Brownian Dynamics (BD) simulations are a standard tool for understanding the dynamics of polymers in and out of equilibrium. Quantitative comparison can be made to rheological measurements of dilute polymer solutions, as well as direct visual observations of fluorescently labeled DNA. The primary computational challenge with BD is the expensive calculation of hydrodynamic interactions (HI), which are necessary to capture physically realistic dynamics. The full HI calculation, performed via a Cholesky decomposition every time step, scales with the length of the polymer as O(N). This limits the calculation to a few hundred simulated particles. A number of approximations in the literature can lower this scaling to O(N - N), and explicit solvent methods scale as O(N); however both incur a significant constant per-time step computational cost. Despite this progress, there remains a need for new or alternative methods of calculating hydrodynamic interactions; large polymer chains or semidilute polymer solutions remain computationally expensive. In this paper, we introduce an alternative method for calculating approximate hydrodynamic interactions. Our method relies on an iterative scheme to establish self-consistency between a hydrodynamic matrix that is averaged over simulation and the hydrodynamic matrix used to run the simulation. Comparison to standard BD simulation and polymer theory results demonstrates that this method quantitatively captures both equilibrium and steady-state dynamics after only a few iterations. The use of an averaged hydrodynamic matrix allows the computationally expensive Brownian noise calculation to be performed infrequently, so that it is no longer the bottleneck of the simulation calculations. We also investigate limitations of this conformational averaging approach in ring polymers.

Coarse-grained (CG) molecular dynamics (MD) simulations have become popular for investigating systems on multiple length and time scales ranging from atomistic to mesoscales. In CGMD, several atoms are mapped onto a single CG bead and the effective interactions between CG beads are determined. Iterative coarse-graining methods, such as iterative Boltzmann inversion (IBI), are computationally expensive and can have convergence issues. In this paper, we present a direct and computationally efficient theoretical procedure for coarse-graining based on the Ornstein-Zernike (OZ) and hypernetted chain (HNC) integral equation theory. We demonstrate the OZ-HNC-based CG method by coarse-graining a bulk water system, a water-methanol mixture system, and an electrolyte system. We show that the accuracy of the CG potentials obtained from the OZ-HNC-based coarse-graining is comparable to iterative systematic coarse-graining methods. Furthermore, we show that the CG potentials from OZ-HNC can be used to reduce the number of iterations and hence the computational cost of the iterative systematic coarse-graining approaches, like IBI and relative entropy minimization.