Convergence in time to equilibrium for the mollified Vlasov–Boltzmann equation

Establish convergence-in-time of solutions to the mollified Vlasov–Boltzmann equation ∂tμ_t(x,v)+Lμ_t(x,v)=\widetilde{Q}μ_t(x,v) toward the equilibrium μ*(x,v)∝exp(−f(x)−|v|^2/2), under appropriate assumptions on the potential f and the delocalized collision kernel \widetilde{q}(x,v,y,w,n).

Background

The classical (local) Vlasov–Boltzmann equation admits μ*∝exp(−f(x)−|v|^2/2) as equilibrium and has known convergence results under certain conditions. The proposed sampling algorithms rely on a mollified, delocalized collision operator \widetilde{Q} that is implementable with particle methods.

While equilibrium characterization carries over, a rigorous proof of convergence in time to μ* for the mollified equation is not yet available, which is essential to fully justify the Boltzmann-based samplers.