  Current Lagrangian (particle-tracking) algorithms used to simulate
diffusion-reaction equations must employ a certain number of particles to
properly emulate the system dynamics---particularly for imperfectly-mixed
systems. The number of particles is tied to the statistics of the initial
concentration fields of the system at hand. Systems with shorter-range
correlation and/or smaller concentration variance require more particles,
potentially limiting the computational feasibility of the method. For the
well-known problem of bimolecular reaction, we show that using kernel-based,
rather than Dirac-delta, particles can significantly reduce the required number
of particles. We derive the fixed width of a Gaussian kernel for a given
reduced number of particles that analytically eliminates the error between
kernel and Dirac solutions at any specified time. We also show how to solve for
the fixed kernel size by minimizing the squared differences between solutions
over any given time interval. Numerical results show that the width of the
kernel should be kept below about 12% of the domain size, and that the analytic
equations used to derive kernel width suffer significantly from the neglect of
higher-order moments. The simulations with a kernel width given by least
squares minimization perform better than those made to match at one specific
time. A heuristic time-variable kernel size, based on the previous results,
performs on a par with the least squares fixed kernel size.
