Theoretical justification of the dependence on the mollification parameter in Boltzmann simulators

Develop a theoretical explanation quantifying how the mollification parameter ε in the delocalized collision kernel \widetilde{q}(x,v,y,w,n)=((ε√π)^{-d})exp(−|x−y|^2/ε^2) affects the convergence speed of the mollified Vlasov–Boltzmann PDE and the number of particles required by the Nanbu and Bird sampling algorithms.

Background

The authors use a Gaussian mollifier in the collision kernel for simplicity and observe empirically that larger ε appears to accelerate PDE convergence and reduce the particle count needed for accurate simulation, suggesting better overall algorithmic performance.

However, no theoretical analysis currently explains this dependence on ε, leaving open how to select ε to balance accuracy, stability, and computational efficiency in practice.