  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 free
(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.
