Boltzmann-type models with uncertain binary interactions

TL;DR

This paper investigates Boltzmann-type models with uncertain binary interactions, comparing deterministic and stochastic approaches, deriving Fokker-Planck equations, and providing numerical evidence for asymptotic behavior in gas and aggregation systems.

Contribution

It introduces a unified framework for analyzing uncertainty in Boltzmann-type models, deriving related Fokker-Planck equations, and applying structure-preserving uncertainty quantification methods.

Findings

Deterministic and stochastic models exhibit different asymptotic behaviors.

Derived Fokker-Planck equations provide detailed insights into distribution trends.

Numerical methods confirm theoretical asymptotic predictions.

Abstract

In this paper we study binary interaction schemes with uncertain parameters for a general class of Boltzmann-type equations with applications in classical gas and aggregation dynamics. We consider deterministic (i.e., a priori averaged) and stochastic kinetic models, corresponding to different ways of understanding the role of uncertainty in the system dynamics, and compare some thermodynamic quantities of interest, such as the mean and the energy, which characterise the asymptotic trends. Furthermore, via suitable scaling techniques we derive the corresponding deterministic and stochastic Fokker-Planck equations in order to gain more detailed insights into the respective asymptotic distributions. We also provide numerical evidences of the trends estimated theoretically by resorting to recently introduced structure preserving uncertainty quantification methods.