Computationally efficient Brownian dynamics via wavelet Monte Carlo

We develop a Brownian dynamics algorithm which evolves soft matter systems by spatially correlated Monte Carlo moves rather than the usual decomposition of the mobility tensor. The algorithm uses vector wavelets as its basic moves and produces hydrodynamics in the low Reynolds number regime according to the Oseen tensor. When small moves are removed the correlations instead closely approximate the Rotne-Prager tensor, itself widely used to correct for deficiencies in Oseen. We also include plane wave moves to provide the longest range correlations, which we detail for systems in both an infinite and periodic box. The computational cost of the algorithm scales competitively with system size and has a small prefactor due to the method's simplicity. Homogeneous systems of $N$ particles and fixed concentration exhibit cost scaling of $N\ln N$ that, in a rough comparison, would only become more expensive than an established lattice Boltzmann algorithm when $N$ is many orders of magnitude larger than in any currently feasible simulation. For dilute systems, the cost scales as $N$ and the comparisons are even more favourable. We also validate the algorithm by checking it reproduces the correct dynamics in simple single polymer systems, as well as verifying the effect of periodicity on the mobility tensor.