### Aleksandar Donev

Courant Institute of Mathematical Sciences

New York University

Fast Reactive Brownian Dynamics

I will describe a particle-based algorithm for reaction-diffusion problems that combines Brownian dynamics with a Markov reaction process. The microscopic model simulated by our Split Reactive Brownian Dynamics (SRBD) algorithm is based on the Doi or volume-reactivity model. This model applies only to reactions with at most two reactants, which is physically realistic. Let us consider the simple reaction A+B->product. In the Doi model, particles are independent spherical Brownian walkers (this can be relaxed to account for hydrodynamic interactions), and while an A and a B particle overlap, there is a Poisson process with a given microscopic reaction rate for the two particles to react and give a product. Our goal is to simulate this complex Markov process in dense systems of many particles, in the presence of multiple reaction channels.

Our algorithm is inspired by the Isotropic Direct Simulation Monte Carlo (I-DSMC) method and the next subvolume method. Strang splitting is used to separate diffusion and reaction; this is the only approximation made in our method. In order to process reactions without approximations, with the particles frozen in place, we use an event-driven algorithm. We divide the system into a grid of cells such that only particles in neighboring cells can react. Each cell schedules the next potential reaction to happen involving a particle in that cell and a particle in one of the neighboring cells, and an event queue is used to select the next cell in which a reaction may happen. Note that, while a grid of cells is used to make the algorithm efficient, the results obtained by the SRBD method are grid-independent and thus free of grid artifacts, such as loss of Galilean invariance and sensitivity of the results to the grid spacing.

I will compare our SRBD method with grid-based methods, such as (C)RDME and a variant of RDME that we call Split Brownian Dynamics with Reaction Master Equation (S-BD-RME), on a problem involving the spontaneous formation of a Turing pattern in a two-dimensional system.