curvedSpaceSim: A framework for simulating particles interacting along geodesics

A large number of powerful, high-quality, and open-source simulation packages exist to efficiently perform molecular dynamics simulations, and their prevalence has greatly accelerated discoveries across a wide range of scientific domains. These packages typically simulate particles in flat (Euclidean) space, with options to specify a variety of boundary conditions. While more exotic, many physical systems are constrained to and interact across curved surfaces, such as organisms moving across the landscape, colloids pinned at curved fluid-fluid interfaces, and layers of epithelial cells forming highly curved tissues. The calculation of distances and the updating of equations of motion in idealized geometries (namely, on surfaces of constant curvature) can be done analytically, but it is much more challenging to efficiently perform molecular-dynamics-like simulations on arbitrarily curved surfaces. This article discusses a simulation framework which combines tools from particle-based simulations with recent work in discrete differential geometry to model particles that interact via geodesic distances and move on an arbitrarily curved surface. We present computational cost estimates for a variety of surface complexities with and without various algorithmic specializations (e.g., restrictions to short-range interaction potentials, or multi-threaded parallelization). Our flexible and extensible framework is set up to easily handle both equilibrium and non-equilibrium dynamics, and will enable researchers to access time- and particle-number-scales previously inaccessible.现有大量高性能、高质量的开源模拟软件包可用于高效开展分子动力学模拟（Molecular Dynamics Simulations），其广泛应用极大推动了众多科学领域的研究突破。这类软件包通常在平直（欧几里得，Euclidean）空间中对粒子进行模拟，并支持配置多种边界条件。尽管此类系统较为特殊，诸多物理系统被约束在弯曲表面上并在其间发生相互作用，例如在地表移动的生物、固定于弯曲流-流界面的胶体，以及构成高度弯曲组织的上皮细胞层。在理想化几何结构（即常曲率曲面）中，距离计算与运动方程更新可通过解析方法完成，但在任意弯曲曲面上高效开展类分子动力学模拟仍极具挑战。本文介绍一款模拟框架，该框架将基于粒子的模拟工具与离散微分几何（Discrete Differential Geometry）领域的最新研究成果相结合，可对以测地线距离（Geodesic Distances）进行相互作用并在任意弯曲曲面上运动的粒子进行建模。我们针对多种曲面复杂度场景，在启用与未启用各类算法优化（例如限制短程相互作用势、多线程并行计算）的情况下，给出了计算成本估算结果。本框架兼具灵活性与可扩展性，可轻松处理平衡态与非平衡态动力学过程，将帮助研究者突破此前难以触及的时间尺度与粒子数尺度限制。