On the reaction-diffusion master equation in the microscopic limit

The RDME will break down in the limit of vanishing voxel sizes, in the sense that contributions from bimolecular reactions will be lost. The problem sets on earlier (for larger voxels), the more diffusion limited the reaction is. This is a problem that has attracted a lot of interest since it was pointed out by Samuel Isaacson  in this paper.
Recently, corrections to the bimolecular rates that are explicitly mesh-dependent has been proposed to deal with the problem. Erban and Chapman finds an expression in 3D that works down to a critical size of the mesh.
In this paper, we use a theorem from Montroll to show that there will always be such a critical mesh size for which no local correction to the RDME can make it agree with the Smoluchowski model in the sense that the mean binding time between two particles should be the same in both models. In the limit of perfect diffusion control, we find analytical values for the critical size in both 2D and 3D. Interestingly, the value we find in 3D agrees with the value found by Erban and Chapman. We also discuss the relationship between the local corrections of Erban and Chapman and ours to those derived by Fange et. al.