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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. 

Flocking phase transitions arise in many aligning active soft matter systems, and an interesting question concerns the role of “topological” vs. “metric” interactions on these transitions. While recent theoretical work suggests that the order–disorder transition in these polar aligning models is universally first order, numerical studies have suggested that topological models may instead have a continuous transition. Some recent simulations have found that some variations of topologically interacting flocking agents have a discontinuous transition, but unambiguous observations of phase coexistence using common Voronoi-based alignment remains elusive. In this work, we use a custom GPU-accelerated simulation package to perform million-particle-scale simulations of a Voronoi–Vicsek model in which alignment interactions stem from an XY-like Hamiltonian. By accessing such large systems on appropriately long time scales and in the time-continuous limit, we are able to show a regime of stable phase coexistence between the ordered and disordered phases, confirming the discontinuous nature of this transition in the thermodynamic limit. 

Tissue interfaces between different populations of cells are sharpened by the cellular topological interactions in a length-scale dependent way. 

Flocking behavior is observed in biological systems from the cellular to superorganismal length scales, and the mechanisms and purposes of this behavior are objects of intense interest. In this paper, we study the collective dynamics of bovine sperm cells in a viscoelastic fluid. These cells appear not to spontaneously flock, but transition into a long-lived flocking phase after being exposed to a transient ordering pulse of fluid flow. Surprisingly, this induced flocking phase has many qualitative similarities with the spontaneous polar flocking phases predicted by Toner-Tu theory, such as anisotropic giant number fluctuations and nontrivial transverse density correlations, despite the induced nature of the phase and the clearly important role of momentum conservation between the swimmers and the surrounding fluid in these experiments. We also find a self-organized global vortex state of the sperm cells, and map out an experimental phase diagram of states of collective motion as a function of cell density and motility statistics. We compare our experiments with a parameter-matched computational model of persistently turning active particles and find that the experimental order-disorder phase boundary as a function of cell density and persistence time can be approximately predicted from measures of single-cell properties. Our results may have implications for the evaluation of sample fertility by studying the collective phase behavior of dense groups of swimming sperm.