Passing to the limit in fuzzy Boltzmann equations

TL;DR

This paper investigates fuzzy Boltzmann equations with spatially delocalized collisions and demonstrates that as the spatial kernel becomes more localized, solutions converge to those of the classical inhomogeneous Boltzmann equations.

Contribution

It establishes a rigorous limit process showing convergence from fuzzy to classical Boltzmann solutions as the spatial kernel approaches a delta distribution.

Findings

Solutions converge to classical Boltzmann solutions as kernel localizes.

Provides a mathematical framework for the limit transition.

Extends understanding of fuzzy kinetic equations.

Abstract

We study a fuzzy Boltzmann equation, where collisions are delocalised and modulated by a spatial kernel. We show that as the spatial kernel converges to a delta distribution, the solutions to these equations converge to renormalised solutions of the inhomogeneous Boltzmann equations.