecular systems at the atomic resolution. Analysis of the resulting molecular trajectories provides us with greater insight into the workings of these complex systems. Soft matter systems such as lipid membranes, proteins and DNA, can vary greatly in their dimensionality from a few nanometres to several millimetres. Computer simulations require a suitable description of their constituting structures and interactions, detailed enough to successfully reproduce their dynamics, yet simple enough for a comprehensive understanding and to costeffectively produce results. High resolution studies call for high computational costs, which, in turn imply limited accessibility to larger systems. This work discusses a methodology to access mesoscale systems with characteristic sizes and time in the order of micrometres and milliseconds, respectively, with molecular resolution. It describes interactions between particles by coupling them to an external density-dependent field potential. The theoretical developments of this study are based on the Hamiltonian formulation of the hybrid particle–field theory (hPF). The seed for its foundation was sowed in 2020, and the current work institutes it by extending its theory into a fully functional molecular dynamics mechanism. In this approach, particle forces are derived from purely mechanical considerations, without resorting to any approximation of statistical mean-field origins. In doing so, rigorous conservation of energy and momentum in grid-converged limits is achieved. Also, by smoothing the densities of the particles, the method is freed from many artefact-causing grid-dependent biases. Alongside constant energy and constant volume simulations, constant pressure simulation techniques are also developed. The current work derives a complete theoretical framework for calculating the internal pressure of a system, and evidences a natural anisotropy in it. Machine learning strategies are employed to optimise the model parameters in order to represent interactions between particles with greater accuracy. In doing so, stress-free conditions are achieved, manifesting in the correct interfacial properties like surface tension, and undulatory motions of lamellar membranes. Achieving a direct description of particle-particle correlations is yet another outcome of the theory. In practice, it means to be able to obtain stable solids and co-existing phases, even in systems with a single-species. Preliminary tests on soft-core Lennard Jones systems indeed show such behaviour. The entire formalism is implemented into the freely available open-source software, HylleraasMD. It uses embarrassingly parallelisable routines written in a combination of Python and Fortran, making it a rather efficient computational approach to achieve equilibrium soft matter systems, for example, self-assembly of lipids both into extended and vesicular structures. Overall, the discussed developments establish a strong foundation in understanding soft mesoscale systems and show several promising avenues to explore in the coming days. It is time to enter into the realm of more realistic and complex biological interfacial systems.